To the 3D stationary Navier-Stokes equations in exterior domains, if the obstacle rotates slowly around the axis and moves also slowly along the same direction to the axis, then there exists a unique strong solution. In particular, we investigate the case when the obstacle moves with a constant speed and succeed to prove the energy inequality for any weak solution provided the external force is in the dual space of homogeneous Sobolev space with the first derivative in L^2. As an application, we can show the uniqueness of weak solutions under the smallness assumption on external forces. On the other hand, in the interior domain with multi-connected boundaries, if the inhomogeneous boundary data satisfies the Leray-Fujita inequality and if the stationary weak solution is close to the extended solenoidal vector field in L^3-norm, then it is asymptotically stable with an exponential convergence rate

We developed the method for studying geometric properties and asymptotic behavior of solutions of parabolic equations, and obtained the asymptotic behavior of hot spots and the optimal decay rates of the Lebesgue norms for the heat equation with a potential. Furthermore, we established a method of obtaining the higher order asymptotic expansions of the solutions behaving like the heat kernel. In addition, we study the location of the blow-up set for a semilinear heat equation by the profile of the solution just before the blow-up time. In particular, we gave a sufficient condition for no boundary blow-up

We investigate the local existence of strong solutions and their blow-up within a finite time in arbitrary dimensional domains. The life-span of local solutions is characterized in terms of the L^1 and L^p-norms of the given initial data. Simultaneously, it is clarified that the total mass and the second momentum of the initial data together with the coefficient of the system of equations have a great influence on the blow-up phenomena. As an application, we prove that the blow-up solution either exhibits a definite blow-up rate determined by p, or oscillates in L^1 with the larger amplitude than the absolute constant. Furthermore, in multi-connected domains, it is still an open question whether there does exist a solution of the stationary Navier-Stoeks equations with the inhomogeneous boundary data whose total flux is zero. The relation between the nonlinear structure of the equations and the topological invariance of the domain plays an important role for the solvability of this problem. We prove that if the harmonic part of solenoidal extensions of the given boundary data associated with the second Betti number of the domain is orthogonal to non-trivial solutions of the Euler equations, then there exists a solution for any viscosity constant. The relation between Leary's inequality and the topological type of the domain is also clarified

We introduced a function space on a domain of the Euclidean space and established its fundamental properties. The function space has several properties similar to the Hardy space on the whole Euclidean space introduced by Fefferman and Stein. In particular, we showed that the change of variables defined through diffeomorphisms, with certain properties, of the basic domains transforms the function space into another function space of the same kind. We used the function space to study classical orthogonal series. We investigated several other function spaces used in the field of time-frequency analysis and obtained several results concerning the operators acting in those spaces

The head investigator, T. Ogawa researched with one of the research collaborator K. Kato that the solution of the semi-linear dispersive equation has a very strong type of the smoothing effect called "analytic smoothing effect" under a certain condition for the initial data. This result says that from an initial data having a strong single singularity such as the Dirac delta measure, the solution for the Korteweb-de Vries equation is immediately going to smooth up to real analytic in both space and time variable. Similar effect can be shown for the solutions of the nonlinear Schroedinger equations and Benjamin-Ono equations.
Also with collaborators H. Kozono and Y. Taniuchi, Ogawa showed that the uniqueness and regularity criterion to the incompressible Navier-Stokes equations and Euler equations. Besides, it is also given that the solution to the harmonic heat flow is presented in terms of the Besov space. Those result is obtained by improving the critical type of the Sobolev inequalities in the Besov space. On the same time, the sharper version of the Beale-Kato-Majda type inequality involving the logarithmic term was obtained by using the Lizorkin-Triebel interpolation spaces.
For the equation appeared in the semiconductor devise simulation, the head organizer Ogawa showed with M. Kurokiba that the solution has a global strong solution in a weighted L-2 space and showed some conservation laws as well as the regularity. Besides, under a special threshold condition, the solution develops a singularity within a finite time.
It is also shown that the threshold is sharp for a positive solutions.
Co-researcher S. Kawashima investigated the asymptotic behavior of the solutions to a general elliptic-hyperbolic system including the equation for the radiation gas. The asymptotic behavior can be characterized by the linearized part of the system and it is presented by the usual heat kernel.
Co-researcher Y.Kagei researched with co-researcher T.Kobayashi about the asymptotic behavior of the solutions to the incompressible Navier-Stokes in the three dimensional half space. They studied on the stability of the constant density steady state for the equation and the showed the best possible decay order of the perturbed solution in the sense of L-2.
Co-researcher K. Ito studied about the intermediate surface diffusion equation and showed that the solution has the self interaction when the diffusion coefficients are going to very large.
Co-researcher N. Kita with T. Wada collaborates on the problem of the asymptotic expansion on the solution of the nonlinear Schroedinger equation when the time parameter goes infinity. They identified the second term of the asymptotic profile of the scattering solution when the nonlinearity has the threshold exponent of the long range interaction.

