In this paper, we consider the motion of incompressible magnetohydrodynamics (MHD) with resistivity in a domain bounded by a free surface. An electromagnetic field generated by some currents in an external domain keeps an MHD flow in a bounded domain. On the free surface, free boundary conditions for MHD flow and transmission conditions for electromagnetic fields are imposed. We proved the local well-posedness in the general setting of domains from a mathematical point of view. The solutions are obtained in an anisotropic space H ((0, T), H ) ∩ L ((0, T), H ) for the velocity field and in an anisotropic space H ((0, T), L ) ∩ L ((0, T), H ) for the magnetic fields with 2 < p < ∞, N < q < ∞ and 2/p + N/q < 1. To prove our main result, we used the Lp-Lq maximal regularity theorem for the Stokes equations with free boundary conditions and for the magnetic field equations with transmission conditions, which have been obtained by Frolova and the second author. 1 1 3 1 2 p q p q p q p q

In this paper, we consider the compressible fluid model of Korteweg type which can be used as a phase transition model. It is shown that the system admits a unique, global strong solution for small initial data in R , 3 ≤ N ≤ 7. In this study, the main tools are the maximal Lp-Lq regularity and Lp-Lq decay properties of solutions to the linearized equations. N

We show the existence of solution in the maximal L −L regularity framework to a class of symmetric parabolic problems on a uniformly C domain in R . Our approach consist in showing R - boundedness of families of solution operators to corresponding resolvent problems first in the whole space, then in half-space, perturbed half-space and finally, using localization arguments, on the domain. Assuming additionally boundedness of the domain we also show exponential decay of the solution. In particular, our approach does not require assuming a priori the uniform Lopatinskii - Shapiro condition. p q 2 n

In this paper, we establish some local and global solutions for the two-phase incompressible inhomogeneous flows with moving interfaces in the maximal Lp - Lq regularity class. Compared with previous results obtained by Solonnikov [Izv. Akad. Nauk SSSR Ser. Mat., 51 (1987), pp. 1065-1087, 1118 (in Russian); translation in Math. USSR-Isz., 31 (1988), pp. 381-405] and by Shibata and Shimizu [Differential Integral Equations, 20 (2007), pp. 241-276], we find the local solutions in the Lp - Lq class in some general uniform W - 1 domain in R by assuming (p, q) ∊]2, ∞[×]N, ∞[or (p, q) ∊]1, 2[×]N, ∞[satisfying 1/p + N/q > 3/2. In particular, the initial data with less regularity are allowed by assuming p < 2. In addition, if the density and the viscosity coefficient are piecewise constant, we can construct the long time solution from the small initial states in the case of the bounded droplet. This is due to some decay property for the corresponding linearized problem. r 2 /r N

In these lecture notes, we study free boundary problems for the Navier–Stokes equations with and without surface tension. The local well-posedness, global well-posedness, and asymptotics of solutions as time goes to infinity are studied in the L in time and L in space framework. To prove the local well-posedness, we use the tool of maximal L –L regularity for the Stokes equations with nonhomogeneous free boundary conditions. Our approach to proving maximal L –L regularity is based on the ℛ-bounded solution operators of the generalized resolvent problem for the Stokes equations with non-homogeneous free boundary conditions and the Weis operator-valued Fourier multiplier. Key to proving global well-posedness for the strong solutions is the decay properties of the Stokes semigroup, which are derived by spectral analysis of the Stokes operator in the bulk space and the Laplace–Beltrami operator on the boundary. We study the following two cases: (1) a bounded domain with surface tension and (2) an exterior domain without surface tension. In studying the latter case, since for unbounded domains we can obtain only polynomial decay in suitable L norms in space, to guarantee the L -integrability of solutions in time it is necessary to have the freedom to choose an exponent with respect to the time variable, thus it is essential to choose different exponents p and q. The basic approach of this chapter is to analyze the generalized resolvent problem, prove the existence of ℛ-bounded solution operators and determine the decay properties of solutions to the non-stationary problem. In particular, R-bounded solution operator and Weis’ operator valued Fourier multiplier theorem and transference theorem for the Fourier multiplier, we derive the maximal L –L regularity for the initial boundary value problem, find periodic solutions with non-homogeneous boundary conditions, and generate analytic semigroups for systems of parabolic equations, including equations appearing in fluid mechanics. This approach is quite new and extends the Fujita–Kato method in the study of the Navier–Stokes equations. p q p q p q q p p q

We consider the initial–boundary value problem for the system of equations describing the flow of compressible isothermal mixture of arbitrary large number of components. The system consists of the compressible Navier–Stokes equations and a subsystem of diffusion equations for the species. The subsystems are coupled by the form of the pressure and the strong cross-diffusion effects in the diffusion fluxes of the species. Assuming the existence of solutions to the symmetrized and linearized equations, proven in Piasecki, Shibata and Zatorska (2019), we derive the estimates for the nonlinear equations and prove the local-in-time existence and maximal L −L regularity of solutions. p q

In this paper, we prove the local and global well-posedness of some nonlinear thermoelastic plate equations with Dirichlet boundary conditions. The main tool for proving the local well-posedness is the maximal Lp-Lq regularity theorem for the linearized equations, and the main tool for proving the global well-posedness is the exponential stability of C0 analytic semigroup associated with linear thermoelastic plate equations with Dirichlet boundary conditions.

We consider the linear thermoelastic plate equations with free boundary conditions in uniform C4-domains, which includes the half-space, bounded and exterior domains. We show that the corresponding operator generates an analytic semigroup in Lp-spaces for all p∈(1,∞) and has maximal Lq-Lp-regularity on finite time intervals. On bounded C4 -domains, we obtain exponential stability.

This paper deals with the local well-posedness of free boundary problems for the Navier-Stokes equations in the case where the uid initially occupies an exterior domain Ω in N-dimensional Euclidian space ℝN.

In this chapter, we provide a review of results on the global well-posedness and optimal decay rate of strong solutions to the compressible Navier-Stokes equations in several type of domains: (1) whole space (Theorems 6, 7, 8, 9, 10, 11, and 12), (2) exterior domains (Theorems 13 and 14), (3) half-space (Theorem 15), (4) bounded domains (Theorem 16), and (5) infinite layers. Global well-posedness for the compressible viscous barotropic fluid motion with nonslip boundary condition was for the first time proved in the early 1980s by Matsumura and Nishida (Commun Math Phys 89:445-464, 1983) under the assumption that the H norm of the initial data is small. In Theorems 1, 2, 3, and 4, we revisit the same problem as in Matsumura and Nishida (Commun Math Phys 89:445- 464, 1983) under the weaker assumptions, namely, that the H norm of initial data is small. This is an improvement of the result in Matsumura and Nishida (Commun Math Phys 89:445-464, 1983) in view of the regularity assumption of the initial data. To show the methods, we perform the proof of Theorems 1, 2, 3, and 4 in all essential details. In this process, the L -L decay properties of solutions to the linearized equations are proved by using the cutoff technique and combining the local energy decay and the result in the whole space. This result was first proved by Kobayashi and Shibata (Commun Math Phys 200:621-659, 1999) under some additional assumption, and in this chapter, this assumption is eliminated by using a bootstrap argument. In the final section of this chapter, the optimal decay rate of the H norm of solution of the nonlinear problem is proved by combining the L -L decay properties of the linearized equations with some energy inequality of exponential decay type under the assumption that the initial data belong to the intersection space of H2 and L1. The main idea of this part of the proof is to combine the L -L decay properties of the Stokes semigroup and some Lyapunov-type energy inequality. 3 2 2 p q p q p q

In this chapter we present the classical energy approach for existence of regular solutions to the equations of compressible, heat-conducting fluids in a bounded three-dimensional domain. Firstly, we provide a state of the art and recall representative results in this field. Next, we give a proof of one of them, concerning Dirichlet boundary conditions for velocity and temperature. The result and thus the proof is divided into two main parts. A local-in-time existence result in high-regularity norms, via a method of successive approximations, occupies the former one. In the latter part, a differential inequality is derived, which allows us to extend the local-in-time solution to the global-in-time solution, provided a certain smallness condition is satisfied. This smallness condition is in fact an equilibrium proximity condition, since it involves differences between data and constants, whereas the data for temperature and density may be large themselves. All our considerations are performed within the L -approach. The proved result is close to that of Valli and Zajaczkowski (Commun Math Phys 103:259-296, 1986), but the techniques used here: the method of successive approximations (instead of a Leray-Schauder fixed-point argument there) as well as a clear continuation argument renders our exposition more traceable. Moreover, one may easily derive now an explicit smallness condition via our approach. Besides, the thermodynamic restriction on viscosities is relaxed, certain technicalities are improved and a possibly useful approach to deal with certain difficulties at the boundary in similar problems is provided. 2

This paper deals with the Lp-Lq decay estimate of the C0 analytic semigroup {T (t)}t≥0 associated with the perturbed Stokes equations with free boundary conditions in an exterior domain. The problem arises in the study of free boundary problem for the Navier-Stokes equations in an exterior domain. We proved that ||δjT(t)f||L p ≤ Cp,q t - j/2 - N/2(1/q - 1/p) ||f||L q (j = 0,1) provided that 1 &lt
q ≤ p ≤ ∞ and q ≠ ∞. Compared with the non-slip boundary condition case, the gradient estimate is better, which is important for the application to proving global well-posedness of free boundary problem for the Navier-Stokes equations. In our proof, it is crucial to prove the uniform estimate of the resolvent operator, the resolvent parameter ranging near zero.

In this paper, we prove the global well-posedness of free boundary problems of the Navier-Stokes equations in a bounded domain with surface tension. The velocity field is obtained in the Lp in time Lq in space maximal regularity class, (2&lt
p&lt
∞, N&lt
q&lt
∞, and 2/p+N/q&lt
1), under the assumption that the initial domain is close to a ball and initial data are sufficiently small. The essential point of our approach is to drive the exponential decay theorem in the Lp-Lq framework for the linearized equations with the help of maximal Lp-Lq regularity theory for the Stokes equations with free boundary conditions and spectral analysis of the Stokes operator and the Laplace-Beltrami operator.

We consider the motion of a viscous incompressible liquid crystal flow in the N -dimensional whole space. We prove the global well-posedness of strong solutions for small initial data by combining the maximal L-p - L-q regularities and L-p - L-q decay properties of solutions for the Stokes equations and heat equations. As a result, we also proved the decay properties of the solutions to the nonlinear equations.

We consider the linear thermoelastic plate equations with free boundary conditions in the L-p in time and L-q in space setting. We obtain unique solvability with optimal regularity for the inhomogeneous problem in a uniform C-4-domain, which includes the cases of a bounded domain and of an exterior domain with C-4-boundary. Moreover, we prove uniform a priori estimates for the solution. The proof is based on the existence of R-bounded solution operators of the corresponding generalized resolvent problem which is shown with the help of an operator-valued Fourier multiplier theorem due to Weis.

In this paper, we proved decay properties of solutions to the Stokes equations with surface tension and gravity in the half space R-+(N) = {(x', x(N)) x' is an element of RN-1, x(N) &gt; 0} (N &gt;= 2). In order to prove the decay properties, we first show that the zero points lambda(+/-) of Lopatinskii determinant for some resolvent problem associated with the Stokes equations have the asymptotics: lambda(+/-) = +/- ic(g)(1/2)vertical bar xi'vertical bar(1/2) - 2 vertical bar xi'vertical bar(2) + O(vertical bar xi'vertical bar(5/2)) as vertical bar xi'vertical bar -&gt; 0, where c(g) &gt; 0 is the gravitational acceleration and xi' is an element of RN-1 is the tangential variable in the Fourier space. We next shift the integral path in the representation formula of the Stokes semi-group to the complex left half-plane by Cauchy's integral theorem, and then it is decomposed into closed curves enclosing lambda(+/-) and the remainder part. We finally see, by the residue theorem, that the low frequency part of the solution to the Stokes equations behaves like the convolution of the (N - 1)-dimensional heat kernel and F-xi'(-1)[e(+/- icg1/2 vertical bar xi'vertical bar 1/2t)](x') formally, where F-xi'(-1) is the inverse Fourier transform with respect to xi'. However, main task in our approach is to show that the remainder part in the above decomposition decay faster than the residue part.

