Updated on 2022/06/30

写真a

 
NISHIHARA, Kenji
 
Affiliation
Faculty of Political Science and Economics
Job title
Professor Emeritus

Education

  •  
    -
    1974

    Waseda University   Graduate School, Division of Science and Engineering  

  •  
    -
    1972

    Waseda University   Faculty of Science and Engineering  

Degree

  • Doctor of Science

Research Experience

  • 1980
    -
     

    School of Political Science and Economics, Waseda University

  • 1975
    -
    1980

    Tokyo National Institute of Technology

Professional Memberships

  •  
     
     

    The Mathematical Society of Japan

 

Research Areas

  • Mathematical analysis

Papers

  • Decay property of solutions for damped wave equations with space-time dependent damping term

    Jiayun Lin, Kenji Nishihara, Jian Zhai

    JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS   374 ( 2 ) 602 - 614  2011.02

     View Summary

    We consider the Cauchy problem for the damped wave equation with space-time dependent potential b(t, x) and absorbing semilinear term vertical bar u vertical bar(rho-1)u. Here, b(t, x) = b(0)(1 + vertical bar x vertical bar(2))(-alpha/2)(1 + t)(-beta) with b(0) > 0, alpha, beta >= 0 and alpha + beta epsilon [0, 1). Using the weighted energy method, we can obtain the L(2) decay rate of the solution, which is almost optimal in the case rho > rho c(N, alpha, beta) := 1+2/(N - alpha). Combining this decay rate with the result that we got in the paper U. Lin, K. Nishihara, J. Zhai, L(2)-estimates of solutions for damped wave equations with space-time dependent damping term, J. Differential Equations 248 (2010) 403-422], we believe that rho c(N, alpha, beta) is a critical exponent. Note that when alpha = beta = 0, rho c(N, alpha, beta) coincides to the Fujita exponent rho(F)(N) := 1 + 2/N. The new points include the estimate in the supercritical exponent and for not necessarily compactly supported data. (C) 2010 Elsevier Inc. All rights reserved.

    DOI

  • Asymptotic profile of solutions for 1-D wave equation with time-dependent damping and absorbing semilinear term

    Kenji Nishihara

    ASYMPTOTIC ANALYSIS   71 ( 4 ) 185 - 205  2011

     View Summary

    We consider the Cauchy problem for the wave equation with time-dependent damping and absorbing semilinear term
    {u(tt) - Delta u + b(t)u(t) + |u|(rho-1) u = 0, (t, x) is an element of R+ x R-N, (u, u(t))(0, x) = (u(0), u(1))(x), x is an element of R-N. (*)
    When b(t) = b(0)( t + 1)(-beta) with -1 < beta < 1 and b0 > 0, we want to seek for the asymptotic profile as t -> infinity of the solution u to (*) in the supercritical case rho > rho(F) (N) := 1 + 2/N. By the weighted energy method we can show the basic decay rates of u, which are almost the same as those to the corresponding linear parabolic equation
    phi(t) - 1/b(t)Delta phi = 0, (t, x). R+ x R-N (**)
    When N = 1, the decay rates of higher order derivatives of u are obtained by the energy method, so that the solution u can be regarded as that of (**) with source term -1/b(t) (u(tt) + |u|(rho-1) u). Thus, we will show
    theta(0)G(B)(t, x) (theta(0): suitable constant)
    to be an asymptotic profile of u, where G(B)(t, x) is the fundamental solution of (**).

    DOI

  • L-2-estimates of solutions for damped wave equations with space-time dependent damping term

    Jiayun Lin, Kenji Nishihara, Jian Zhai

    JOURNAL OF DIFFERENTIAL EQUATIONS   248 ( 2 ) 403 - 422  2010.01

     View Summary

    In this paper, we consider the damped wave equation with space-time dependent potential b(t, x) and absorbing semilinear term vertical bar u vertical bar(rho-1)u. Here, b(t, x) = b(0)(1+vertical bar z vertical bar(2))(-alpha/2) (1+t)(-beta) with b(0) > 0, alpha, beta >= 0 and alpha + beta is an element of [0, 1). Based on the local existence theorem, we obtain the global existence and the L-2 decay rate of the solution by using the weighted energy method. The decay rate coincides with the result of Nishihara [K. Nishihara, Decay properties for the damped wave equation with space dependent potential and absorbed semilinear term. preprint] in the case of beta = 0 and coincides with the result of Nishihara and Zhai [K Nishihara. J. Zhai. Asymptotic behaviors of time dependent damped wave equations, preprint] in the case of alpha = 0. (C) 2009 Elsevier Inc. All rights reserved.

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  • Decay Properties for the Damped Wave Equation with Space Dependent Potential and Absorbed Semilinear Term

    Kenji Nishihara

    COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS   35 ( 8 ) 1402 - 1418  2010

     View Summary

    We consider the Cauchy problem for the damped wave equation with space dependent potential V(x)u(t) and absorbed semilinear term |u|(rho-1)u in R(N). Our assumption on V(x) similar to (1+|x|(2))(-alpha/2) (0 <= alpha < 1) still implies the diffusion phenomena and the decay rates of solutions are expected to be the same as the corresponding parabolic problem. In this paper we obtain two kinds of decay rates of the solution effective for rho > rho(c)(N, alpha) := 1+2/(N - alpha) and for rho < rho(c)(N, alpha). We believe that in the "supercritical" exponent the decay rates obtained are almost the same as those for the linear parabolic problem, while, in the "subcritical" exponent the solution decays faster than that of linear equation, thanks to the absorbed semilinear term. So we believe that rho(c)(N, alpha) is a critical exponent. Note that rho(c)(N, alpha) with alpha = 0 coincides to the Fujita exponent rho(F)(N) := 1 + N/2.