Though the mathematical research on the Navier-Stokes equations that describe the motion of incompressible fluid has a long history, the theory has not completed yet. In particular concerning the problems of the regularity and uniqueness of weak solutions, we have only partial answers. A weak solution satisfies equations only in some weak sense, and therefore it may have singular points. If a solution is smooth, then it preserves energy with respect to time variable. For some weak solutions we can show the non-increasing property of energy, but it is uncertain whether they preserve energy or not. It is called the "energy inequality." For weak solutions the integrability of time-derivative is unclear only from the definition. This is why we cannot show the preservation of energy. This fact suggests that time-derivative is a singular measure with respect to "time". Consequently we must consider the integral of this "singular" part to show the preservation of energy.We know that it is possible to estimate the decrease of energy from below by use of fractional time-derivative for the weak solution constructed by the method of discrete Morse flow. This is a new estimate. The purpose of this research is to study the possibility of such a refinement for any weak solution. For precise, we clarified the following facts. We consider weak solutions as functions which map to the space of square-integrable functions. Then the decrease of energy is related to the limit of the 1/2-time-difference of solutions in the sense of Nikol'skii. If we assume that the limit is zero with or without some speed, then we can show the energy identity with an additional term which compensates the decrease of energy. Furthermore without the assumption of the existence of limit, we can show the energy identity with another additional term of different expression. The difference of expression comes from that of topology of convergence of limit of time-difference

Navier-Stokes方程式の解の安定性の解析にはStokes作用素Aに加えて,変数係数の低階の微分作用素を含んだ項Bを摂動として処理しなければならない.外部問題の場合,よく知られた半群生成の摂動論は役に立たない.何故ならば,作用素A+Bのスペクトルの存在範囲をAのそれを不変にする様に摂動させなければならないからである.その際,関数空間の選択に注意を払う必要がある.定常解の存在と安定性の問題は斉次Sobolev空間における考察でひと段落したものの,3次元外部領域の場合はnet forceがゼロであるという不自然な条件は依然そのままであった.これを克服するためにはStokes作用素が全単射であり,かつスケール不変則を満たすような新たな関数空間を見い出さねばならなかった.その試みとして,まずFourier変換,特異積分作用素が使える全空間R^nにおいてMorrey空間を実補間した空間を導入し,Navier-Stokes方程式を解くことに成功した.これまでは複素補間を用いて、Navier-Stokes方程式の強解(古典解)を構成したが、Riesz-Thorinの不等式に代表される様に,複素補間理論はシャープな補間不等式の係数が得られる反面,両立対の空間は広がらない.このことは,すべてのL^γ(1<γ<∞)において線形化方程式(Stokes方程式)が可解である内部問題に関しては障害とならなかった.一方,実補間空間の利点は、両立対の空間からより広い空間が得られることであり,線形化方程式の可解性に制限のある外部問題に実補間空間理論を導入したことは,画期的な試みであった.応用として,Lorentz空間L^<p,q>(Ω)において外部定常解を構成し,更にnet forceの条件を仮定することなく,その安定性を示した.