In this paper, we proved decay properties of solutions to the Stokes equations with surface tension and gravity in the half space R-+(N) = {(x', x(N)) x' is an element of RN-1, x(N) &gt; 0} (N &gt;= 2). In order to prove the decay properties, we first show that the zero points lambda(+/-) of Lopatinskii determinant for some resolvent problem associated with the Stokes equations have the asymptotics: lambda(+/-) = +/- ic(g)(1/2)vertical bar xi'vertical bar(1/2) - 2 vertical bar xi'vertical bar(2) + O(vertical bar xi'vertical bar(5/2)) as vertical bar xi'vertical bar -&gt; 0, where c(g) &gt; 0 is the gravitational acceleration and xi' is an element of RN-1 is the tangential variable in the Fourier space. We next shift the integral path in the representation formula of the Stokes semi-group to the complex left half-plane by Cauchy's integral theorem, and then it is decomposed into closed curves enclosing lambda(+/-) and the remainder part. We finally see, by the residue theorem, that the low frequency part of the solution to the Stokes equations behaves like the convolution of the (N - 1)-dimensional heat kernel and F-xi'(-1)[e(+/- icg1/2 vertical bar xi'vertical bar 1/2t)](x') formally, where F-xi'(-1) is the inverse Fourier transform with respect to xi'. However, main task in our approach is to show that the remainder part in the above decomposition decay faster than the residue part.

In this paper, we prove the maximal L-p-L-q regularity of the compressible and incompressible two phase flow with phase transition in the model problem case with the help of SI-bounded solution operators corresponding to generalized resolvent problem. The problem arises from the mathematical study of the motion, of two-phase flows having gaseous phase and liquid phase separated by a sharp interface with phase transition.

In this paper, we prove a local in time unique existence theorem for some two phase problem of compressible and compressible barotropic viscous fluid flow without surface tension in the L-p in time and the L-q in space framework with 2 &lt; p &lt; 1 and N &lt; q &lt; infinity under the assumption that the initial domain is a uniform W-q(2-1/q) domain in R-N (N &gt;= 2). After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve the problem by the contraction mapping principle with the maximal L-p-L-q regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key step of our method is to prove the existence of R-bounded solution operator to resolvent problem corresponding to linearized problem. The R-boundedness combined with Weis's operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal L-p-L-q regularity theorem.

In this paper, we prove a global in time unique existence theorem for the compressible viscous fluids in a bounded domain with slip boundary condition in the maximal L-p-L-q regularity class with 2 &lt; p &lt; infinity and N &lt; q &lt; infinity under the assumption that initial data are small enough and orthogonal to rigid motions if domain is rotationally symmetric, To prove the global well-posedness, we use the prolongation argument based on the maximal L-p-L-q regularity estimate of exponentially decay type. The same problem was first treated by Kobayashi and Zajaczkowski [5] in the L-2 framework by using the energy method; our approach is completely different from Kobayashi and Zajaczkowski [5]. (C) 2015 Elsevier Inc. All rights reserved.

In this paper, we prove the local well-posedness of the free boundary problems of Navier-Stokes equations in a general domain Omega subset of R-N (N &gt;= 2). The velocity field is obtained in the maximal regularity class W-q,p(2,1)(Omega x (0,T)) = L-p((0,T),W-q(2) (Omega) (N)) boolean AND W-p(1)((0,T); L-q (Omega) (N)) (2 &lt; p &lt; infinity and N &lt; q &lt; infinity) for any initial data satisfying certain compatibility conditions. The assumption of the domain Omega is the unique existence of solutions to the weak Dirichlet-Neumann problem as well as some uniformity of covering of the closure of Omega. A bounded domain, a perturbed half space, and a perturbed layer satisfy the conditions for the domain, and therefore drop problems and ocean problems are treated in the uniform manner. Our method is based on the maximal L-p-L-q regularity theorem of a linearized problem in a general domain.

In this paper, we prove a global in time unique existence theorem for the free boundary problem of a compressible barotropic viscous fluid flow without surface tension in the L-p in time and L-q in space framework with 2 &lt; p &lt; infinity and N &lt; q &lt; infinity under the assumption that the initial domain is bounded and initial data are small enough and orthogonal to rigid motions. Such global well-posedness was proved by Zajaczkowski in 1993 in the L-2 framework, and our result is an extension of his result to the maximal L-p-L-q regularity setting. We use the maximal L-p-L-q regularity theorem for the linearlized equations and the exponential stability of the corresponding analytic semigroup, which is a completely different approach than Zajaczkowski (1993).

In this paper, we consider the boundary value problem of Stokes operator arising in the study of free boundary problem for the Navier-Stokes equations with surface tension in a uniform W3−1/r r domain of N-dimensional Euclidean space ℝN (N ⩾ 2, N &lt
r &lt
∞). We prove the existence of R-bounded solution operator with spectral parameter λ varying in a sector Σε,λ0 = {λ ∈ ℂ | | arg λ| ⩽ π − ε, |λ| ⩾ λ0} (0 &lt
ε &lt
π/2), and the maximal Lp-Lq regularity with the help of the R-bounded solution operator and the Weis operator valued Fourier multiplier theorem. The essential assumption of this paper is the unique solvability of the weak Dirichlet-Neumann problem, namely it is assumed the unique existence of solution p ∈ W1 q (Ω) to the variational problem: (∇p,∇ϕ)Ω = (f,∇ϕ)Ω for any ϕ ∈ W1 q′(Ω) with 1 &lt
q &lt
∞ and q′ = q/(q − 1), where W1 q (Ω) is a closed subspace of Ŵ1 q,Γ (Ω) = {p ∈ Lq,loc(Ω) | ∇p ∈ Lq(Ω)N, p|Γ = 0} with respect to gradient norm ∥∇ · ∥Lq( Ω) that contains a space W1 q, Γ (Ω) = {p ∈ W1 q (Ω) | p|Γ = 0}, and Γ is one part of boundary on which free boundary condition is imposed. The unique solvability of such weak Dirichlet-Neumann problem is necessary for the unique existence of a solution to the resolvent problem with uniform estimate with respect to spectral parameter varying in (λ0,∞), which was proved in Shibata [13]. Our assumption is satisfied for any q ∈ (1,∞) by the following domains: half space, perturbed half space, bounded domains, layer, perturbed layer, straight cube, and exterior domains with W1 q (Ω) = Ŵ1 q,Γ (Ω).

This paper concerns the free boundary problem for the Navier-Stokes equations without surface tension in the Lp in time and Lq in space setting with 2&lt
p&lt
∞ and N&lt
q&lt
∞. A local in time existence theorem is proved in a uniform Wq2-1/q domain in the N-dimensional Euclidean space RN (N≥2) under the assumption that the weak Dirichlet-Neumann problem is uniquely solvable. Moreover, a global in time existence theorem is proved for small initial data under the additional assumption that Ω is bounded. This was already proved by Solonnikov [25] by using the continuation argument of local in time solutions which are exponentially stable in the energy level under the assumption that the initial data is orthogonal to the rigid motion. We also use the continuation argument and the same orthogonality condition for the initial data. But, our argument about the continuation of local in time solutions is based on some decay theorem for the linearized problem, which is a different point than [25].

This paper concerns the free boundary problem for the Navier-Stokes equations without surface tension in the L-p in time and L-q in space setting with 2 &lt; p &lt; infinity and N &lt; q &lt; infinity. A local in time existence theorem is proved in a uniform w(q)(2-1/q) domain in the N-dimensional Euclidean space R-N (N &gt;= 2) under the assumption that the weak Dirichlet Neumann problem is uniquely solvable. Moreover, a global in time existence theorem is proved for small initial data under the additional assumption that Omega is bounded. This was already proved by Solonnikov [25] by using the continuation argument of local in time solutions which are exponentially stable in the energy level under the assumption that the initial data is orthogonal to the rigid motion. We also use the continuation argument and the same orthogonality condition for the initial data. But, our argument about the continuation of local in time solutions is based on some decay theorem for the linearized problem, which is a different point than [25]. (C) 2015 Published by Elsevier Inc.

Given a uniform W-q(3-1/q)-domain Omega subset of R-N for N &lt; q &lt; infinity, we consider a simplified Ericksen-Leslie system modeling the flow of compressible nematic liquid crystals based on Lin and Liu [Comm. Pure Appl. Math., 48 (1995), pp. 501-537]. We show the unique existence of local-intime strong solutions. Furthermore, if Omega is bounded and initial data are chosen suitably small, we obtain global-in-time strong solutions. Our approach is based on maximal regularity estimates of the compressible Navier-Stokes equation by Enomoto, von Below, and Shibata [Ann. Univ. Ferrara, 60 (2014), pp. 55-89] and maximal regularity estimates for the Neumann problem as a consequence of Weis's 2001 vector-valued Fourier multiplier theorem.

The situation of this paper is that the Stokes equation for the compressible viscous fluid flow in the upper half-space is coupled via inhomogeneous interface conditions with the Stokes equations for the incompressible one in the lower half-space, which is the model problem for the evolution of compressible and incompressible viscous fluid flows with a sharp interface. We show the existence of R-bounded solution operators to the corresponding generalized resolvent problem, which implies the generation of analytic semigroup and maximal Lp-Lq regularity for the corresponding time dependent problem with the help of the Weis' operator valued Fourier multiplier theorem. The problem was studied by Denisova (Interfaces Free Bound. 2(3):283-312, 2000) under some restriction on the viscosity coefficients and one of our purposes is to eliminate the assumption in (Denisova in Interfaces Free Bound. 2(3):283-312, 2000).

The situation of this paper is that the Stokes equation for the compressible viscous fluid flow in the upper half-space is coupled via inhomogeneous interface conditions with the Stokes equations for the incompressible one in the lower half-space, which is the model problem for the evolution of compressible and incompressible viscous fluid flows with a sharp interface. We show the existence of R-bounded solution operators to the corresponding generalized resolvent problem, which implies the generation of analytic semigroup and maximal Lp-Lq regularity for the corresponding time dependent problem with the help of the Weis' operator valued Fourier multiplier theorem. The problem was studied by Denisova (Interfaces Free Bound. 2(3):283-312, 2000) under some restriction on the viscosity coefficients and one of our purposes is to eliminate the assumption in (Denisova in Interfaces Free Bound. 2(3):283-312, 2000).

In this paper, we consider the boundary value problem of Stokes operators with variable viscosity in the case of free boundary condition in a uniform W-r(2-1/r) domain of N-dimensional Euclidean space R-N (N &gt;= 2, N &lt; r &lt; infinity). We prove the R-boundedness of solution operators with spectral parameter A varying in a sector Sigma(epsilon),(lambda 0) = {lambda is an element of C : vertical bar arg lambda vertical bar &lt;= pi - epsilon, vertical bar lambda vertical bar &gt;= lambda(0)}, from which we can deduce the L-p-L-q maximal regularity as well as the generation of analytic semigroup for the time dependent problem. The problem of this type arises in the mathematical study of the incompressible viscous fluid flow with free surface. The essential assumption of this paper is the unique solvability of the weak Dirichlet-Neumann problem, namely, it is assumed the unique existence of solution p is an element of W-q(1)(Omega) to the variational problem: (del p, del phi)(Omega) = (f, del phi)(Omega) for any phi is an element of W-q(1), (Omega) with 1 &lt; q &lt; infinity and q' = q/(q - 1), where W-q(1)(Omega) is a closed subspace of (W) over cap (1)(q Gamma)(Omega) = {p is an element of L-q,(loc)(Omega) : del(p) is an element of L-q(Omega)(N), p vertical bar Gamma = 0} with respect to gradient norm parallel to del.parallel to(Lq(Omega)) that contains a space W-q,Gamma(1)(Omega) = {p is an element of W-q(1)(Omega) : p vertical bar Gamma = 0}, and Gamma is one part of boundary on which free boundary condition is imposed. The unique solvability of such weak Dirichlet-Neumann problem is necessary for the unique existence of a solution to the resolvent problem with uniform estimate with respect to spectral parameter varying in (lambda(0), infinity), which was proved in Shibata [28]. Our assumption is satisfied for any q is an element of (1, infinity) by the following domains: half space, perturbed half space, bounded domains, layer, perturbed layer with W-q(1)(Omega) = (W) over cap (1)(q,Gamma)(Omega), and by exterior domains with W-q,Gamma(1)(Omega) = the closure of W-q,Gamma(1)(Omega) with respect to the gradient norm. Combining the result in this paper with that in a forthcoming paper about the nonlinear problems, we can conclude that the unique existence of solutions to weak Dirichlet-Neumann problem implies a local in time unique existence theorem of strong solutions to the free boundary problem without surface tension taken into account for the Navier-Stokes equations in a uniform W-r(2-1/r) domain.