    DOI

  • 消散型波動方程式のコーシー問題の解の拡散現象

    西原 健二

    数学   62 ( 2 ) 164 - 181  2010

    CiNii

  • Asymptotic behaviors of solutions for time dependent damped wave equations

    Kenji Nishihara, Jian Zhai

    JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS   360 ( 2 ) 412 - 421  2009.12

     View Summary

    in this paper, we consider the Cauchy problem for the wave equation with time dependent damping b(t)u(t) and absorbed semilinear term vertical bar u vertical bar(rho-1)u. Here, b(t) = b(0)(1+ t)(-beta) with -1 < beta < 1 and b(0) > 0. Using the weighted energy method, we obtain the L(1) and L(2) decay rates of the solution. which coincide to those for self-similar solutions to the corresponding parabolic equation when 1 < rho < rho(F)(N): = 1 + 2/N @ 2009 Elsevier Inc. All rights reserved.

    DOI

  • Asymptotic behavior of solutions for the damped wave equation with slowly decaying data

    Takashi Narazaki, Kenji Nishihara

    JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS   338 ( 2 ) 803 - 819  2008.02

     View Summary

    We consider the Cauchy problem for the damped wave equation
    u(tt) - Delta u + u(t) = vertical bar u vertical bar(rho-1)u, (t, x) epsilon R+ x R-N
    and the heat equation
    phi(t) - Delta phi = vertical bar phi vertical bar(rho-1)phi, (t, x) epsilon R+ x R-N
    If the data is small and slowly decays likely c(1) (1 + vertical bar x vertical bar)(-kN), 0 < k <= 1, then the critical exponent is rho(c)(k) = 1 + 2/kN for the semilinear heat equation. In this paper it is shown that in the supercrifical case there exists a unique time global solution to the Cauchy problem for the semilinear heat equation in any dimensional space R-N, whose asymptotic profile is given by
    Phi(0)(t, x) = integral(RN)e(-vertical bar x-y vertical bar 2/4t)/(4 pi t)(N/2) c(1)/(1+vertical bar y vertical bar(2))(kN/2)dy
    provided that the data phi(0) satisfies lim(vertical bar x vertical bar ->infinity)< x >(kN)phi(0)(x) = c(1) (not equal 0). Even in the semilinear damped wave equation in the supercritical case a time global solution u with the data (u, u(t))(0, x) = (u(0), u(1))(x) is shown in low dimensional spaces R-N, N = 1, 2, 3, to have the same asymptotic profile Phi(0)(t, x) provided that lim(vertical bar x vertical bar ->infinity)< x >(kN)(u(0) + u(1))(x) = c(1) (not equal 0). Those proofs are given by elementary estimates on the explicit formulas of solutions. (C) 2007 Elsevier Inc. All rights reserved.

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  • Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption

    R Ikehata, K Nishihara, HJ Zhao

    JOURNAL OF DIFFERENTIAL EQUATIONS   226 ( 1 ) 1 - 29  2006.07

     View Summary

    We consider the Cauchy problem for the damped wave equation with absorption
    u(tt) - Delta u + u(t) + vertical bar u vertical bar(p-1) u = 0, (t,x) is an element of R+ x R-N.
    The behavior of u as t -> infinity is expected to be same as that for the corresponding heat equation phi(t) - Delta phi + vertical bar phi vertical bar(p-1) phi = 0, (t, x) is an element of R+ x R-N. In the subcritical case 1 < p < p(c) (N): = 1 + 2/N there exists a similarity solution W-b (t, x) with the form t(-1/(p-1)) f (x/root t) depending on b = lim(vertical bar x vertical bar ->infinity vertical bar x vertical bar)(2/(p- 1)) f (vertical bar x vertical bar) >= 0. Our first aim is to show the decay rates
    (parallel to u(t)parallel to(L2), parallel to u(t)parallel to(Lp+1), parallel to del u(t)parallel to(L2)) = 0 (t (-) 1/p-1+N/4, t (-) 1/p-1+N/2(p+1), t (-) 1/p-1-1/2+N/4) (**)
    provided that the initial data without initial data size restriction spatially decays with reasonable polynomial order. The decay rates (**) are sharp in the sense that they are same as those of the similarity solution. The second aim is to show that the Gauss kernel is the asymptotic profile in the supercritical case, which has been shown in case of one-dimensional space by Hayashi. Kaikina and Naumkin [N. Hayashi, E.I. Kaikina, P.I. Naumkin, Asymptotics for nonlinear damped wave equations with large initial data, preprint, 2004]. We C-energy show this assertion in two- and three-dimensional space. To prove our results, both the weighted L-2-energy method and the explicit formula of solutions will be employed. The weight is an improved one originally developed in [Y. Todorova, B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001) 464-489]. (c) 2006 Elsevier Inc. All rights reserved.