1. Heat convection problem :
In order to investigate the global structure of the solution space of the nonlinear PDE's and to treat the global bifurcation curves in it, we worked on the analytical method combined with the computational analysis and computer assisted proof. We proposed criterions to prove the existence of solutions which correspond to parameter values as computer assited proof. Using the method we showed the existence of global bifurcation curves on which the roll-type solutions exist that correspond to large Rayleigh numbers.
In the case of 3-dimension we investigated numerically the pattern formation of roll-type, rectangle-typpe and hexagonaltype solutions and their stability, and we clarified the global bifurcation diagram which is not seen from the local bifurcation theory.
2. Taylor problem :
We considered the stability of Couette flow when the two cylinder rotate in the opposite directions. It is reduced to the eigenvalue problem for the system of ordinary differential equations and it can be treated by our computer assisted proof to see the exact critical Taylor number, at which the stationary or Hopf bifurcation occurs. The bifurcation point with multiplicity is one of our future subject.
3. The existence theorem for stationary solution of Navier-Stokes equation is proved by our numerical verification method at least for small Reynols number.
4. Dynamical systems :
We know that when the degeneracy of singular points of vector field increases, the behavior of dynamics becomes more complex and the global phenomena become more included. We investigated the singular point with codimension 3 and proved analytically that the hetero-clinic cycle bifurcates and also chaotic attractor does.
5. For the 3-dimensional exterior problem of stationary Navier-Stokes equation, we introduced a real interpolation of Morrey spaces to solve N-S equation and succeeded to construct the exterior stationary solution and to prove its stability without the unnatural zero net force conditions.

In a domain Ω⊂R^n, consider a weak solution u of the Navier-Stokes equations in the class u∈L^∞ (0, T ; L^n (Ω)). If lim sup_<t-t_*-0>‖u (t) ‖^n_n-‖u (t_*) ‖^n_n is small at each point of t_*∈ (0, T), then u is regular on Ω^^-× (0, T). As an application, we give a precise characterization of the singular time, i.e., we show that if a solution u of the Navier-Stokes equations is initially smooth and loses its regularity at some later time T_*<T, then either lim sup_<t-T_*-0>‖u (t) ‖_<L^n (Ω) >= +∞, or u (t) oscillates in L^n (Ω) around the weak limit w-lim_<t-T_*-0>u (t) with sufficiently large amplitude. Furthermore, we prove that every weak solution u of bounded variation on (0, T) with values in L^n (Ω) becomes regular.
Consider the nonstationary Navier-Stokes equations in Ω× (0, T), where Ω is a domain in R^3. We show that there is an absolute constant ε_0 such that every weak solution u with the property sup_<t∈ (a, b) >‖u (t) ‖^3_W (D) 【less than or equal】ε_0 is necessarily of class C^∞ in the space-time variables on any compact subset of D× (a, b), where D ⊂⊂Ω and 0<a<b<T.As an application, we prove that if the weak solution u behaves around (x_0, t_0) ∈Ω× (0, T) like u (x, t) =o (|x-x_0|^<-1>) as x→x_0 uniformly in t in some neighborhood of t_0, then (x_0, t_0) is a removable singularity of u.
Consider weak solutions w of the Navier-Stokes equations in Serrin's class
w∈L^α (0, ∞ ; L^q (Ω)) for 2/α + 3/q = 1 with 3<q【less than or equal】∞,
where Ω is a general unbounded domain in R^3. We shall show that although the inital and exteral disturbances from w are large, every perturbed flow u with the energy inequality converges asymptotically to w as
‖υ (t) -w (t) ‖_<L^2 (Ω) >→0, ‖▽υ(t) -▽w (t) ‖_<L^2 (Ω) >=O (t^<-1/2>) as t→∞.

The purpose of this project is to investigate the spectral and scattering theory for Schrodinger operators in general. Moreover, it is also intended to explore new area of problems in quantum physics and related topics. Quite a few reserch results has been obtained in the project, and only a selected results by the head investigator and collaborators are presented here.
1. By employing the theory of phase space tunneling, it is proved that the exponential decay rate of eigenfunctions for Schrodinger operator is larger in the semiclassical limit in the presence of constant magnetic field.
2. Semiclassical asymptotics of the scattrering is investigated. In particular, it is shown that the spectral shift function has a rapid jump (of the size 2π times integer) near each quantum resonance.
3. It is shown that the coefficients of the scattering matrix corresponding to the interaction between two nonintersecting energy surfaces decay exponentially in the semiclassical limit. A new method to analyze the phase space tunneling is developed and employed (joint work with A.Martinez, V.Sordoni).
4. The Lifshitz tail for the integrated density of states is proved for 2 dimensional discrete Schrodinger operators and continuous Schrodinger operators (arbitrary dimension) with Anderson-type random magnetic fields.
5. A new proof of the Wegner estimate based on the theory of the spectral shift function is developed. The Wegner estimate plays crucial role in the proof of Anderson localization for random Schrodinger operators (joint work with J.M.Combes, P.D.Hislop).