In this paper, we consider a generalized resolvent problem for the linearization system of the Navier-Stokes equations describing some free boundary problem of a compressible barotropic viscous fluid flow without taking the surface tension into account. We prove the existence of the R-bounded solution operators, which drives not only the generation of analytic semigroup but also the maximal L-p-L-q regularity by means of Weis' operator valued Fourier multiplier theorem for the corresponding time dependent problem that enable us to prove a local in time existence theorem of the free boundary problem for a compressible barotropic viscous fluid flow in the L-p in time and L-q space setting (cf. Annali dell Universita di Ferrara 60 (2014), 55-89). The results in this paper were given in the PhD thesis [Three topics in fluid dynamics: viscoelastic, generalized Newtonian, and compressible fluids, 2012, TU Darmstadt] by the first author under supervision of the second author. Here we present a slightly different method of deriving a concrete form of solutions to the model problem. In this paper, one of the essential points is to show the invertibility of a 2 x 2 Lopatinski matrix function. The corresponding system in [Three topics in fluid dynamics: viscoelastic, generalized Newtonian, and compressible fluids, 2012, TU Darmstadt] is a 3 x 3 matrix, so that the method presented here is slightly simpler.

In this paper, we consider a generalized resolvent problem for the linearization system of the Navier-Stokes equations describing some free boundary problem of a compressible barotropic viscous fluid flow without taking the surface tension into account. We prove the existence of the R-bounded solution operators, which drives not only the generation of analytic semigroup but also the maximal L-p-L-q regularity by means of Weis' operator valued Fourier multiplier theorem for the corresponding time dependent problem that enable us to prove a local in time existence theorem of the free boundary problem for a compressible barotropic viscous fluid flow in the L-p in time and L-q space setting (cf. Annali dell Universita di Ferrara 60 (2014), 55-89). The results in this paper were given in the PhD thesis [Three topics in fluid dynamics: viscoelastic, generalized Newtonian, and compressible fluids, 2012, TU Darmstadt] by the first author under supervision of the second author. Here we present a slightly different method of deriving a concrete form of solutions to the model problem. In this paper, one of the essential points is to show the invertibility of a 2 x 2 Lopatinski matrix function. The corresponding system in [Three topics in fluid dynamics: viscoelastic, generalized Newtonian, and compressible fluids, 2012, TU Darmstadt] is a 3 x 3 matrix, so that the method presented here is slightly simpler.

In this paper, we prove a local in time unique existence theorem for the freeboundary problem of a compressible barotropic viscous fluid flow without surface tensionin the Lp in time and Lq in space framework with 2 &lt
p &lt
∞ and N &lt
q &lt
∞ under the assumption that the initial domain is a uniform W2-1/qq one in ℝN (N ≥ 2).After transforming a unknown time dependent domain to the initial domain by theLagrangian transformation, we solve problem by the Banach contraction mappingprinciple based on the maximal Lp-Lq regularity of the generalized Stokes operatorfor the compressible viscous fluid flowwith free boundary condition. The key issue forthe linear theorem is the existence of R-bounded solution operator in a sector, whichcombined with Weis's operator valued Fourier multiplier theorem implies the generationof analytic semigroup and the maximal Lp-Lq regularity theorem. The nonlinearproblem we studied here was already investigated by several authors (Denisova andSolonnikov, St. Petersburg Math J 14:1-22, 2003
J Math Sci 115:2753-2765, 2003
Secchi, Commun PDE 1:185-204, 1990
Math Method Appl Sci 13:391-404, 1990
Secchi and Valli, J Reine Angew Math 341:1-31, 1983
Solonnikov and Tani, Constantincarathéodory: an international tribute, vols 1, 2, pp 1270-1303,World ScientificPublishing, Teaneck, 1991
Lecture notes in mathematics, vol 1530, Springer, Berlin,1992
Tani, J Math Kyoto Univ 21:839-859, 1981
Zajaczkowski, SIAM JMath Anal25:1-84, 1994) in the L2 framework and Hölder spaces, but our approach is differentfrom them. © 2013 Università degli Studi di Ferrara.

In this paper, we prove the R-sectoriality of the resolvent problem for the boundary value problem of the Stokes operator for the compressible viscous fluids in a general domain, which implies the generation of analytic semigroup and the maximal L-p-L-q regularity for the initial boundary value problem of the Stokes operator. Combining our linear theory with fixed point arguments in the Lagrangian coordinates, we have a local in time unique existence theorem in a general domain and a global in time unique existence theorem for some initial data close to a constant state in a bounded domain for the initial boundary value problem of the Navier-Stokes equations describing the motion of compressible viscous fluids. All the results obtained in this paper were announced in Enomoto-Shibata [12].

In this paper, we prove the R-sectoriality of the resolvent problem for the boundary value problem of the Stokes operator for the compressible viscous fluids in a general domain, which implies the generation of analytic semigroup and the maximal L-p-L-q regularity for the initial boundary value problem of the Stokes operator. Combining our linear theory with fixed point arguments in the Lagrangian coordinates, we have a local in time unique existence theorem in a general domain and a global in time unique existence theorem for some initial data close to a constant state in a bounded domain for the initial boundary value problem of the Navier-Stokes equations describing the motion of compressible viscous fluids. All the results obtained in this paper were announced in Enomoto-Shibata [12].

In this paper, we prove unique existence of solutions to the generalized resolvent problem of the Stokes operator with first order boundary condition in a general domain of the N-dimensional Eulidean space . This type of problem arises in the mathematical study of the flow of a viscous incompressible one-phase fluid with free surface. Moreover, we prove uniform estimates of solutions with respect to resolvent parameter varying in a sector , where and . The essential assumption of this paper is the existence of a unique solution to a suitable weak Dirichlet problem, namely it is assumed the unique existence of solution to the variational problem: for any . Here, is the closure of by the semi-norm , and is the boundary of . In fact, we show that the unique solvability of such a Dirichlet problem is necessary for the unique existence of a solution to the resolvent problem with uniform estimate with respect to resolvent parameter varying in . Our assumption is satisfied for any by the following domains: whole space, half space, layer, bounded domains, exterior domains, perturbed half space, perturbed layer, but for a general domain, we do not know any result about the unique existence of solutions to the weak Dirichlet problem except for q = 2.

In this paper, we proved the local energy decay and some L-p-L-q decay properties of solutions to the initial-boundary value problem for the Stokes equations of compressible viscous fluid flow in a 2-dimensional exterior domain. Kobayashi (1997) [19] and Kobayashi and Shibata (1999) [21] treated the same problem in a 3-dimensional exterior domain, and our results obtained in this paper are an extension of results in Kobayashi (1997) [19] and Kobayashi and Shibata (1999) [21] to the 2-dimensional case. The fundamental solution to the resolvent equations for the Stokes equations in R-2 has a logarithmical singularity at lambda = 0, lambda being a resolvent parameter, while it is continuous up to lambda = 0 in R-3. This difference requires us a new idea to prove the local energy decay estimate. Once getting the local energy decay estimate, the required L-p-L-q decay estimates in the exterior domain are obtained by combining the local energy estimate and the L-p-L-q estimates in R-2 by a cut-off technique. (C) 2012 Elsevier Inc. All rights reserved.

The basic model for incompressible two-phase flows with phase transitions is derived from basic principles and shown to be thermodynamically consistent in the sense that the total energy is conserved and the total entropy is nondecreasing. The local well-posedness of such problems is proved by means of the technique of maximal Lp-regularity in the case of equal densities. This way we obtain a local semiflow on a well-defined nonlinear state manifold. The equilibria of the system in absence of external forces are identified and it is shown that the negative total entropy is a strict Ljapunov functional for the system. If a solution does not develop singularities, it is proved that it exists globally in time, its orbit is relatively compact, and its limit set is nonempty and contained in the set of equilibria.

In this paper, we proved the local energy decay and some L-p-L-q decay properties of solutions to the initial-boundary value problem for the Stokes equations of compressible viscous fluid flow in a 2-dimensional exterior domain. Kobayashi (1997) [19] and Kobayashi and Shibata (1999) [21] treated the same problem in a 3-dimensional exterior domain, and our results obtained in this paper are an extension of results in Kobayashi (1997) [19] and Kobayashi and Shibata (1999) [21] to the 2-dimensional case. The fundamental solution to the resolvent equations for the Stokes equations in R-2 has a logarithmical singularity at lambda = 0, lambda being a resolvent parameter, while it is continuous up to lambda = 0 in R-3. This difference requires us a new idea to prove the local energy decay estimate. Once getting the local energy decay estimate, the required L-p-L-q decay estimates in the exterior domain are obtained by combining the local energy estimate and the L-p-L-q estimates in R-2 by a cut-off technique. (C) 2012 Elsevier Inc. All rights reserved.

The basic model for incompressible two-phase flows with phase transitions is derived from basic principles and shown to be thermodynamically consistent in the sense that the total energy is conserved and the total entropy is nondecreasing. The local well-posedness of such problems is proved by means of the technique of maximal Lp-regularity in the case of equal densities. This way we obtain a local semiflow on a well-defined nonlinear state manifold. The equilibria of the system in absence of external forces are identified and it is shown that the negative total entropy is a strict Ljapunov functional for the system. If a solution does not develop singularities, it is proved that it exists globally in time, its orbit is relatively compact, and its limit set is nonempty and contained in the set of equilibria.

In this paper, we proved the generalized resolvent estimate and the maximal L-p-L-q regularity of the Stokes equation with first order boundary condition in the half-space, which arises in the mathematical study of the motion of a viscous incompressible one phase fluid flow with free surface. The core of our approach is to prove the R boundedness of solution operators defined in a sector Sigma(epsilon,gamma 0) = {lambda is an element of C \ {0} vertical bar vertical bar arg lambda vertical bar &lt;= pi - epsilon, vertical bar lambda vertical bar &gt;= gamma(0)} with 0 &lt; epsilon &lt; pi/2 and gamma(0) &gt;= 0. This R boundedness implies the resolvent estimate of the Stokes operator and the combination of this R boundedness with the operator valued Fourier multiplier theorem of L. Weis implies the maximal L-p-L-q regularity of the non-stationary Stokes. For a densely defined closed operator A, we know that what A has maximal L-p regularity implies that the resolvent estimate of A in lambda is an element of Sigma(epsilon,gamma 0), but the opposite direction is not true in general (cf. Kalton and Lancien [19]). However, in this paper using the R boundedness of the operator family in the sector Sigma(epsilon,lambda 0), we derive a systematic way to prove the resolvent estimate and the maximal L-p regularity at the same time.

In this paper we consider the set of equations describing Oldroyd-B fluids in exterior domains. It is shown that these equations admit a unique, global solution defined in a certain function space provided the initial data and the coupling constant are small enough. (C) 2011 Elsevier Inc. All rights reserved.