    DOI

  • Asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity

    Kenji Nishihara

    ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK   57 ( 4 ) 604 - 614  2006.07

     View Summary

    We consider the Cauchy problem for the nonlinear dissipative evolution system with ellipticity on one dimensional space
    psi(t) = -(1 - alpha) psi - theta(x) + alpha psi(xx), (t, x) is an element of (0, infinity) x R theta(t) = -(1 - alpha) theta + nu(2)psi(x) + alpha theta(xx) + 2 psi theta(x),
    with 0 < nu(2) < 4 alpha(1 - alpha), 0 < alpha < 1. S. Q. Tang and H. Zhao [4] have considered the problem and obtained the optimal decay property for suitably small data. In this paper we derive the asymptotic profile using the Gauss kernel G(t, x), which shows the precise behavior of solution as time tends to infinity. In fact, we will show that the asymptotic formula
    [GRAPHICS]
    holds, where D-0, beta(0) are determined by the data. It is the key point to reformulate the system to the nonlinear parabolic one by suitable changing variables.

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  • Global asymptotics for the damped wave equation with absorption in higher dimensional space

    Kenji Nishihara

    JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN   58 ( 3 ) 805 - 836  2006.07

     View Summary

    We consider the Cauchy problem for the damped wave equation with absorption
    u(tt) - Delta u + u(t) + \u\(rho-1) u = 0, (t, x) is an element of R+ x R-N, (*)
    with N = 3,4. The behavior of u as t --> infinity is expected to be the Gauss kernel in the supercritical case rho > rho(c) (N) := 1 + 2/N. In fact, this has been shown by Karch [12] (Studia Math., 143 (2000), 175-197) for rho > 1+ 4/N (N = 1, 2,3), Hayashi, Kaikina and Naumkin [8] (preprint (2004)) for rho > rho(c) (N)(N = 1) and by Ikehata, Nishihara and Zhao [11] (J. Math. Anal. Appl., 313 (2006), 598-610) for rho(c)(N) < rho <= 1 + 4/N (N = 1, 2) and rho(c) (N) < rho < 1 + 3/N (N = 3). Developing their result, we will show the behavior of solutions for rho(c) (N) < rho <= 1 + 4/N (N = 3), rho(c) (N) < rho < 1 + 4/N (N = 4). For the proof, both the weighted L-2-energy method with an improved weight developed in Todorova and Yordanov [22] (J. Differential Equations, 174 (2001), 464-489) and the explicit formula of solutions are still usefully used. This method seems to be not applicable for N = 5, because the semilinear term is not in C-2 and the second derivatives are necessary when the explicit formula of solutions is estimated.

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  • Existence and nonexistence of time-global solutions to damped wave equation on half-line

    K Nishihara, HJ Zhao

    NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS   61 ( 6 ) 931 - 960  2005.06

     View Summary

    On the half-line R+ = (0, infinity) the initial-boundary value problems with null-Dirichlet boundary for both the semilinear heat equation and damped wave equation are considered. The critical exponent p(c) (N, k) of semilinear term for the existence and nonexistence about the semilinear heat equation on the halved space D-N,D-k = R-+(k) = RN-k is given by p(c)(N, k) = 1 + 2/(N + k) (J. Appl. Math. Phys. 39 (1988) 135-149; Arch. Rational Mech. Anal. 109 (1990) 63-71). Since the damped wave equation is expected to be close to the heat equation (J. Differential Equations 191 (2003) 445-469; Math. Z. 244 (2003) 631-649), the critical exponent for the semilinear damped wave equation is expected to be same as that of the semilinear heat equation. However, there is no blow-up result on the halved space for the damped wave equation. In this paper, the exponent pc(l, 1) = 2 is shown to be critical for the existence and nonexistence of time-global solution to both the semilinear heat equation and damped wave equation on the half-line R+, together with the derivation of the blow-up time. For the proof the explicit formulas of solutions are used in a similar fashion to those in Li and Zhou (Discrete Continuous Dynamic Systems 1 (1995) 503-520). (c) 2005 Elsevier Ltd. All rights reserved.

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  • Local Energy Decay for Wave Equations with Initial Data Decaying Slowly Near Infinity

    R. Ikehata, K. Nishihara

    Math. Sci. Appl.   22   265 - 275  2005

  • Global stability of strong rarefaction waves of the Jin-Xin relaxation model for the p-system

    K Nishihara, HJ Zhao, YC Zhao

    COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS   29 ( 9-10 ) 1607 - 1634  2004

     View Summary

    This paper is concerned with global stability of strong rarefaction waves of the Jin-Xin relaxation model for the p-system. The proofs are given by an elementary energy method and the existence of a positively invariant region obtained by Serre [Serre, D. (2000). Relaxations semi-lineaire et cinetique des systemes de Lois de conservation. Ann. Inst. H. Poincare Anal. Non Lineaire 17(2):169-1921 plays an important role in our analysis.

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  • Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations

    K Nishihara, T Yang, HJ Zhao

    SIAM JOURNAL ON MATHEMATICAL ANALYSIS   35 ( 6 ) 1561 - 1597  2004

     View Summary

    This paper is concerned with the time-asymptotic behavior toward strong rarefaction waves of solutions to one-dimensional compressible Navier-Stokes equations. Assume that the corresponding Riemann problem to the compressible Euler equations can be solved by rarefaction waves (V-R, U-R, S-R)(t, x). If the initial data (v(0), u(0), s(0))(x) to the nonisentropic compressible Navier-Stokes equations is a small perturbation of an approximate rarefaction wave constructed as in [ S. Kawashima, A. Matsumura, and K. Nishihara, Proc. Japan Acad. Ser. A, 62 (1986), pp. 249-252], then we show that, for the general gas, the Cauchy problem admits a unique global smooth solution (v, u, s)(t, x) which tends to (V-R, U-R, S-R)(t, x) as t tends to infinity. A global stability result can also be established for the nonisentropic ideal polytropic gas, provided that the adiabatic exponent gamma is close to 1. Furthermore, we show that for the isentropic compressible Navier-Stokes equations, the corresponding global stability result holds, provided that the resulting compressible Euler equations are strictly hyperbolic and both characteristical fields are genuinely nonlinear. Here, global stability means that the initial perturbation can be large. Since we do not require the strength of the rarefaction waves to be small, these results give the nonlinear stability of strong rarefaction waves for the one-dimensional compressible Navier-Stokes equations.