We had studied the following two subjects for the period of July, 1997-March, 2000.
We first studied the well-posedness of the Cauchy problem for the system of nonlinear wave equations with different propagation speeds. One of the most important problems in the field of partial differential equations is to look for the largest possible function space in which the wave equations with quadratic nonlinearity is well-posed. This problem is closely related to the Lorentz invariant for the wave equation. When we consider the system of nonlinear wave equations with different propagation speeds, the discrepancy of propagation speeds breaks the Lorentz symmetry. We classified the quadratic nonlinear terms from a point of view of the time local well-posedness.
Second, we studied the unique solvability of the Cauchy problem for the Korteweg-de Vries equation with stochastic forcing term. The stochastic forcing term is regarded as a nonsmooth perturbation from a mathematical point of view. Especially, in general, the inverse scattering method is inapplicable to the Korteweg-de Vries equation with forcing term. We investigated the time local existence of solution for a natural class of stochastic forces.

In this research, we study a behavior of solutions of Yang-Mills heat flow on compact 4-manifolds. We obtain result which is a type of "small data global existence".
It is well-known that there exists the lower bound of the energy of smooth connections over compact 4-manifolds. The bound is determined by the topological invariant of the pricipal bundle. In this research, we prove that if the energy of the initial data is close to the bound, then there exists a time-global smooth solution of the Yang-Mills heat flow.

Navier-Stokes方程式の解の安定性の解析にはStokes作用素Aに加えて、変数係数をもった低階の微分を含んだ項Bを摂動として処理しなければならない。外部問題の場合、よく知られた半群生成の摂動論は役に立たない。何故ならば、作用素A+Bのスペクトルの存在範囲をAのそれを不変にする様に摂動させなければならないからである。その際関数空間の選択に注意を払う必要がある。定常解の存在と安定性の問題は斉次Sobolev空間における考察でひと段落したものの、3次元外部領域の場合はnet forceがゼロであるという不自然な条件は依然そのままであった。これを克服するためにはStokes作用素が全単射であり、かつスケール不変則を満たすような新たな関数空間を見い出さねばならなかった。その試みとして、まずFourier変換、特異積分作用素が使える全空間R^NにおいてMorrey空間を実補間した空間を導入し、でNavier-Stokes方程式を解くことに成功した。これまでは複素補間を用いて、Navier-Stokes方程式の強解 (古典解)を構成したが、Riesz-Thorinの不等式に代表される様に、複素補間理論はシャープな補間不等式の係数が得られる反面、両立対の空間は広がらない。このことは、すべてのL^r(1<r<∞)において線形化方程式 (Stokes方程式)が可解である内部問題に関しては障害とならなかった。一方、実補間空間の利点は、両立対の空間からより広い空間が得られることであり、線形化方程式の可解性に制限のある外部問題に実補間空間理論を導入したことは、画期的な進歩であった。応用として、Lorentz空間P^<p,q>(Ω)において外部定常解を購成し、更にnet forceの条件を仮定することなく、その安定性を示した。

1. Concerning a system of nonlinear dispersive equations arose from the water wave theory, T.Ogawa discovered a different kind of a smoothing effect mainly due to the special structures of nonlinear coupling and established the local Well-posedness of the solution in a weaker initial data.
2. H.Kozono studied the uniqueness problem for the Leray -Hopff weak solution to the Navier-Stokes equation and showed the uniqueness holds for the critical case, C(O, T ; L^n), suppose that the solution satisfies the small gap condition.
3. M.Kawashita considered unique existence of the strong solutions of the Cauchy prob- lems of the compressible Navier-Stokes equations. These equations are well known as explaining motions of fluid that density may change in time and space variables.
4. K.Kato worked with Dr. P.Pipolo about the solitary wave solutions to general- ized Kadomtsev-Petviashvili equations (KP equations) and proved that solutions are real analytic. Also in a joint work with N.Hayashi and P.Naumkin he studies that there exist scattering states to small initial data for some nonlinear Schr_dinger equations and Hartree equations by using some class of Gevrey functions.
5. S.Kawashima proved the existence and asymptotic stability of shock waves for the simplest model system of a radiating gas. Also, we showed the existence of global solutions to a class of hyperbolic-elliptic coupled systems and obtained the decay estimate of the solutions.
6. Y.Kagei introduced a new approximation to the Oberbeck-Boussinesq equation and showed the existence and uniqueness of solution. Also the stability is discussed.