In this paper we prove the generalized resolvent estimate and maximal L-p-L-q regularity of the Stokes equation with and without surface tension and gravity in the whole space with flat interface. We prove II boundedness of solution operators defined in a sector Sigma(epsilon,gamma 0) = {lambda is an element of C\ {0} vertical bar vertical bar arg lambda vertical bar &lt;= pi - epsilon vertical bar lambda vertical bar &gt;= gamma(0)] with 0 &lt; epsilon &lt; pi/2 and gamma(0) &gt;= 0, which combined with the Fourier multiplier theorem of S.G. Mihlin and the operator valued Fourier multiplier theorem of L Weis yields the required generalized resolvent estimate and maximal L-p-L-q regularity at the same time. One of the character of the paper is to introduce special function spaces E-q((R) over dot(n),Sigma(epsilon,gamma 0) and E-p,E-q,E-gamma 0 ((R) over dot(n) x R) (cf. (1.7) and (1.8)), which is necessary to treat the situation that the normal component of velocity fields jumps across the interface. Such spaces never appear in the study of the Stokes equations with other boundary conditions like non-slip condition, Navier slip condition, Robin condition or pure Neumann condition appearing in the study of one phase problem (cf. Desch et al., 2001 [12], Farwig and Sohr, 1994 [13], Saal, 2003 [22], Shibata and Shimada, 2007 [23], Shibata and Shimizu 2008 [25], 2009 [26], in press [27]), because the normal component of the velocity fields vanishes at the boundary which is physical requirement that the flow does not go out and come in through the rigid boundary. (C) 2011 Elsevier Inc. All rights reserved.

We consider the free boundary problem of the Navier-Stokes equation with surface tension. Our initial domain is one of a bounded domain, an exterior domain, a perturbed half-space or a perturbed layer in n (n epsilon 2). We report a local in time unique existence theorem in the space [image omitted] with some T 0, 2 p and n q for any initial data which satisfy compatibility condition. Our theorem can be proved by the standard fixed point argument based on the Lp-Lq maximal regularity theorem for the corresponding linearized equations. Our results cover the cases of a drop problem and an ocean problem that were studied by Solonnikov (Solvability of the evolution problem for an isolated mass of a viscous incompressible capillary liquid, Zap. Nauchn. Sem. (LOMI) 140 (1984) pp. 179-186 (in Russian) (English transl.: J. Soviet Math. 32 (1986), pp. 223-238)), Solonnikov (Unsteady motion of a finite mass of fluid, bounded by a free surface, Zap. Nauchn. Sem. (LOMI) 152 (1986), pp. 137-157 (in Russian) (English transl.: J. Soviet Math. 40 (1988), pp. 672-686)), Solonnikov (On nonstationary motion of a finite isolated mass of self-gravitating fluid, Algebra Anal. 1 (1989), pp. 207-249 (in Russian) (English transl.: Leningrad Math. J. 1 (1990), pp. 227-276)), Solonnikov (Solvability of the problem of evolution of a viscous incompressible fluid bounded by a free surface on a finite time interval, Algebra Anal. 3 (1991), pp. 222-257 (in Russian) (English transl.: St. Petersburg Math. J. 3 (1992) 189-220)), Beale (Large time regularity of viscous surface waves, Arch. Rat. Mech. Anal. 84 (1984), pp. 307-352) and Tani (Small-time existence for the three-dimensional incompressible Navier-Stokes equations with a free surface, Arch. Rat. Mech. Anal. 133 (1996), pp. 299-331).

We consider the initial-boundary value problem in hyperbolic thermoelasticity with second sound in a three-dimensional exterior domain. The low frequency expansion of solutions to the corresponding stationary resolvent problem is given and the limit to the classical thermoelastic problem is investigated.

We consider the initial-boundary value problem in hyperbolic thermoelasticity with second sound in a three-dimensional exterior domain. The low frequency expansion of solutions to the corresponding stationary resolvent problem is given and the limit to the classical thermoelastic problem is investigated.

Consider the Navier-Stokes equations in the rotational framework. It is proved that these equations possess a unique global mild solution for arbitrary speed of rotation provided the initial data u(0) is small enough in the H(sigma)(1/2)(R(3))-norm.

Consider the Navier-Stokes equations in the rotational framework. It is proved that these equations possess a unique global mild solution for arbitrary speed of rotation provided the initial data u(0) is small enough in the H(sigma)(1/2)(R(3))-norm.

In this paper we prove frequency expansions of the resolvent and local energy decay estimates for the linear thermoelastic plate equations:
u(tt) + Delta(2)u + Delta theta = 0 and theta(t) - Delta theta - Delta u(t) = 0 in Omega x (0, infinity).
subject to Dirichlet boundary conditions: u vertical bar Gamma = D(v)u vertical bar Gamma = theta vertical bar Gamma = 0 and initial conditions (u, u(t), theta)vertical bar(t=0) = (u(0), v(0), theta(0)) Here Omega is an exterior domain (domain with bounded complement) in R(n) with n = 2 or n = 3, the boundary F of which is assumed to be a C(4)-hypersurface.

In this paper we prove frequency expansions of the resolvent and local energy decay estimates for the linear thermoelastic plate equations:
u(tt) + Delta(2)u + Delta theta = 0 and theta(t) - Delta theta - Delta u(t) = 0 in Omega x (0, infinity).
subject to Dirichlet boundary conditions: u vertical bar Gamma = D(v)u vertical bar Gamma = theta vertical bar Gamma = 0 and initial conditions (u, u(t), theta)vertical bar(t=0) = (u(0), v(0), theta(0)) Here Omega is an exterior domain (domain with bounded complement) in R(n) with n = 2 or n = 3, the boundary F of which is assumed to be a C(4)-hypersurface.

In this paper, we show the unique existence of solutions to the nonstationary problem for the modified Oseen equation with rotating effect in Omega:
D(t)u-Delta u + kD(3)u + M(a)u + del p = 0, div u = 0 in Omega x (0, infinity),
(OS)
u vertical bar partial derivative Omega = 0, u vertical bar(t=0) = f,
where Omega is an exterior domain in R-3, M(a)u = -a(e(3) x x). del u + ae(3) x u, x = (x(1),x(2),x(3))is an element of R-3 and e(3) = (0, 0, 1). This problem arises from a linearization of the Navier Stokes equations describing an incompressible viscous fluid flow past a rotating obstacle. If 1 &lt; q &lt; infinity and initial data f satisfies the conditions: f is an element of W-q(2)(Omega), div f = 0 in Omega, f vertical bar partial derivative Omega = 0 and M(a)f is an element of L-q(Omega), then problem (OS) admits a unique solution (u(t), p(t)) which satisfies the following conditions:
u(t) is an element of C-1([0, infinity), L-q (Omega)) boolean AND C-0, ([0, infinity), w(Q)(2) (Omega)), p (t) is an element of C-0([0, infinity), (W) over cap (1)(q)(Omega)),
parallel to(u(t), t(1/2)del u(t), t del(2)u(t), del p(t))parallel to(Lq(Omega)) &lt;= C-a0,C-k0,C-gamma E-gamma t parallel to f parallel to(Lq(Omega)) ,
t((1/2)-(1+q)))parallel to-p(t) parallel to(Lq(Omega b)) &lt;= C-a0,E-k0,b,gamma(gamma t)parallel to f parallel to(Lq(Omega)),
parallel to D(t)u(t)parallel to(Lq(Omega)) + parallel to u(t)parallel to (Wq2(Omega)) + parallel to del p(t)parallel to(Lq(Omega)) &lt;=
C-a0,k(0,gamma)E(gamma t) (parallel to f parallel to(Wq2(Omega)) + parallel to M(a)f parallel to(Lq(Omega)))
for any t &gt; 0 and large positive gamma, where b is any number such that B-b superset of R-3 \ Omega and Omega(b) = B-b boolean AND Omega with B-b = {x is an element of R-3 vertical bar vertical bar x vertical bar &lt; b}. The estimate for pressure term p is new and important for further researches of the corresponding full nonlinear problem. We also prove the generation of a continuous semigroup associated with problem (OS), which has been announced in Shibata [20, Theorem 1.1]. The result

The paper is concerned with linear thermoelastic plate equations in the half-space R-+(n) = {x = (x(1),...,x(n)) | x(n) &gt; 0}:
u(tt) + Delta(2)u + Delta theta = 0 and theta(t) - Delta theta - Delta u(t) = 0 in R-+(n) x (0,infinity),
subject to the boundary condition: u|(xn=0) = D(n)u(|xn=0) = theta|(xn=0) = 0 and initial condition: (u,D(t)u,theta)|(t=0) = (u(0),v(0) theta(0)) is an element of H-p = W-p,D(2) x L-p x L-p, where W-p,D(2) = {u is an element of W-p(2)| u|(xn=0) = D(n)u|(xn=0) = 0}. We show that for any p is an element of (1,infinity), the associated sernigroup {T(t)}(t &gt;= 0) is analytic in the underlying space H-p. Moreover, a solution (u, theta) satisfies the estimates:
parallel to del(j)(del(2)u(.,t),u(t)(.t),theta(.t))parallel to(Lp(R+n)) &lt;= C(p,q)t(-j/2-n/2(1/p-1/q))parallel to(del(2)u(0),v(0),theta(0))parallel to(Lp(R+n)) (t &gt; 0)
for j = 0, 1, 2 provided that 1 &lt; p &lt;= q &lt;= infinity when j = 0, 1 and that 1 &lt; p &lt;= q &lt;= infinity when j = 2, where del(j) stands for space gradient of order j.

We consider the motion of a viscous fluid filling the whole three-dimensional space exterior to a rotating obstacle with constant angular velocity. We develop the L (p) -L (q) estimates and the similar estimates in the Lorentz spaces of the Stokes semigroup with rotation effect. We next apply them to the Navier-Stokes equation to prove the global existence of a unique solution which goes to a stationary flow as t -&gt; a with some definite rates when both the stationary flow and the initial disturbance are sufficiently small in L (3,a) (weak-L (3) space).

The paper is concerned with linear thermoelastic plate equations in a domain Omega:
u(tt) + Delta(2)u + Delta theta = 0 and theta(t) - Delta theta - Delta u(t) = 0 in Omega x (0, infinity),
subject to the Dirichlet boundary condition u|(Gamma) = D(nu)u|(Gamma) = theta|(Gamma) = 0 and initial condition (u, u(t), theta)|(t=0) = (u(0),v(0),theta(0)) is an element of W-p,D(2)(Omega) x L-p x L-p. Here, Omega is a bounded or exterior domain in R-n (n &gt;= 2). We assume that the boundary Gamma of Omega is a C-4 hypersurface and we define W-p,D(2) by the formula W-p,D(2) = {u is an element of W-p(2) : u|(Gamma) = D(nu)u|(Gamma) = 0}. We show that, for any p is an element of (1, infinity), the associated semigroup {T(t)}(t &gt;= 0) is analytic. Moreover, if Omega is bounded, then {T(t)}(t &gt;= 0) is exponentially stable.

In this paper we consider a resolvent problem of the Stokes operator with some boundary condition in the half space, which is obtained as a model problem arising in evolution free boundary problems for viscous, incompressible fluid flow. We show standard resolvent estimates in the L(q) framework (1 &lt; q &lt; infinity), applying some kernel estimates to concrete solution formulas. The Volevich trick in [21] plays a fundamental role in estimating solutions. (C) 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

In this paper, we prove the L-p-L-q maximal regularity of solutions to the Neumann problem for the Stokes equations with non-homogeneous boundary condition and divergence condition in a bounded domain. The result was first stated by Solonnikov [ 17], but he assumed that p = q &gt; 3 and considered only the finite time interval case. In this paper, we consider not only the case: 1 &lt; P, q &lt; infinity but also the infinite time interval case. Especially, we obtain the L-p-L-q maximal regularity theorem with exponential stability on the infinite time interval.