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  • The L-p-L-q estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media

    P Marcati, K Nishihara

    JOURNAL OF DIFFERENTIAL EQUATIONS   191 ( 2 ) 445 - 469  2003.07

     View Summary

    We first obtain the L-p-L-q estimates of solutions to the Cauchy problem for one-dimensional damped wave equation
    V-tt - V-xx + V-t = 0, (V, V-t)\(t=0) = (V-0, V-1)(x), (x, t) cis an element of R x R+,
    corresponding to that for the parabolic equation
    phi(t) - phi(xx) = 0 phi\(t=0) = (V-0 + V-1)(x).
    [GRAPHICS]
    etc. for 1less than or equal toqless than or equal topless than or equal toinfinity. To show (*), the explicit formula of the damped wave equation will be used. To apply the estimates to nonlinear problems is the second aim. We will treat the system of a compressible flow through porous media. The solution is expected to behave as the diffusion wave, which is the solution to the porous media equation due to the Darcy law. When the initial data has the same constant state at +/- infinity, a sharp L-p-convergence rate for pgreater than or equal to2 has been recently obtained by Nishihara (Proc. Roy. Soc. Edinburgh, Sect. A, 133A (2003), 1-20) by choosing a suitably located diffusion wave. We will show the L-1 convergence, ;applying (*). (C) 2003 Elsevier Science (USA). All rights reserved.

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  • L-p-L-q estimates of solutions to the damped wave equation in 3-dimensional space and their application

    K Nishihara

    MATHEMATISCHE ZEITSCHRIFT   244 ( 3 ) 631 - 649  2003.07

     View Summary

    It has been asserted that the damped wave equation has the diffusive structure as t --> infinity. In this paper we consider the Cauchy problem in 3-dimensional space for the linear damped wave equation and the corresponding parabolic equation, and obtain the L-p - L-q estimates of the difference of each solution, which represent the assertion precisely. Explicit formulas of the solutions are analyzed for the proof. The second aim is to apply the L-p - L-q estimates to the semilinear damped wave equation with power nonlinearity. If the power is larger than the Fujita exponent, then the time global existence of small weak solution is proved and its optimal decay order is obtained.

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  • Asymptotics toward the diffusion wave for a one-dimensional compressible flow through porous media

    西原健二

    Proc. Roy. Soc. Edinburgh   133A   177 - 196  2003

  • Diffusion phenomenon for second order linear evolution equations

    R Ikehata, K Nishihara

    STUDIA MATHEMATICA   158 ( 2 ) 153 - 161  2003

     View Summary

    We present an abstract theory of the diffusion phenomenon for second order linear evolution equations in a Hilbert space. To derive the diffusion phenomenon, a new device developed in Ikehata-Matsuyama [5] is applied. Several applications to damped linear wave equations in unbounded domains are also given.

  • $L^p$-$L^q$ estimates for 3-D damped wave equation and their application to the semilinear problem

    西原健二

    Seminar Notes of Mathematical Science, Ibaraki Univ.   6   69 - 83  2003

  • Convergence rates to viscous shock profile for general scalar viscous conservation laws with large initial disturbance

    K Nishihara, HJ Zhao

    JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN   54 ( 2 ) 447 - 466  2002.04

     View Summary

    This paper is concerned with the convergence rates to viscous shock profile for general scalar viscous conservation laws. Compared with former results in this direction, the main novelty in this paper lies in the fact that the initial disturbance can be chosen arbitrarily large, This answers positively an open problem proposed by A. Matsumura in [12] and K. Nishihara in [16]. Our analysis is based on the L-1-stability results obtained by H. Freistuhler and D. Serre in [1].

  • Asymptotic behavior of a one-dimensional compressible viscous gas with free boundary

    西原健二

    SIAM J. Math. Anal.   34 ( 2 ) 273 - 292  2002

    DOI

  • Large-time behaviors of solutions to an inflow problem in the half space for a one-dimensional system of compressible viscous gas

    A Matsumura, K Nishihara

    COMMUNICATIONS IN MATHEMATICAL PHYSICS   222 ( 3 ) 449 - 474  2001.09

     View Summary

    The "inflow problems" for a one-dimensional compressible barotropic flow on the half-line R+ = (0, +infinity) is investigated. Not only classical waves but also the new wave, which is called the "boundary layer solution", arise. Large time behaviors of the solutions to be expected have been classified in terms of the boundary values by [A. Matsumura, Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas, to appear in Proceedings of IMS Conference on Differential Equations from Mechanics, Hong Kong, 1999]. In this paper we give the rigorous proofs of the stability theorems on both the boundary layer solution and a superposition of the boundary layer solution and the rarefaction wave.