The purpose of this study is to develop analytical and statistical theories of turbulence that are practically applicable not only to isotropic, but also to anisotropic turbulence, and to verify them by numerical simulations.
The results of the study include the followings :
(1) An analytical and statistical theory is developed for turbulent diffusion in strongly stratified turbulence. It is based on a linearized approximation (Rapid Distortion Theory) and Corrsin's conjecture, and clarifies an mechanism of suppression of vertical turbulent diffusion by strong stratification. It is also shown that an anomalous turbulent diffusion that could not be explained by existing turbulence models, may occur in turbulence with a mean flow of simple shearing motion. These theoretical results were confirmed by numerical simulations.
(2) By applying the Lagrangian Renormalized Approximation (LRA) , it is shown that a new kind of energy spectrum in two-dimensional turbulence may exist, which is not explained by the theory of Kraichnan-Batchelor-Leith (KBL) , but still exhibits the k^3 spectrum as predicted by KBL.By constructing a Lagrangian statistical theory for the beta-plane turbulence, which is a well known model for the fluid motion on a rotating sphere such as a planet, an analysis is made of the phase shift due to the interaction between waves and turbulence. The theoretical results were shown to be in good agreement with direct numerical simulations (DNS).
(3) A new computational method is proposed for efficiently estimating Lagrangian two-time correlations that play the key roles in the study of mass and heat transfer in turbulence. The method uses Taylor expansions in powers of time and corresponding Pade approximations. It is shown thet the method works well and its results are in good agreement with DNS, not only for single but also two-particle correlation, and not only for isotropic but also anisotropic axisymmetric turbulence.

3次元ユークリッド空間上のヤング・ミルズ・ヒッグス場の流れの方程式に関しても考察を行なった.この研究に関しては,以下のような結果を得た.3次元ユークリッド空間上のヤング・ミルズ・ヒッグス場の流れの方程式の滑らかな解の正則性に関する指標として,ユークリッド空間の無限遠点として捉えられる2次元球面上のある種の積分が小さい限り,その解は滑らかに延長できる.この性質は,コンパクト多様体上の非線形放物型方程式では良く知られている性質であるが,コンパクトでない空間上の方程式に関しては,全く新しいタイプの結果で,無限遠点へのエネルギーの集中という現象を観察することができた.また,3次元ユークリッド空間上のヤング・ミルズ・ヒッグス場の流れの方程式の解の爆発点における漸近的な挙動もほぼ観察できることがわかった.さらに,流れの方程式の爆発時間におけるエネルギーの挙動を調べ,流れの方程式が時間大域的な弱会を持つことを証明した.
以上の研究は慶応義塾大学理工学部の前田吉昭氏と名古屋大学多元数理科学研究科の小薗英雄氏との共同研究である.

The purpose of this project is to study Gevrey property and asymptotic analysis for divergent power series solutions of analytic partial differential equations.
The first result is that we established the Toeplitz operator method in analytic partial differential equations which enables us to give a precise notion of Fredholmness in the Goursat problem in various Gevrey spaces which has not been studied in explicite way in early studies. We proved further that an index formula for ordinary differnetial operator on Gevrey space is nothing but the geometrical index fornmula for a Toeplitz sumbol associated with the Gevrey filtration n the ring of ordinary differential operators.
The second result is that we gave a necessary and sufficient condition for the Borel summability for divergent power series solution of the Cauchy problem of the heat equation, and we proved that the Borel sum is just expressed by an integral with the heat kernel. The condition for the C data we obtained is the well known condition for the uniqueness of solutions of the Cauchy problem. In proving this we made clear that the problem of Borel summability in partial differential equations is not local property of solutions, whereas the notion of the Borel summability is only local one. We also made clear that this problem provides a new kind of problems in partial differential equations.