In this paper, we Study a stationary and a nonstationary problem of the Ginzburg-Landau-Maxwell equations with Coulomb gauge in the LP framework. First we prove a unique existence of stationary solution near the constant state with a small external magnetic field. Moreover, we prove a globally in time existence of solutions to the time dependent Ginzburg-Landau-Maxwell equations with simill initial data and external magnetic field, and we show its convergence to the corresponding stationary Solution when time tends to infinity. The key of our approach iS to use various L-p-L-q estimates of the analytic sernigrOUP generated by the linearized problem. Especially our initial data belong to L-3 without any additional regularity. (c) 2005 Published by Elsevier Inc.

We prove a generalized resolvent estimate of Stokes equations with nonhomogeneous Robin boundary condition and divergence condition in the L-q framework (1 &lt; q &lt; infinity) in. a domain of R-n (n &gt;= 2) that is a bounded domain or the exterior of a bounded domain. The Robin condition consists of two conditions: v (.) u = 0 and alpha u + beta(T(u,p)v - &lt; T(u,p)v, v &gt; v) = h on the boundary of the domain with alpha, beta &gt;= 0 and alpha + beta = 1, where u denotes a velocity vector, p a pressure, T(u, p) the stress tensor for the Stokes flow, and v the unit outer normal to the boundary of the domain. It presents the slip condition when 3 = 1 and the non-slip one when a = 1, respectively.

We consider a free boundary problem for the Navier-Stokes equation in R-n (n &gt;= 2). We prove a local in time unique existence theorem for any initial data and a global in time unique existence theorem for some small initial data. The problem we consider in this paper was already treated by V. Solonnikov [15]. But, recently in [10] we proved an L-p-L-q maximal regularity theorem for the Stokes equation with Neumann boundary condition which is a linearized version of the free boundary problem for the Navier-Stokes equation treated in this paper. Our proof is based on this theorem. Therefore our solution is obtained in the space W-q,p(2,1) (2 &lt; p &lt; infinity and n &lt; q &lt; infinity) while a solution in [15] is in W-q(2,1) = W-q,q(2,1) (n &lt; q &lt; infinity) Moreover, our proof of the global in time existence theorem is much simpler than [15], because in [10] we established a maximal regularity theorem on the whole time interval (0, infinity) with exponential stability. The results obtained in this paper were already announced in Shibata-Shimizu [11].

We consider a compressible viscous fluid affected by external forces of general form which are small and smooth enough in suitable norms in R-3. In Shibata and Tanaka [Y. Shibata, K. Tanaka, On the steady flow of compressible viscous fluid and its stability with respect to initial disturbance, J. Math. Soc. Japan 55 (2003) 797-826], we proved the unique existence and some regularity of the steady flow and its globally in-time stability with respect to a small initial disturbance in the H-3 -framework. In this paper, we investigate the rate of the convergence of the non-stationary flow to the corresponding steady flow when the initial data are small enough in the H3 and also belong to L-6/5. (C) 2007 Elsevier Ltd. All rights reserved.

In this paper, we obtain local energy decay estimates and L-p-L-q estimates of the solutions to the Stokes equations with Neumann boundary condition which is obtained as a linearized equation of the free boundary problem for the Navier-Stokes equations. Comparing with the non-slip boundary condition case, we have a better decay estimate for the gradient of the semigroup because of the null force at the boundary.

We prove the L p-L q maximal regularity of solutions to the Neumann problem for the Stokes equations with non-homogeneous boundary condition and divergence condition in a bounded domain. And as an application, we consider a free boundary problem for the Navier-Stokes equation. We prove a locally in time unique existence of solutions to this problem for any initial data and a globally in time unique existence of solutions to this problem for some small initial data.

We discuss L-p-L-q type estimates of the Stokes semigroup and their application to the Navier-Stokes equation in a perturbed half-space. Especially, we have the L-p-L-q type estimate of the gradient of the Stokes semigroup for any p and q with 1 &lt; p &lt;= q &lt; infinity, while the same estimate holds only for the exponents p and q with 1 &lt; p &lt;= q &lt;= n in the exterior domain case, where n denotes the space dimension, and, therefore, we can get better results concerning the asymptotic behaviour of solutions to the Navier-Stokes equations compared with the exterior domain case.
Our proof of the L-p-L-q type estimate of the Stokes semigroup is based on the local energy decay estimate obtained by investigation of the asymptotic behavior of the Stokes resolvent near the origin. The order of asymptotic expansion of the Stokes resolvent near the origin is one half better compared with the exterior domain case, because we have the reflection principle on the boundary in the half-space case unlike the whole space case, and, then, such better asymptotics near the boundary are also obtained in the perturbed half-space by a perturbation argument. This is one of the reasons why the result in the perturbed half-space case is essentially better compared with the exterior domain case.

This paper is concerned with the stationary Navier-Stokes equations in exterior domains of dimension n ≥ 3, and provides a sufficient condition on the external force for the unique solvability. This condition is valid both in the case with small but nonzero velocity at infinity, and in the case with zero velocity at infinity. As a result it is proved that, if the external force satisfies this condition, the solution with nonzero velocity at infinity converges to the solution with zero velocity at infinity with respect to the weak-∗ topology of appropriate function spaces.© 2005, The Mathematical Society of Japan. All rights reserved.

This paper is concerned with the study of the system of Laplace operators with perfect wall condition in the L-p framework. Our study includes a bounded domain, an exterior domain and a domain having noncompact boundary such as a perturbed half space. A direct application of our study is to prove the analyticity of the semigroup corresponding to the Maxwell equation of parabolic type, which appears as a linearized equation in the study of the nonstationary problem concerning the Ginzburg-Landau-Maxwell equation describing the Ginzburg-Landau model for superconductivity, the magnetohydrodynamic equation and the Navier-Stokes equation with Neumann boundary condition. And also, our theory is applicable to some solvability of the stationary problem of these nonlinear equations in the L-p framework.

The linearized initial boundary value problem describing the motion of the viscous compressible fluid is studied under Dirichlet zero condition in bounded and unbounded domains. The resolvent estimate for the corresponding operator is proved in the L, framework and the sharp inner estimate of the resolvent set is obtained. Copyright (C) 2004 John Wiley Sons, Ltd.

In this paper, we prove a local energy decay of the Oseen semigroup in the n-dimensional exterior domain (n greater than or equal to 3). In the three dimensional exterior domain case, Kobayashi and Shlbata [16] already proved the local energy decay. Our theorem is not only an extension of the results due to Kobayashi and Shlbata to the n-dimensional case but also the complete study of the local energy decay theorem for the Oseen equation with optimal time decay rates. The local energy decay gives us a crucial step to obtain the L-p-L-q estimates of the Oseen semigroup, which enable us to prove the unique existence of globally in time solutions to the Navier-Stokes equation in an exterior domain with small initial data in the L-n framework, and their properties of time decay.

We consider the exterior problem in the plane for the wave equation with a Neumann boundary condition and study the asymptotic behavior of the solution for large times. For possible application we are interested to show a decay estimate which does not involve weighted norms of the initial data. In the paper we prove such an estimate, by a combination of the estimate of the local energy decay and decay estimates for the free space solution. (C) 2003 Elsevier Inc. All rights reserved.

We consider a compressible viscous fluid effected by general form external force in R-3. In part 1, an analysis of the linearized problem based on the weighted-L-2 method implies a condition on the external force for the existence and some regularities of the steady flow. In part 2, we study the stability of the steady flow with respect to the initial disturbance. What we proved is: if H-3-norm of the initial disturbance is small enough, then the solution to the non-stationary problem exists uniquely and globally in, time.

Obtained is the L-p estimate of solutions to the resolvent problem for the Stokes system with interface condition in a bounded domain in R-n. It is the first step to consider the free boundary value problem. (C) 2003 Elsevier Science (USA). All rights reserved.

This paper is concerned with the standard L-p estimate of solutions to the resolvent problem for the Stokes operator on an infinite layer.

We consider the L-p-L-q estimates of solutions to the Cauchy problem of linearized compressible Navier-Stokes equation. Especially, we investigate the diffusion wave property of the compressible Navier-Stokes flows, which was studied by D. Hoff and K. Zumbrum and Tai- P. Liu and W. Wang.

Tosio Kato's method and principle for evolution equations in mathematical physics (Sapporo, 2001)

We consider the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in R-3. We give the L-q-L-p estimates fur solutions to the linearized equations and show an optimal decay estimate for solutions to the nonlinear problem.

We proved L-q - L-r type estimates of the Stokes semigroup in a two dimensional exterior domain. Our proof is based on the local energy decay estimate obtained by investigation of the asymptotic behavior of the resolvent of the Stokes operator near the origin.

We consider the initial-boundary value problem for a linear thermoelastic plate equation and we prove that the energy associated to the system decays exponentially to zero as time goes to infinity. (C) 1997 by B. G. Teubner Stuttgart-John Wiley & Sons Ltd.

A new approach to the existence theory for the Navier-Stokes equations, using a technique of KATO [15], further developed in combination with estimates for Oseen's equation by KOBAYASHI & SHIBATA [17] and SHIBATA [24], has made possible the solution of a long-standing open problem often referred to as Finn's ''starting problem''. This paper provides the solution.

We prove a global in time existence theorem of classical solutions of the initial boundary value problem For a non-linear thermoviscoelastic equation in a bounded domain for very smooth initial data, external forces and heat supply which are very close to a specific constant equilibrium state. Our proof is a combination of a local in time existence theorem and some a priori estimates of local in time solutions. Such a priori estimates are proved basically for suitable linear problems by using some multiplicative techniques. An exponential stability of the constant equilibrium state also follows from our proof of the existence and regularity theorems.

We prove the global existence of analytic solutions to the non-linear viscoelastic equations of the following type: u(tt) = phi(\\del(x)u\\2)DELTAu + psi(\\D(x)beta1u\\2,...,\\D(x)betaN)\\)DELTAu(t), where phi and psi are continuous and non-negative functions satisfying some additional conditions. The method of the analytic energy estimates is used.

The Neumann problem for the linear thermoelastic plate equations is studied in the bounded domain of any space dimensions. The exponential decay property is investigated as well as the existence theorem.

We are mainly concerned with the Dirichlet initial boundary value problem in one-dimensional nonlinear thermoelasticity. It is proved that if the initial data are close to the equilibrium then the problem admits a unique, global, smooth solution. Moreover, as time tends to infinity, the solution is exponentially stable. As a corollary we also obtain the existence of periodic solutions for small, periodic right-hand sides.

The global existence of smooth solutions to the equations of nonlinear hyperbolic system of 2nd order with third order viscosity is shown for small and smooth initial data in a bounded domain of n-dimensional Euclidean space with smooth boundary. Dirichlet boundary condition is studied and the asymptotic behaviour of exponential decay type of solutions as t tending to infinity is described. Time periodic solutions are also studied. As an application of our main theorem, nonlinear viscoelasticity, strongly damped nonlinear wave equation and acoustic wave equation in viscous conducting fluid are treated.

The global existence of smooth solutions to the equations of nonlinear thermoelasticity is shown for a one-dimensional homogeneous reference configuration. Dirichlet boundary conditions are studied and the asymptotic behaviour of the solutions as t --&gt; infinity is described.