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  • Asymptotic behavior of solutions to the system of compressible adiabatic flow through porous media

    K Nishihara, M Nishikawa

    SIAM JOURNAL ON MATHEMATICAL ANALYSIS   33 ( 1 ) 216 - 239  2001.06

     View Summary

    Hsiao and Serre in [Chinese Ann. Math. Ser. B, 16B (1995), pp. 1-14] showed the solution to the system
    [GRAPHICS]
    with initial data
    (v, u, s) (0, x) = (v(0), u(0), s(0)) (x) ((v) under bar, u +/-, (s) under bar) as x --> +/-infinity
    tends to the following nonlinear parabolic equation time-asymptotically:
    [GRAPHICS]
    In this paper we find its convergence rate, which will be optimal.

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  • Boundary effect on a stationary viscous shock wave for scalar viscous conservation laws

    K Nishihara

    JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS   255 ( 2 ) 535 - 550  2001.03

     View Summary

    The initial-boundary value problem on the negative half-line
    [GRAPHICS]
    is considered, subsequently to T.-P. Liu and K. Nishihara (1997, J. Differential Equations 133, 296-320). Here, the flux f is a smooth function satisfying flu,) = 0 and the Oleinik shock condition f(phi) < 0 for u(+) < phi < u(-) if u(+) < u(-) or f(phi) > 0 for u(+) > phi > u(-) if u(+) < u(-). In this situation the corresponding Cauchy problem on the whole line R = (-<infinity>,infinity) to (*) has a stationary viscous shock wave phi (x + x,) for any fixed x,. Our aim in this paper is to show that the solution U(x, t) to (*) behaves as phi (x + d(t)) with d(t) = O(In t) as t --> infinity under the suitable smallness conditions. When f = u(2)/2, the fact was shown by T.-P. Liu and S.-H. Yu (1997, Arch. Rational Mech. Anal. 139, 57-82), based on the Hopf-Cole transformation. Our proof is based on the weighted energy method. (C) 2001 Academic Press.

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  • Large time behavior of solutions to the Cauchy problem for one-dimensional thermoelastic system with dissipation

    K Nishihara, S Nishibata

    JOURNAL OF INEQUALITIES AND APPLICATIONS   6 ( 2 ) 167 - 189  2001

     View Summary

    In this paper we investigate the large time behavior of solutions to the Cauchy problem on R for a one-dimensional thermoelastic system with dissipation. When the initial data is suitably small, (S. Zheng, Chin. Ann. Math. 8B(1987), 142-155) established the global existence and the decay properties of the solution. Our aim is to improve the results and to obtain the sharper decay properties, which seems to be optimal. The proof is given by the energy method and the Green function method.

  • Asymptotic behaviors of solutions to viscous conservation laws via L^2-energy method

    西原健二

    Adv. Math.   30 ( 4 ) 293 - 321  2001

  • Global asymptotics toward the rarefaction wave for solutions of viscous p-system with boundary effect

    A Matsumura, K Nishihara

    QUARTERLY OF APPLIED MATHEMATICS   58 ( 1 ) 69 - 83  2000.03

     View Summary

    The initial-boundary value problem on the half-line R+. = (0, infinity) for a system of barotropic viscous flow b(t) - u(x) = 0, u(t) + p(v)(x) = mu ux/v)(x) is investigated. where the pressure p(v) = v(-gamma) (gamma greater than or equal to 1) for the specific volume v > 0. Note that the boundary value at x = 0 is given only for the velocity u, say u(-), and that the initial data (v(0), u(0)) (x) have the constant states (v(+), u(+)) at x = +infinity with v(0)(x) > 0, v(+) > 0. If u(-) < u(+), then there is a unique v(-) such that (v(+),u(+)) is an element of R-2(v(-),u(-)) (the 2-rarefaction curve) and hence there exists the 2-rarefaction wave (v(2)(R), u(2)(R)) (x/T) connecting (v(-), u(-)) with (v(+), u(+)). Our assertion is that, if u(-) < u(+), then there exists a global solution (v, u) (t, x) in C-0 ([0, infinity); H-1 (R+)), which tends to the 2-rarefaction wave (v(2)(R), u(2)(R))(x/t)\(x greater than or equal to 0) as t --> infinity in the maximum norm, with no smallness condition on \u(+) - u(-)\ and parallel to(v(0) - v(+), u(0) - u(+))parallel to(H1), nor restriction on gamma (greater than or equal to 1). A similar result to the corresponding Cauchy problem is also obtained. The proofs are given by an elementary L-2-energy method.

  • L-p-convergence rate to nonlinear diffusion waves for p-system with damping

    K Nishihara, WK Wang, T Yang

    JOURNAL OF DIFFERENTIAL EQUATIONS   161 ( 1 ) 191 - 218  2000.02

     View Summary

    In this paper, we study the p-system with frictional damping and show that the solutions time-asymptotically tend to the nonlinear diffusion waves governed by the classical Darcy's law. By introducing an approximate Green function we obtain the optimal L-p, 2 less than or equal to p less than or equal to + infinity, convergence rate of the solution, which is a perturbation of the nonlinear diffusion wave, to the hyperbolic system. (C) 2000 Academic Press.

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  • Asymptotic stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas with boundary

    T Pan, HX Liu, K Nishihara

    JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS   16 ( 3 ) 431 - 441  1999.10

     View Summary

    This paper is concerned with asymptotic behavior of solutions of a one-dimensional barotropic flow governed by v(t) - u(x) = 0, u(t) + p(v)(x) = mu(u(x)/v)(x) on R-+(1) with boundary. The initial data of (v, u) have constant states (v(+), u(+)) at +infinity and the boundary condition at x = 0 is given only on the velocity u, say u_. By virtue of the boundary effect the solution is expected to behave as outgoing wave. Therefore, when u_ < u(+), v(-) is determined as (v(+), u(+)) is an element of R-2(v(-), u(-)), 2-rarefaction curve for the corresponding hyperbolic system, which admits the a-rarefaction wave (v(r), u(r))(x/t) connecting two constant states (v(-), u(-)) and (v(+), u(+)). Our assertion is that the solution of the original system tends to the restriction of (v(r), u(r))(x/t) to R-+(1) as t --> infinity provided that both the initial perturbations and \(v(+) - v(-), u(+) - u(-))\ are small. The result is given by an elementary L-2 energy method.