流体運動に伴うさまざまな不連続面のなかで、流体力学的に最も基礎的なものの一つに渦面と呼ばれる接線速度の不連続面がある。これまで、2次元的な渦面のダイナミックスについては多くの研究がなされており、初期に解析的な形を持った渦面が自発的にその解析性を失う過程などが解明されてきた。しかしながら、3次元的ダイナミックスについては、現実の流れは一般に3次元的であるにも関わらず、ほとんど未解明であった。本研究では、2次元における渦面の運動を支配するBirkoff-Rottの式を3次元化の場合に一般化した表現を用いて、初期に周期的な撹乱を与えられた渦面の非線形ダイナミックスの解析および数値計算を行った。その結果、渦度が集中する領域の時間および空間的特異性を解析および数値的に明らかにすることができた。とくに、2次元の場合には直線的でしかあり得ないその領域の形状が、3次元の場合には蛇行し、その蛇行が簡単な三角関数によって近似されることが分かった。
現実の流体中では、しばしば、速度だけでなく密度や電導度などの物性値も不連続となる界面が現れる。このような界面の運動には渦面のそれと共通点が多い。そのような界面の典型例としては、一様電場中の電導度の違う2種の流体の界面があり、雷雲中の帯電液滴の変形あるいは雷の発生と関連してG.I.Taylorらによって実験的研究がなされてきた。本研究では簡単のため運動は2次元的であるとしてこのような界面の運動を解析した。また、そのための等角写像および高速フーリエ変換を用いる効率的な数値解析法を開発した。その結果、界面がある代数べきで記述できる突起(特異性)を自発的に形成することが分かった。

Navicr-Stokcs方程式の外部問題は、内部問題と比較して、単に取り扱いがより複雑であるというとには留まらず、解の無限遠方での漸近挙動の与え方により、方程式の適切性が著しく異なるので、大変興味深い結果が得られた。「境界で静止しかつ無限遠方で一定の速度で動くようなStokcs方程式の解は存在しない」という有名なStokcsのパラドックスはその典型である。本研究では、このパラドックスがStokcs作用素A^1を一階の偏導関数がr-乗可積分(∇u∈L´)であるhomogeneous Sobolev空間において考察し、その核が自明なものに限るか?という問題と同値であることに着目し、2次元だけではなく多次元においても同様なパラドックスが成り立つことを証明した。この事実はNavicr-Stokcs方程式の外部問題を線形Stokcs方程式の摂動として捉えることの限界を示している。実際、A^1が全単射である必要十分条件は3次元外部領域の場合、3/2<r<3であることは特異な現象を生じることが解明された。すなわち、∇u∈L^<3/2>なる解が存在するための必要十分条件は領域Ωの境界Γによるdrag forccがゼロ“∫r Dcf u・v dS=O"である。ここにDcf uは速度場uによるdcformation tcnsorであり、vは境界Γの単位法線ベクトルである。一方、Navier-Stokcs方程式はスケール則があり、上記の∇u∈L^<3/2>なるクラスはスケール変換で不変な関数空間である。従ってこの空間で解を求めることは重要である。いささか物理的には不自然であるが、このクラスの安定性を示すことに成功した。その際、代表者による関数空間の以前の研究が大きな役割を果たした。

本研究ではナビエ・ストークス方程式に従う流体の運動を渦運動に着目して解析的及び数値的に調べた。
まず、最も簡単で基本的渦面、すなわち平面状の渦面の運動についてその非線形の時間発展を解析した。この渦面は微小攪乱に対して不安定(Kelvin-Helmholtz不安定)であり、2次元の場合についてはこれまで多くの研究がなされてきた。本研究では、Mooreによってなされた非線形解析を3次元の場合に拡張して、小さいけれど有限の大きさの攪乱の3次元的時間発展を解析した。その結果、初期に滑らかな渦面が非線形ダイナミックスによって自発的に滑らかさを失い、2次元では存在しない渦の伸展効果に起因する、3次元固有の特異性を生じることを見いだし、その特異性の性質を明らかにした。大きな攪乱の時間発展については、その解析的取扱いは現在困難であるので、数値的に調べた。そのために、まず2重周期(空間について2方向への周期を持つ)グリーン関数の効率的計算法を開発し、それを用いて3次元的渦分布の性質を明らかにした。
さらに、渦面の形が平面状以外の場合についても、ラグランジュ的視点から一般的に安定性解析が行なえる新しい方法を開発した。また、一般的な流れの場に置かれた渦糸の定常な形についての新しい知見を得た。
その他、線形化作用素のスペクトル解析によって、非有界領域における非圧縮生粘性流体のエネルギーが時間的に減衰することを示し、その臨界指数を求めた。また、同方法と摂動法により漸近安定である外部定常解のクラスを提唱し、抵抗との関係を解明した。