A particle method based on the GENERIC formalism which is a theoretical framework for non-equilibrium thermodynamics has been studied. Indeed, a particle method based on the Poisson bracket defined on the state space of the Eulerian description of fluid flow has been developed and some numerical experiments have been done. In addition, the two-dimensional vorticity equation has been formulated within the GENERIC formalism and discretized using the discrete variational derivative method. The thus-obtained numerical method preserves or dissipates the kinetic energy and enstrophy depending on whether the flow is inviscid or viscid

This research is concerned with the Navier-Stokes equations on either the whole plane or two-dimensional exterior domains. It was shown that, if there exists a small stationary external force with strong symmetry, the equation has a small stationary solution decaying rapidly at infinity. It was also shown that, if the stationary solution above is sufficiently small, it is stable under initial perturbation without restriction on the size. The Navier-Stokes equaions in an infinite layer is also studied. It is shown that, if the equation is treated in the Besov spaces, nontrivial solutions with no external forces exist if p is infinite, and that these solutions correspond to the Poiseuille flows

The head investigator. Nakao. has considered the stabilization problem for the nonlinear wave equations in interior and exterior domains and also the behaviours of solutions for nonlinear heat equations.
For exterior problems. we first proved the local energy decay for linear wave equations. and on the basis of this we have derived L^p estimates. Further. by use of these estimates we have discussed on the global existence of semilinear wave equations. We note that in our argument no geometrical conditions on the shape of obstacles.
For interior problems. we have proved global existence of smooth solutions for the quasilinear wave equations with a very weak dissipative term. In this procedure we have showed a unique continuation property for the wave equation with a variable coefficient. Nakao's inequality was used for the decay estimate. which is an originality of this paper.
Concernibg nonlinear heat equations we have treated meancurvature type and m-laplacian type quasilinear equations under various nonlinear perturbations. We have derived sharp estimates of solutions including asymptotics as t → ∞ and smoothing effects near t = O.
Investigator Kawashima has mainly treated the equations concerning gas dynamics and shown many interesting results. Investigator Shibata has considered the visco-elastic wave equations and also exterior problems concerning fluid dynamicsand prove many new results by use of spectral analysis. Investigator Kato has discussed on the global solutions of a non-Newtonian flow equation.

We studied the initial-boundary value problem for Navier-Stokes equation in 3 dimensional exterior domain which describes the motion of incompressible viscous fluids. In 1930, J. Leray proved the existence of its weak solutions without uniqueness. But, we can not get any qualitative property of the motion of fluid from Leray's solutions. In 1950, R. Finn started to study the qualitative property of soltuions to the stationary Navier-Stokes equation, which was termed physically reasonable solutions (prs) by him. He proved the unique existence theorem of prs for small external force and the velocity uィイD2∞ィエD2 at infinity. After Finn, Heywood proved the stability of prs in the LィイD22ィエD2 framework, which now become one of the most important base of numerical investigation of the fluid motion. But, prs does not belong to LィイD22ィエD2 space, and therefore we have to study the stability of prs in the class to which prs belongs. This problem remained more than 30 years. In our present study, we solved this problem. We used the following argument. We took the Oseen approcimation in the 3 dim. Exterior domain, and then we investigated the local energy estimate near the boundary of optimal order of the Oseen equation by showing the fractional differentiablity of Oseen resolvent near the origin. Combinig this and LィイD2pィエD2-LィイD2qィエD2 estimate in the whole space by cut-off technique, we proved the optimal LィイD2pィエD2-LィイD2qィエD2 estimate of the Oseen semigroup in 3 dim. Exterior domain. The most important point is that all the constant appearing in the estimate is independent of uィイD2∞ィエD2. By using this estimate, we could solve the stability problem of Finn's prs.
(2) In 2 dimensinal case, we know the unique existence of Leray's weak solutions. But, since the Stokes fundamental solution has logarihmic singularity, we know less property of solution to NS in the exterior domain or even the whole space compared with 3 dim. Case. We could obtain the asymptotic expansion of Stokes resolvent near the origin and we found that the logarithmic singularity is canceled out by the reflection phenomenon near the bounday, and therefore we obtained the optimal LィイD2pィエD2-LィイD2qィエD2 estimate (1< q≦ p≦ ∞) of Stokes semigroup in a 2 dim. Exterior domain by using the similar argument to the 3 dim. Case. Applying this estimate, we could show the best convergent rate of weak solutions to NS when time goes to infinity.
(3) We considered the motion of the compressible viscous fluid too. Extending the method developed in study (1), we obtained the optimal LィイD2pィエD2-LィイD2qィエD2 estimate of solutions to linearized eqautions in the 3 dim. Exterior domain, which is applied to obtain the optimal convergent rate of solutions to the original nonlinear probolem at time infinity.

First, we investigated the theory of hyperfunctions from a viewpoint of (classical) analysis, and proved fundamental results from this viewpoint. In doing so, we considered the (inverse) Fourier transform S'_<epsilon> of {exp [epsilon <xi>] u (xi) ; u*S'}, and regarded the space of hyperfunctions as the local space of *_<epsilon>0>S'_<epsilon>. Here S denotes the Schwartz space. Furthermore, we established the (classical) analytical theory of pseudodifferential operators and microlocal analysis for hyper-functions, generalizing calculus of pseudodifferential operators and Fourier integral operators in S' to calculus in S'_<epsilon>. In the studies of partial differential operators (and pseudodifferential operators), we could apply the same arguments to S'_<epsilon>, especially the space of hyperfunctions, as used in the category of distributions. And we made it possible to investigate, with unified treatments, partial differential operators in the spaces of distributions, ultradistributions (and Gevrey), and hyperfunctions (and analytic functions). For example, we treated the problems, deriving a priori (energy) estimates. In particular, we obtained results on propagation of analytic singularities and analytic hypoellipticity for analytic pseudodifferential operators from a priori estimates. We also proved that the relation between hypoellipticity and local solvability in the category of hyperfunctions is the same as in the category of distributions. And we obtained several results on local solvability in the space of hyperfunctions from a priori estimates.
We studied partial differential operators from a viewpoint of derivation of a priori estimates. And we obtained a priori estimates for various problems. The related problems were studied by the investigators of this project. We believe that the results obtained here are of great use for the studies on partial differential operators.

1)弾性体の運動を記述する数学的理論はいわゆる非線形熱粘弾性方程式(nonlinear thermoviscoelastic equations)といわれる微分方程式の初期値境界値混合問題で与えられる。本年度は初期値が十分小かつ滑らかな場合に時間大域的に滑らかな解が存在することを示した。
2)粘性流体中を物体が運動する様子は数学的にはナヴィアーストークス(Navier-Stokes)方程式の外部問題として定式化される。本年度は無限遠でのスピードがゼロでない場合の3次元流の強解の存在をL_3の枠組みで示した。さらに、その解の漸近挙動を示した。いわゆる、Wake regionの数学的解析を精密に行った。これは、流体の運動を数値解析的に解明しジェット機やロケットなどの合理的な設計においても将来有効な理論を与えていると信じる。こうして、この研究はさらに物理や工学の流体の専門家や数値解析の専門家とともにより大きな組織として続ける必要がおおいにある。
さらに、この研究の一つの帰結として、1950年代にR.Finnによって提出されたstarting problem(粘性流体中の物体を徐々に加速していき有限時間後加速することを止めたならば、漸近的に一定の速度の運動に物体はなるか、又そのときの物体と回りの流体の漸近挙動を求めよ)を完全に解くことができた。
この方面の研究は2次元流においては全くなされていないので、つぎの課題としては、3次元流での方法をヒントに2次元の場合にさらに発展することが考えられる。しかし、技術的に簡単に3次元のやりかたが2次元に拡張出来ない。これも今後の課題である。
3)1次元の熱弾性体の数学的理論に現れる、いわゆる、1次元熱弾性体方程式の初期値境界値問題の解の存在とその漸近挙動を外力が時間に依存しない場合に示した。今後の課題は、外力が時間に依存する場合、特に外力が周期的な場合に解の存在をしめすことである。

非線形偏微分方程式において、解の特異性が非線形性によってどのような相互作用をするのかということを調べるのは、極めて重要な問題の一つである。空間1次元の特殊な3次の非線形性を持つ非線形シュレデインガー方程式は完全可積分系となり、その解の性質はよく調べられている。特に、完全可積分系の非線形シュレデインガー方程式に対しては、時刻無限大でも非線形効果が消えず、解は摂動を受けていない自由解には近付かないことが知られている。空間1次元の場合、3次の非線形性は線形散乱理論で云うところの長距離ポテンシャルに相当しており、この事実自体は自然なことである。しかし最近、非線形波動方程式について、従来長距離ポテンシャルに相当すると考えられていた場合でも、ある特別な非線形項に対しては解の特異性が相殺し、時刻無限大で解は自由解に近付くことが分かってきた。非線形シュレデインガー方程式に対しても、特別な非線形項の場合は波動方程式の時と同様、解の特異性が相殺し時刻無限大で非線形効果が消えることが期待される。そこで、今年度は、どのような3次の非線形項に対して、解の特異性が相殺し時刻無限大で解が自由解に近付くかを調べた。その結果、そのような非線形項はシュレデインガー方程式のゲージ不変性と密接な関係があることが分かった。
また、熱弾性プレートを伝わる波を記述する方程式の線形化問題について、熱散逸効果による解のエネルギーの時間減衰の速さを調べた。熱弾性波は近年様々な分野で注目を集めており、熱弾性波の非線形問題の数学的解析が切望されている。線形化問題の解析は、それ自身興味深い問題であるばかりでなく、非線形熱弾性波の問題を研究する際に有用である。

線型及び非線型の双曲型方程式に対する初期値問題の解の存在及び解の性質を次の問題点について研究した。
(i)関数解析的な手法を用いて解の局所的及び大域的存在についての研究
(ii)超関数解の特異点の作る集合の幾何学的な特徴付けを行う。
(iii)測度を係数に持つ双曲型方程式の解の存在及び解の性質を調べる。
(iv)幾何光学から生じる散乱問題及び波動伝播問題の研究。
今年の研究成果として上にのべたテーマに関して部分的ではあるが成果が得られた。

1.一般2n+1次元接触多様体上の拡張されたモノポールの位相不変量が,Higgs場と曲率から定まる2n-形式の曲面積分で与えられることがわかった。
2.この位相不変量はHiggs場についてホモトピー不変であり,許容されるどのようなHiggs場をとっても位相不変量が零となるケースが考えられる。この現象は3次元モノポールと対比して特異的である。
3.一般次元(≧5),拡張されたモノポールに特有なもうひとつの現象として,エネルギー汎関数(作用積分)の値が位相不変量をとりえないということがある。
4.5次元トーラスが接触構造を許容するか否かという懸案の問題に対して今後の研究の新たな展開として接触構造から定まる可積分な2次元部分多様体とそのうえのベクトル束の研究が注目される。

弾性体方程式、熱弾性体方程式、Schrodmger方程式等を含む,非線形発展方程式の解の局所的な意味での存在,大域的な意味での存在をかなりの一般論を含む形で確立することができた。更に解の漸近挙動の解明,解の特異性の表われる様相を多くの場合に調べることが出来た。手法はいくつかあるが,いわゆるenergy method,microlocal analysis等をたくみに用いた。もう少しく列挙することにする.
(1)線形化した方程式の解の存在,解の挙動をいわゆるenergy methodといわれる部分積分に基礎をおく方法で,いくつかのすぐれたmultspliersをみつけて、示すことに成功した。また、あわせてmicrolocalな手法を用いることで,解の正則性の伝播の状態を解明することも出来た。これらは世界的に先駆となる成果をいくつも含んでいる。また,定常的な方程式の固有値の分布状態も,確率的手法を用いることで,より詳しく調べることが出来ることを示し,古典的軌道の解の安定性に関する影響を調べる上で,より詳しく分かることが示せた。これらは勿論初めて試みを多く含んでいる。
(2)Picardの遂似近似法のアイディアを必要に応じて拡張し、非線形発展方程式の解の局所的及び大域的な存在を示した。(1)に及べた線形化した方程式の結集が更に応用され、非線形の場合の解の漸近挙動,特異性の様相が解明された。

1.effectively hyperbolicの一般化としての双曲型作用素のクラスを定義し、それに対するCauchy問題のC^∞適切性を示した。また、2重特性的である双曲型作用素に対するCauchy問題を考え、新しい結果を得た
2.一意接続定理について研究し、新しい方法によるアプローチを試みた。この方法によって新しい結果を得ることは、今後の課題の一つである。
3.Gevrey準楕円性に対して、いくつかの興味ある例を与え、その準楕円性を研究した。また 解析的準楕円性を我々の立場(問題を超局所アプリオリ評価の導出に帰着する)から研究するための基礎となる諸結果を得た。
4.擬微分作用素の有界性及び正値性に関しては、十分な結果は得られなかったが、今後の研究のための準備は整ったと思う。
5.1に関連して、超局所アプリオリ評価に関するいくつかの結果が、得られた。
6.確率論の立場から、Schrodinger作用素の研究、極限定理の研究及びマルコフ過程の検究を推し進めた。
7.microhyperbolicな擬微分作用素のいくつかのクラスに対して、C^∞の特異性の伝播定理を示した。
8.非線形の弾性体の方程式に対する混合問題の解の存在及び一意性に関して、いくかの結果が得られた。
9.非線形波動方程式の解の大域的存在及び爆発に関して研究を進め、初期値がコンパクトな台を持たない場合も考察した。