  • Porous media中の一次元圧縮性流の漸近挙動について

    日本数学会秋季総合分科会、広島大学    1999.09

  • Nonlinear waves with boundary effect

    Summer Workshop on Conservation Laws, Stanford Univ., USA    1999.08

  • Boundary effect on asymptotic behaviour of solutions to the p-system with linear damping

    K Nishihara, T Yang

    JOURNAL OF DIFFERENTIAL EQUATIONS   156 ( 2 ) 439 - 458  1999.08

     View Summary

    We consider the asymptotic behaviour of solutions to the p-system with linear damping on the half-line R+ = (0, infinity),
    v(t) - u(x) = 0, u(t) + p(v)(x) = -alpha u,
    with the Dirichlet boundary condition u/(x=0) =0 or the Neumann boundary condition u(x)/(x=0) =0 The initial date (v(0),u(0))(x) has the constant state (v(+),u(+)) at x = infinity. L. Hsiao and T.-P. Liu [ Commun. Math. Phys. 143 (1992), 599-605] have shown that the solution to the corresponding Cauchy problem behaves like diffusion wave, and K. Nishihara [J. Differential Equations 131 (1996), 171-188; 137 (1997), 384-395] has proved its optimal convergence rate. Our main concern in this paper is the boundary effect. In the case of null-Dirichlet boundary condition on u, the solution (u, u) is proved to lend to (v(+), 0) as t tends to infinity. Its optimal convergence rate is also obtained by using the Green function of the diffusion equation with constant coefficients. In the case of null-Neumann boundary condition on u, v(0, t) is conservative and v(0, t) equivalent to v(0)(0) by virtue of the first equation, so that v(x, t) is expected to tend to the diffusion wave (v) over bar(x, t) connecting v(0)(0) and u,. In fact the solution in (v, u)(x, t) is proved to tend to ((v) over bar(x, t), 0) In the special case v(0)(0)=v(+), the optimal convergence rate is also obtained. However, this is not known in the case v(0)(0) not equal v(+). (C) 1999 Academic Press.

  • Asymptotic behaviors of solutions to viscous conservation laws via L2-energy method

    1999年全国数学研究生暑期学校、復旦大学、上海    1999.07

  • Boundary effect on stationary viscous shock wave for scalar viscous conservation laws

    K.Nishihara

    International Conference on Applied Partial Differential Equations, Tongji Univ., 上海    1999.07

  • Asymptotics toward the planar rarefaction wave for viscous conservation law in two space dimensions

    M. Nishikawa, K. Nishihara

    Trans. Amer. Math. Soc./American Mathematical Society   352 ( 3 ) 1203 - 1215  1999

  • Asymptotic behavior of solutions to the p-system with linear damping

    Proc. of 23rd Sapporo Symposium on Partial Differential Equations    1998.07

  • Behaviors of solutions for the Burgers equation with boundary corresponding to rarefaction waves

    SIAM J. Math. Anal.   29;2  1998.03

  • Asymptotic behaviors of solutions for viscous conservation laws with boundary effect—System case—1998.3

    Special Program on Partial Differential Equations in Liu Bue Centre, City Univ. of Hong Kong    1998.03

  • Asymptotic behaviors of solutions for viscous conservation laws with boundary effect—Scalar case—1998.3

    Special Program on Partial Differential Equations in Liu Bue Centre, City Univ. of Hong Kong    1998.03

  • Global asymptotics toward the rarefaction wave for solutions of viscous p-system on the quarter plane

    Workshop“Qualitative Properties for Nonlinear Hyperbolic Operators. Degeneracies, Nonlocal Terms and Global Solvability”(於: Seiffen, Germany)    1998.03

  • Asymptotic Behaviour of Solutions to the Korteweg-de Vries-Burgers Equation

    Differential and Integral Equations/KHAYYAM Publishing   11;1  1998.01

  • 半直線上のViscous p</>-systemの解の希薄波への漸近、1997年10月(with 松村昭孝)

    日本数学会秋季総合分科会(東京大学)    1997.10

  • Asymptotic behavior of solutions of quasilinear hyperbolic equations with linear damping

    J. Differential Equations /Academic Press   137;2  1997.07

    DOI

  • Nonlinear stability of traveling waves for one-dimensional visco-elastic materials with non-convex nonlinearity

    Tokyo J. Math.   20;1  1997.06

  • Linear dampingを持つ準線形双曲型方程式の解の漸近挙動

    第4回応用解析研究会(於 熱海)    1997.02

  • Asymptotic behavior for scalar viscous conservation laws with boundary effect

    J.Differential Equations/Academic Press   133;2  1997.01

  • Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping

    J.Differential Equations/Academic Press   131;2  1996.11

    DOI

  • Dampingを持つ双曲型保存系の解のDiffusion Waveへの漸近の速さについて

    日本数学会秋季総合分科会(於 都立大学)    1996.09

  • Asymptotic behavior of solutions for scalar viscous conservation laws with boundary

    6th International Conference on Hyperbolic Problem Theory.Numerics.Applications(於 City Univ of Hong Kong)    1996.06

  • Asymptotics toward the planar rarefaction wave for viscous conservation laws in two space dimensions

    日本数学会年会(於 新潟大学)    1996.04

  • Behaviors of solutions of the Burgers equation with boundary corresponding to the rarefaction wave

    日本数学会年会(於 新潟大学)    1996.04

  • Behaviors of solutions of the Burgers equation with boundary corresponding to the viscous shock wave

    日本数学会年会(於 新潟大学)    1996.04

  • Asymptotic behavior of solutions for the Burgers equation with boundary conditions

    非線型偏微分方程式研究会(於中央大学)    1996.01

  • 境界条件付きの単独双曲型保存則の方程式の解の挙動

    微分方程式研究会(於早稲田大学)    1995.12

  • Stability of Traveling Waves with Degenerate Shock for System of One-Dimensional Viscoelastic Model

    J. Differential Equations/Academic Press   120;2  1995.10

  • On a Nonlinear Degenerate Integro-Differential Equation of Hyperbolic Type with a Strong Dissipation

    Adv. Math. Sci. Appl./Gakkotosho   5  1995

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Books and Other Publications

  • 非線形微分方程式の大域解---圧縮性粘性流の数学解析

    松村昭孝, 西原健二

    日本評論社  2004.07 ISBN: 4535783861

Research Projects

  • Integrated Study for Nonlinear Evolution Equations and Nonlinear Elliptic Equations

  • Time global structure of solutions of nonlinear conservation law with viscosity and relaxation

  • Diffusion phenomenon of solutions for the damped wave equation

  • Synthetic study of nonlinear evolution equation and its related topics

  • 双曲型性と放物型性との間に横たわる階層構造の解明

    科学研究費助成事業(早稲田大学)  科学研究費助成事業(挑戦的萌芽研究)

  • 消散型波動方程式の解の拡散現象と波動現象

    科学研究費助成事業(早稲田大学)  科学研究費助成事業(基盤研究(C))

  • 非線形放物型方程式系と関連する楕円型方程式系の研究

    科学研究費助成事業(早稲田大学)  科学研究費助成事業(基盤研究(C))

  • 非線形楕円型方程式とその周辺に関する研究

    科学研究費助成事業(早稲田大学)  科学研究費助成事業(基盤研究(C))

  • Study of nonlinear prabolic systems and related elliptic systems

  • Asymptotic Behavior of solutions to viscous hyperbolic conservation laws

  • Study on Nonlinear Evolution Equations and Nonlinear Elliptic Equations

  • Stability of nonlinear waves for hyperbolic conservation laws will viscosity

  • Large time behavior of solutions to equations for viscous gas over multi-dimensional half space

  • Research on global solutions in time and their asymptotic behaviors for viscous and relaxation models of conservation laws

  • Stability of nonlinear waves in viscous conservation system together with diffusion phenomena of solutions of damped wave equation

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Specific Research

  • 粘性的双曲型保存系の非線型波の安定性

    2001  

     View Summary

    Porous Media 中の圧縮性流は摩擦による効果が大きく、線型強制項をもつ連立双曲型の方程式系で表される。この系の初期値問題の解は、Darcy の法則から得られる放物型方程式の解--それは散逸波と呼ばれる非線型波の一つである--に漸近することが予想され、実際、Hsiao-Liu により、初めて示された。 本年度の研究では、その漸近の速さについて、散逸波の Location を与えられた初期値に対して一意に定めることにより、sharp な漸近の速さを得て、その論文は Proc. Royal Soc. Edinburgh, Section A に掲載が決まっている。方法は、エネルギー法とグリーン関数の方法を組み合せて得られる。しかしながら、この方法では、L^1 空間におけるノルムでの漸近は得られない。そこで、新たに、二階消散型波動方程式の解表示を詳しく解析して、L^1-ノルムにおいても sharp な漸近の速さを得て、その論文は作成準備中である。

  • 粘性的双曲型保存側の方程式の非線型波の安定性(Ⅲ)

    1998  

     View Summary

     今年度の研究課題のもと、まず、線型dampingを持つp-systemの、半直線上の初期値境界値問題の解の漸近挙動について考察した。Cauchy問題の解は、diffusion waveと呼ばれる波に漸近することが、Hsiao, Liu氏等の研究で初めて知られ、定数状態の周りでは、具体的なGreen関数の表示を用いた,報告者による最良の漸近のオーダーも得られている。本年の研究では、半空間における境界の効果を調べ、初期値境界値問題の解も、Cauchy問題と同様の漸近をすることが判った。定数状態の周りとなる場合は、やはりGreen関数の具体的表示を使って最良の漸近オーダーも得た。これは、T. Yang氏との共同研究としてまとめられ、J. Differential Equationsに掲載予定である。 一方、Cauchy問題で、定数状態の周りとならない場合は、変数係数の熱方程式のGreen関数を扱う必要があり、最近、Liu氏によって導入された近似Green関数の方法を用いて、最良の漸近オーダーが得られた。これは、W. Wang、T. Yang氏との共著論文として投稿中である。 更に、エントロピーも考慮したsystemに対して、定数状態の周りのCauchy問題の解の漸近挙動を考察し、西川雅堂氏との共著論文として作成準備中である。この場合は、空間変数のみに依る変数係数の熱方程式および線型強制項を持つ2階準線型波動方程式を考えねばならず、注意深いエネルギー評価と近似Green関数を組み合わせて、最適の漸近挙動を得た。