完全可積分系の典型例である3次の非線形項を持つ非線形シュレディンガー方程式は、物理的にはザハロフ方程式というシュレディンガー方程式と波動方程式が非線形に連立した方程式系において、イオン音波速度が非常に速いときの近似として得られると考えられている。3次の非線形項を持つ非線形シュレディンガー方程式は幾何学題にきれいな対称性を豊富に持ち、単独のソリトン解やN-ソリトン解のような物理的に重要な意味を持つ解を持つことが知られている。イオン音波速度が非常に速い時に、3次の非線形項を持つ非線形シュレディンガー方程式の解が実際に元のザハロフ方程式の解を近似しているかどうかは重要な問題である。この問題は数学的には特異摂動の問題となり、イオン音波速度定数を無限大にした時に、ザハロフ方程式の解において初期時刻の近傍でいわゆる初期層と呼ばれる特異性が発生する。この特異性の解析も重要な問題である。今回は、ザハロフ方程式においてイオン音波速度を無限大にした時、ザハロフ方程式の解は非線形シュレディンガー方程式の解に収束することを示し、初期層の精密な解析を行った。
また、空間2次元のべき乗型の非線形項を持つ非線形波動方程式に対しては、散乱作用素が構成できる下限の非線形項の指数が予想されていたが、実際にその下限の指数まで散乱作用素が構成できることを示した。
さらに、現実の物理現象では、なんらかの理由で減衰効果が働くことが多い。そこで、完全可積分系のような保存系に、減衰効果が加わったときに、解はどの様な振舞いをするようになるのかという問題を研究することは重要である。今回、粘性項を持つある種の弾性体の方程式に対して、定常解の近傍で初期値を与えると、時刻無限大では解はその定常解に漸近的に近づくということを示した。

1.研究目的:一次元拡散過程は、W.Feller,E.B.Dynkin,H.P.McKean JR.,伊藤清等によって解析的にも確率論的にも完全に解明されている。しかしながら、多次元拡散過程は、解析学における偏微分方程式論、関数解析学、複素解析学、確率論と密接に関連する重要な研究課題であるにも拘わらず、一次元の場合と比べると完全な解明からは未だ遠い状態にある。本研究の目的は、多次元拡散過程を解析学の立場から総合的に研究することである。
2.研究実績:研究代表者及び分担者は、多次元拡散過程の境界問題を、最新の偏微分方程式的手法及び関数解析的手法を用いて研究するアプロ-チを発展させることにより、楕円型とは限らない一般の境界条件の場合を詳しく考察することができた。さらに、この研究を通じて、偏微分方程式論における各種の十分条件を、「拡散粒子の運動」という具体的なイメ-ジを通じて、直観的に解釈することを試みた。解析学の偏微分方程式論、関数解析学、確率論の三分野の接点に光を当てる、この研究結果は、「Boundary Value Problems and Markov Processes」(境界値問題とマルコフ過程)、「On the Existence of Feller Semigroups with Boundary Conditions」(境界条件付きのフェラ-半群について)と題してそれぞれ講義録及び特集論文として、シュブリンガ-社(ドイツ)、アメリカ数学会から出版された。これは、この方面で現在最も進んだ研究結果として、欧米を中心に高く評価されたことの証左であり、本研究の目的は十分に達成することができたといえる。
一次元拡散過程の研究の例を見るまでもなく、多次元拡散過程及び関連する諸問題を多分野から総合的に研究して行くことは、今後の解析学の発展にとって益々重要になるものと思われる。

函数空間、特に、函数および超函数の可微分性の度合いをL_pーノルムで測るSobolev空間とBesov空間を解析学と幾何学に応用することを目標とし研究を進めた。
(イ)Aを解析的半群の生成作用素とするとき、Banach空間における抽象的常微分方程式:du/dt-Au=f(t)の強解の存在について。よくしられているように、外力項fについての強連続性の仮定のみで強解の存在は言えない。我々はfが局所的にBesov空間B〓に属するという強解の存在を証明した。これはHolder連続性やCrandall-Pazyの条件より弱い条件で決定的な結果であり、準線型方程式へ応用できる。また、Aがweak singularityを持つ半群の生成作用素の場合についても同様な結果が得られている。
(ロ)擬微分作用素のBesov空間における有界性について。表象a(x,ξ)のxおよびξについての最小な微分可能性の仮定のもとで有界性定理を得た。微分可能性の度合いを測るために我々は重複次数のBesov空間を使った。この重複次数のBesov空間についての一般的理論も得られている。この結果はBesov空間論が偏微分方程式など解析学の各分野で有用なことを示す例になっている。
(ハ)杉本充はHardy空間を使って双曲型方程式に関連するフ-リ乗法作用素のL_Pー有界性に関するよい結果を得た。
(ニ)その他、解析学の各分野、特に双曲型偏微分方程式や確率過程をポテンシャルに持つSchrodinger作用素についての研究成果も得られている。
(ホ)本研究は位相幾何学と微分幾何学の研究者と協力を必要とし、また微分幾何学への応用も目指している。

1.森本はY字型の柱状領域においてNavier-Stokes方程式の定常問題を考えた.領域が軸対称であることを仮定し,境界値も解も軸対称であることを仮定して解の存在を示した.これは,アミックによる有界領域での結果のY字領域への拡張である.領域が非有界であることから,アミックの場合と本質的にことなる困難が生じた.藤田により開発された,仮想の溝を掘って流れを流すという方法を拡張し,無限遠方でのポアズイ流の構成を行うことで,この困難を解消することが出来た.2.菱田は物体が非圧縮性流体中にある物体が回転している場合の流体の流れについての数学的解析を行った.通常のNavier-Stokes方程式にx×∇なる形の変係数でしかも非有界な係数のつく,取り扱いに困難な作用素である.今までにこの様な作用素の解析は無く,新しい解析を要求される問題である.本年度までの研究では,部分積分することにより得られる,保存量と全空間での精緻な解析を駆使して,少なくとも対応する線形問題がL_p枠においてC_0半群をなすことを示し,対応する非線形問題を時間局所的に解いた.さらに,解の正則性についての考察を行った.3.柴田は,清水とともに,弾性体の方程式のレゾルベント問題で開発したL_p評価を求める方法を拡張してStokes方程式のレゾルベント問題のNeumann型の境界値問題に関するL_p評価を行った.また,院生の秋山とともに,Ginzburg-Landau方程式の定数定常解の安定性を磁場がある場合に示した.さらに,院生の阿部と2枚の板の間を流れる非圧縮性粘性流体を記述するStokes方程式の粘着性境界条件のもとでのレゾルベント問題を考え,正則半群が生成される事を示した.さらにこれは指数的な安定性をもつ事を示し,対応する非線形問題の初期値問題を解いた.これらの解法は実解析的手法に基づいており,さらなる発展が粘性流体の自由境界値問題などへ見込まれる

We have obtained a lot of results. We refer here some of the main results.1. For 2 x 2 systems with two independent variables, we obtained a necessary and sufficient condition in order that the Cauchy problem is well posed. The condition is expressed using the Newton polyhedron. In this study we found a peculiar example which is strictly hyperbolic apart from the initial plane for that the Cauchy problem is not well posed for any lower order term.2. We introduced a new notion "pseudo-symmetric hyperbolic systems" which extends the symmetrizable hyperbolic systems. We proved that the Cauchy problem for pseudo-symmetric hyperbolic systems with one space variable is well posed. The question is still open for pseudo-symmetric systems with several space variables.3. We succeeded in obtaining a necessary condition on lower order terms for the Cauchy problem is well posed for general, hyperbolic systems using the determinant on a non commutative field where the localization lives : the. leading part of the non commutative determinant of the localization of the total symbol coincides with the principal part of the classical determinant of the principal symbol.4. The symmetrizability of the frozen system at every space point implies the symmetrizability of the original systm if the reduced dimension is enough high. In particular if the every frozen system is stringly hyperbolic then the original system is also strongly hyperbolic if the reduced dimension is high

1.Stability of the Oseen flow in the n-dimensional exterior domain (n>2).2.Stability of the Couette flow and the Poiseuille flow in the infinite layer.3.Rate of convergence of the non-stationary flow to the stationary flow of compressible viscous fluid.4.Resolvent estimate of solutions to the Stokes equation with Neumann boundary condition

In a joint work with Yoshihiro Shibata, we obtained a sufficient condition on time-independent external forces for the unique existence of a stationary solution in a certain class of the Navier-Stokes equation in exterior domains of dimension n greater than or equal to 3 by using the duality between the Lorentz spaces and real interpolation. Our class is a natural generalization of the so-called physically reasonable solutions, and our suffirient condition gives a unified view for the case with zero velocity at infinity and the case with non-zero velocity at infinity.Next, in a joint work with Yuko Enomoto and Yoshihiro Shibata, we verified the stability in the weak-Ln space of the stationary solution above for time-evolution under small initial perturbation in the weak-Ln space, and showed that the smallness above can be taken uniformly in the velocity at infinity of the stationary solution.Furthermore, by using real interpolation for sublinear operators, we generalized these results for time-dependent external forces, and obtained a sufficient condition for the unique existence of the corresponding time-periodic or almost periodic solutions. We also showed the stability of these solutions in weak-Ln spaces under perturbations on the external forces and initial data uniform in the velocity at infinity of the solutions.On the other hand, as a preparation for generalized the results above for general unbounded domains, we generalized the Lp-theory on the boundary value problem for the Stokes equation in a layer domain, in a joint work with Takayuki Abe for higher-order Sobolev spaces and Besov spaces, and obtained a sufficient condition on the external forces for the unique existence of the solution of the boundary value problem. In particular, we showed that the uniqueness of the solution fails in the case p=infinity, and that the Poiseuille flow can be characterized as the solution with zero as the external forces and boundary values

The main purpose of this research is concerned with the exterior problem for the quasi-linear wave equations. For this problem we have been successful in proving the global existence of smooth solutions under the effect of localized dissipation. We have achieved the results through two ways; one is based on the local energy decay and L^p estimates of solutions for linear equation, and the other one is the method to utilize total energy decay for the llinearized equation. Both ways are intended to make the effects of dissipation as weaker as possible, but, we have made no geometrical conditions on the shape of the boundary.Concerning another problem on the energy decay for the equation with nonlinear dissipations we introduced anew concept ‘Half linear' and has been successful in deriving very delicate decay estimates of energy and applied them to the existence of global solutions for the equations with a nonlinear source term.As related problems we have considered the existence and stability of periodic solutions for the nonlinear wave equations in bounded domains with some nonlinear localized dissipations. Further, we have considered the Kirchhoff type nonlinear wave equations in exterior domains. Under a nonlinear dissipations we have proved various results on global solutions. For the wave equation in exterior domains with a Neumann type boundary dissipation we have derived a new energy decay estimate.Investigator Kawashima has derived many interesting results concerning Boltzman equations and hyperbolic conservation equations. Investigator Shibata has derived by the method of spectral analysis, many interesting results concerning the exterior problem for the compressive Navier-Stokes equations. Investigator Ogawa has proved precise estimates of solutions concerning behaviors and regularities of solutions for the nonlinear wave equations, nonlinear Shroadinger equations and some harmonic evolution equation

In this research, we consider the Stokes equation with Neumann boundary condition which is obtained as a linearized equation of the free boundary problem for the Navier-Stokes equation. We analyzed this problem by the following procedure : (1) Analysis of the resolvent problem (2) Generation of Analytic semigroups (3) L_p-L_q estimates(1)Obtained is the L_p estimate of solutions to the resolvent problem for Stokes system with Neumann type boundary condition in a bounded or exterior domain in R^n. The result has been obtained by Grubb and Solonnikov by the systematic use of theory of pseudo-differential operators. In this paper, we give an essentially different proof from theirs. The core of my approach is to estimate the solutions in the whole space and half-space case. We apply the Fourier multiplier theorem to solution of the model problems.(2)First we introduce the Helmholtz decomposition. Then we delete pressure trem and reduce to the problem only including velocity vector. Then we generated analytic semigroup to this reduced Stokes equation.(3)We obtained local energy decay estimates and L_p-L_q estimates of the solutions to the Stokes equation with Neumann boudary condition. Comparing with the non-slip (Dirichlet) boundary condition case, we have a better decay estimate for the gradient of the semigroup because of the null net force at the boundary

We study the spectral analysis of Stokes equations based on the recent development of the real analysis, Fourier analysis and functional analysis and its application to the Naveir-Stokes equations in several different situations arising from the mathematical physics.1) We studied an inncompressible viscous flow past a rigid body, which is mathematically described by Oseen equations. We studied the decay properties of the Oseen semigroup in the exterior domain and showed global in time stability Navier-Stokes flow past a rigid body2) We studied an inncompressible viscous flow in a perturbed half-space which describes for example flow past high buildings. We studied an optimal decay properties of solutions to the Stokes equations in a perturbed half-space and proved a global in time unique existence of solutions to the Navier-Stokes equations in a perturbed half-space with small initial data.3) We proved the maximal regularity of solutions to the Stokes equation with the Neumann boundry condition in a bonded domain. We use some recent development of the operator-valued Fourier analysis by Weis and Denk-Hieber-Pruss. Our method is very simple compared with previous results and seems to be applicable to linear evoulution equations of parabolic type. Moreover, we proved local in time unique existence of strong solutions with arbitrary initial data and global in time unique existence of strong solutions with some small initial data of the free boundary problem of Navier Stokes equations which describes the transient motion of an isolated volume of viscous incompressible fluid4) We studied an inncompressible viscous flow past a rotating rigid body. This problem was already studied by Galdi and Galdi and Silvestre in the L_2 framework. Our main contribution is to show the decay estimate of the continuous semigroup associated with linearized problem. The main difficulty comes from first order differential system with polynomially growing coefficients which is not subordinated by the Laplacian. We developed new technique to investigate the high frequency part of the spectrum

1. Researches on environmental fluids which are typical examples of multicomponent multiphase fluids. The continuous models have been obtained as coupled systems of Navier-Stokes systems and convective reaction-diffusion systems. Numerical models which are consistent with the continuous models were formulated as high precision, upwind and TVD schemes, and numerical simulations based on those models have been performed.2. Fluid flow analysis around structures in environmental fluids. Numerical models for convective reaction-diffusion phenomena in environmental fluids have been developed. In particular, numerical analysis of air flows and motion of environmental pollutants in the over complex geographical topographies have been made.3. Studies in time-dependent nonlinear perturbations of integrated semigroups. Time-dependent semilinear evolution equations are treated from the point of view of nonlinear evolution operator theory and have been applied to various mathematical models formulated as large-scale semilinear systems of partial differential equations.4. The mathematical approach to HIV infection process and HAART therapies. Mathematical models of HIV disease progressions are formulated from the point of microbiology and physiology of HIV infection process in the immune system of an individual infected patient. The results of computer simulations based an the model agree with clinical data.5. Researches on time-dependent nonlinear perturbations of analytic semigroups. A general theory for time-dependents nonlinear perturbation of analytic semigroups have been advanced and a characterization of the existence of nonlinear evolution operators has been obtained in terms of their semilinear generators. As a particular application of this theory, a bone remodeling model was completely solved

In this research, we consider a free boundary problem for the Navier-Stokes equation which describes the motion of an isolated finite volume of viscous incompressible fluid without taking surface tension into account By using the Lagrange coordinates, free boundary problem is written by the quasi-linear equations on the fixed boundary. Our purpose is to prove a local in time unique existence theorem for any initial data and a global in time unique existence theorem for some small initial data.To treat quasi-linear equations, first we prove the Lp-Lq maximal regularity of solutions to the linearized problem, which is described by the Neumann problem for the Stokes equation. We consider this problem by analytic semigroup approach. Our main issues is to use R-boundedness and operator valued Fourier multiplier theorem which are recently developed by Weis ('01, Math.Ann.) and Denk-Hieber-Pruss ('03, Mem.AMS).Based on the Lp-Lq maximal regularity result of the linearized problem, by using the contraction mapping principle, we proved a local in time unique existence theorem for any initial data and external force and a global in time unique existence theorem for some small initial data which are orthogonal to the rigid space in the case where external force vanishes

Establishment of Verified Numerical Computation We have studied verified numerical computations for partial differential equations and systems of linear equations using digital computers. Calculating sum of a vector and dot product of two vectors with guaranteed high accuracy is ubiquitous in scientific computing. We have developed such algorithms for accurate sum and dot product, which are known to be the fastest so far. As applications, we have applied the fast and accurate algorithms to sparse matrix computations, computational geometry and so forth. Moreover, we have succeeded in proving the existence and uniqueness of a solution of a partial differential equation, and in calculating an error bound of its approximate solution

The main object of this project is to study the asymptotic behaviours of solutions of nonlinear wave equations through the investigation of global attractors. As related problems we also intended to investigate the energy decay problem for the wave equations and global attractors for nonlinear parabolic equations.First we considered the problem for the equations in bounded domains and established new results concerning the existence, sire and some absorbing properties of global attractors.Secondly, we considered the exterior problem fix Klein-Gordon type nonlinear wave equations and established a parallel results to the problem in bounded domains. In exterior domains the Sobolev spaces are not embedded I compactly into $1,^p$ spaces. This difficulty was overcome by the discover y that the local energy of solutions are controlled as small as we can near infinity when time also goes to infinity.In a joint work with Professor Y. Zhijiag from China we proved the existence and some exponential type absorbing of global attractors for some quasi-linear wave equations. This result generalize a known one for one space dimension to general dimensions.As related problems we give several results on global attractors for degenerate type quasi-linear parabolic equations which include estimates on smoothing effects. These are joint works with Prof C. Chen from China and Dr NT, Aris from Indonesia

研究成果：１．回転しながら平行移動する一つの剛体の周りの非圧縮性粘性流体の定常流の安定性を示した．証明の鍵は線形化問題に対応する連続半群の生成と，この半群の時間無限遠での漸近挙動を求めることにある．この線形化問題は多項式増大する係数を摂動項としてもつオゼン作用素の外部領域における初期値・境界値問題として定式化される．このように多項式増大する摂動項はこれまで全く扱われていない困難な問題であった．証明の鍵はレゾルベントをセクトリアル作用素の部分とセクトリアルではないが無限遠方でよい挙動をする部分に分けることにあった.２．種々の領域での非圧縮性粘性流体の自由境界問題の時間局所解の一意存在を示した．これまではヒルベルト空間の枠で考えられていた問題を，Lp－Lq最大正則性原理を示すことで，一般のバナッハ空間で示すことに成功した．これは自由境界問題では全く行われていない，ヴィエ．ストークス方程式をスケーリング不変な空間で扱うということを実現するために不可欠な考察である．非圧縮性粘性流体の自由境界問題の研究に新しい展開を与えた．またこれまでは液滴の落下や海面の運動に対応する数学的解析しか行われなかったのを，有界領域，外部領域，摂動半空間，摂動層領域，チューブ領域など物理的に自然に現れるあらゆる領域に対応する理論を構築した．３．板の運動と熱力学第２法則による内部エネルギィ変化を考慮した，分散型と放物型の混合系に関する初期値．境界値問題が解析半群を生成することをLp枠で示した．この問題はこれまでヒルベルト空間枠でしか扱われていなかったのを，バナッハ空間枠での扱いの拡張した．これにより方程式系のもつ分散性と放物性の両面の共存状態を完全に解明した．

(1)３次元圧縮性粘性流体中を運動する物体の安定性を、極限状態が定数である場合に考えた。この場合の解の存在、一意性、またL2枠での解の安定性は村松―西田により10年以上前に示されていた。また、初期値がL2空間に属しているときの解の各点での評価はDeckelnikにより示されていた。しかしこの評価は最良ではなかった。私は九州工科大学の小林氏との共同研究により、各点ごとの解の最良な評価を行った。その評価はdiffusion problemのものと結果としては、同じである。(2)　(1)での研究のなかで、質量項は双曲型方程式を本質的に満たす。こうして、全体としていわゆるdiffusion waveといわれる現象を数学的にとらえる方法を発見した。さらにそれを明確にするために、viscoelastic equationに対するCauchy問題を考えその解のL∞－L/I>1評価を行った。この証明中に、diffusion wave processを数学的にとらえる新しい方法を開発した。これは、双曲型と放物型の効果がいりまじった多くの方程式の解析に新しい視点と解析の方法を与えるものである。(3)　解の安定性を研究するのにあたり、線形化した問題の解のdecay estimateを行うことが問題となる。解はFourier変換をかいして表記されることが多い。そこで、関数とそのFourier変換像との関係を厳密に調べることは大事である。その観点にたち、古典的なBochner定理の拡張を行い、その応用としてStokes semigroupのfractional derivativeのdecay estimateを行った。結果自体は知られているものもあるが、照明方法は新しく、従来のものに比して簡単、明瞭であり、多くの新しい応用が期待できる。以上が特定課題研究助成費を用いて行った1998年度の研究の概要である。

２次元外領域でのNavier-Stokes方程式の漸近挙動について次の結果を得た。　線形化して得られるStokes作用素Aのレゾルベント（λ＋A）-1にたいして、λ＝0 の近傍で（λ＋A）-1＝　つぎに、レゾルベントの展開を応用してStokes semigroup｛etA｝の時刻t→∞ でのdecay評価を次のように求めた。∥etAf∥≦Cpqt-o（1/q-1/p）∥f∥q 1＜q≦p≦∞∥▽etAf∥Lp≦Cpqt-(1/q-1/p)-1/2∥f∥Lp1＜q≦p≦2t-1/q∥f∥Lp 　1＜q≦p, 2＜p≦∞　この事実を用い Kato argument を応用して Navier-Stokes 方程式の解u(t,x)のt→∞での挙動が次のようになることを示した。∥u(t,･)∥Lp＝o(t-(1/2-1/p)), 2≦p≦∞ ∥▽u(t,･)∥L2＝o(t-1/2)２次元問題の解の存在は Leray-Hopf, 一意性は Lions-Prodi により示されていた。また、∥u(t,･)∥L2→0t→∞は Masuda により示された。しかし、この収束の order を求めることは解の性質を知る上で重要な問題であった。はじめにこれに挑戦したのは、1993年の Kozono-Ogawa の論文であった。しかしそこでは、L∞ノルムの評価が optimal ではなかった。また、Stokes 方程式の漸近挙動を調べてもいなかった。この研究では、特にL∞ノルムの評価の optimal な評価を与えたこと。そして、Stokes semigroup の decay 評価を与えたことが特に注目すべき点である。また方法論的に言えば、レゾルベントの評価等を求める方法は解の性質を知るうえで色々な応用が見込まれる重要なものである。研究成果の発表1997年７月J. Mu&#209;　oz Rivera and Y. Shibata.Weiley & Sons Ltd.Mathematical Methods in the Applied Sciences, A linear thermoelastic plate equations with Dirichlet boundary condition1997年８月P. Galdi, J. Heywood and Y. ShibataSpringer-VerlagArchiv Rational Mechanics and AnalysisOn the global existence and convergence to steady state of Navier-Stokes flow past an obstacle that is started from rest1997年11月Y. ShibataGakkoutoshoMathematical Sciences and ApplicationsAn initial boundary value problem for some hyperbolic-parabolic coupled system1997年11月Y. ShibataGakkoutoshoMathematical Sciences and ApplicationsAn exterior initial boundary value problem for the Navier-Stokes equation