  • 粘性的双曲型保存則の方程式の非線型波の安定性(Ⅱ)

    1997  

     View Summary

    今年度の研究では、一次元粘性流の方程式系であるviscous p-systemに対する初期境界値問題の解の希薄波への漸近に関する成果を得た。右半空間において、右遠方の速度u+が、左側境界上における速度u-より大なるとき、対応するRiemannの問題の考察から、右方に進む希薄波(2-rarefaction wave)に漸近することが予想され、実際、それを証明した。 まず、u+-u-が小さく初期擾乱も小さいとき、希薄波の安定性を示した。証明は、通常の方法で、線型化された方程式に対するエネルギー法による。希薄波は、滑らかでないので、滑らかな近似を構成する必要があるが、作り方から、perturbationの境界値が零とすることが出来、Cauchy問題とほとんど同様の評価が可能となる。 更に、u+-u-の大きさに制限がなく、任意に与えられた初期擾乱に対しても、初期境界値問題の解が希薄波に漸近することが示された。これは、任意のデータに対する結果で、このような非線型の問題では最良といえるであろう。このときも希薄波の滑らかな近似を構成する必要があるが、上記と同じように作ることができない。従って、エネルギー評価をするとき境界からの未知の値が出て、ひとつひとつ丁寧に評価しなければならない。また、流体の比体積vが零にならないことを示すことも重要で、かなり注意深い評価を必要とする。また、ここでの方法はCauchy問題にも適用できて、松村―西原の結果(Commun. Math.Phys. 144(1992))も改良した。 これらの結果は、下記2つの論文に収められ、投稿中である。研究成果の発表T. Pan, H. Liu and K. Nishihara, Asymptotic stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas with boundary, to appear in Jpn J. Ind. Appl. Math.A. Matsumura and K. Nishihara, Global asymptotics toward the rarefaction wave for solutions of viscous p-system with boundary effect, to appear in Quart. Appl. Math.

  • 粘性的双曲型保存則の方程式の非線型波の安定性

    1996  

     View Summary

     粘性的双曲型保存則の方程式の持つ非線型波--粘性的衝撃波、希薄波--の安定性は、1960 年、Il'in-Oleinik により単独方程式の場合に示され、連立の場合には、1985 年、松村氏と共同で申請者はその部分的解答を得た(J. Goodman も独立に同様の結果を得た)。それを契機に、散逸波、接触不連続波も含めて、盛んに研究がなされてきた。 本研究では、Porous media 中の圧縮性流をモデルとする、Damping 付きの双曲型保存系の散逸波の安定性に関し、いくつかの結果を得、それらは、次の二つの論文にまとめられた: 1. Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping, J. Differential Equations 131 (1996), 171-188. 2. Asymptotic behavior of solutions of quasilinear hyperbolic equations with linear damping, J. Differential Equations 137(1997),384-395. 1. では、Hsiao-Liu によって示された散逸波の安定性の結果について、その漸近の速さを改良した。方法は、エネルギー法と熱核を利用した。得られた結果はほぼ最良と思われる。2. では、1. の方法を更に進め、Linear dampingを持つ二階準線型波動方程式の解の漸近形が散逸波によって表されることを示した。 また、1, 2 で考察された方法は、Damping 付きの Thermoelastic system に対しても適用可能と思われるが、考察の余地が残っており検討中である。

  • 粘性をもつ双曲保存系における非線形型波の安定性

    1995  

     View Summary

    (粘性をもつ)双曲型保存則の方程式は,基本的な非線型波として,(粘性的)衝撃波と希薄波をもつ。これらの非線型波の安定性が本研究の目的であり,特に,2つの非線型波-粘性的衝撃波と希薄波-の重ね合せが期待される場合の考察を目標としている。 さて,異なるタイプの波の重ね合せが期待される場合には,波と波の相互作用があるのでその考察が肝要である。それについての予想は共同研究者との間で得られているが証明には未だ相当の準備が必要と思われる。一つの準備として,それぞれの波と境界との相互作用を考察し,得られた結果をまとめたものが次の2つの論文である。:・Asymptotic Behaviour for Scalar Viscous Conservation Laws with Boundary Effect(Prof. T.-P. Liuと共著)・Behaviors of Solutions for the Burgers Equation with Boundary Corresponding to Rarefaction Waves(Prof. T.-P. Liu,Prof. A. Matsumuraと共著) 前者の論文では,粘性的衝撃波が境界にはいり込む場合,及び境界から離れていく場合に,単独Burgers方程式の解の挙動を考察した。後者の論文では,希薄波に対して前者と同様の考察を行った。この場合には,境界及び無限大における特性速度の正負によって,波が(1)境界に入り込んだり,(2)境界から離れたり,(3)境界近くの波は境界にはいり込み遠方では益々離れていったりする。(1)の場合,境界との相互作用によって実は波は粘性的衝撃波となる。(2)は希薄波である。その結果,(3)の場合は,希薄波と粘性的衝撃波の重ね合せが漸近状態となって,ある意味で目標に近い状態での安定性も得られた。しかし,境界がない場合には粘性的衝撃波のlocationの決定が本質的で,ここでは,境界がlocationを決定しており,その難しさは避けられている。本来の目標は今後の研究課題である。 方程式系に対する粘性的衝撃波の安定性は次の論文で得られている。: Nonlinear Stability of Travelling Waves for One-Dimensional Viscoelastic Materials with Non-ConvexNonlinearity (with Dr. Ming Mei)