Jonesの指数理論は因子環、部分因子環の対に付随した細かい構造の研究を可能にした。対の研究に表れる様々な不変量のうち最も重要な物はおそらく(2種の)相対可換子環のタワーであろう。これらは2種類のグラフ(principalグラフ及びdual principalグラフと呼ばれる)により記述される。有限群が因子環に(外部的に)作用している時、群と部分群から生じる接合積を考える事により得られる因子環、部分因子環の対に対するprincipal及びdual principalグラフの計算のアルゴリズムを得る事が出来た。このアルゴリズムはMackey流の群、部分群の既約表現の間のinduction-restrictionグラフとして記述される。
因子環、部分因子環の対に対する(外部的な)自己同型の研究を行った。(有限な)群Gが対M〓Nに作用している時、接合積の対M×G〓N×Gは元の対M〓Nとどの位異なるかという問題は興味深い。たとえばこれら2つの対はいつ違うグラフを持つだろうか。この種の問題を考えるには、作用が普通の意味でより強い意味で外部的であるかどうかという事が問題となる。作用が強い意味で外部的となる為の完全な特徴付けを得る事が出来た。Longoにより導入されたsector理論のおもしろい応用であり、sector理論の重要性がますます明らかになった。
上の特徴付けを使いながらsetorのfusion ruleを調べる事により様々な因子環、部分因子環の対に対するグラフが計算可能となった。III型部分因子環の研究への様々な応用がこれからの研究課題であると思われる。更に各分担者も独自の研究を発展させた事を付け加えておく。

Hyperfunctionの理論(佐藤理論)は、佐藤幹夫先生によって創始された代数的手法によって大きく発展して来た。我々の目的はまずこの理論を純粋に解析的な(微積分の)手法によって再構成し、新しい応用を試みることである。最も基礎となるのは全てのhyperfunctionを熱方程式の解の初期値とみることである。我々はこれによってhyperfunctionの新しい特微づけを得た。すなわち熱方程式の解とhyperfunctionを1:1に対応させることができる。この基本定理をもとにして、次々と理論を展開させていくことができる。第1は偏微分方程式の解の存在及至非存在の問題への応用である。Schwartz超関数からhyperfunctionの空間、更にそれよりも広い空間での可解性の問題などかなり統一的に取り扱うことが可能になって来た。第2は解の局所的性質の解析、いわゆるMicro-local analysisへの熱方程式の基本解からのapproachであるがこれもかなり見通しがでて来た。現在は主としてこの2つの問題について新しい結果を得つつありいくつかの論文として発表する予定である。

It is well known that there are close relations between the regularity of solutions and the geometric invariant for nonlinear wave equations. Especially, the null condition introduced by Klainerman and Christodoulou often plays an important role in the case of relativistic nonlinear wave equations. From this point of view, in the acadimic years of 2000 and 2001, we studied the global existence of solutions for the Cauchy problem of the Dirac-Proca equations and the Maxwell-Higgs equations. We first showed that the Proca equation has a null condition structure, which led to the global existence of solution for small and smooth initial data. We next discovered that the Maxwell-Higgs equations generically have a null condition structure, which enabled us to show the global existence of solution for small and smooth initial dataIn the academic year of 2002, we studied the well-posedness of the Cauchy problem for the modified KdV equation in a weak class. It is known that when s< 1/2, the solution map is not in C^2, while it is in C^∞ for s > 1/2. We studied what kind of structure for the modified KdV equation breaks down the well-posedness in a weak clas

研究実績は以下のとおり.研究代表者の小川は研究分担者の加藤と共に,非線型分散系の方程式についてBenjamin-Ono方程式の初期値問題の解がその初期値に一点のみSobolev空間H^S(s>3/2)程度の特異点を持つ場合に、対応する弱解が時間が立てば、時間、空間両方向につき実解析的となるsmoothing effectを持つことを示した。その過程で、無限連立のBenjamin-Ono型連立系の時間局所適切性を証明した。またKdV方程式とBenjamin-Ono方程式の中間的な効果を表すBenjaminのoriginal方程式に関して、その初期値問題が負の指数をも許すSobolev空間H^s(R)(s>-3/4)で時間局所的に適切となることを示した。さらに、谷内と共同で臨界型の対数形Sobolevの不等式(Brezis-Gallouetの不等式)を斉次,非斉次Besov空間に拡張した。またそれを用いて非圧縮性Navier-Stokes方程式、Euler方程式、及び球面上への調和写像流の解の正則延長のための十分条件をこれまでに知られているSerrin型の条件よりも拡張した。これらの結果を元に、韓国ソウル国立大学数学科のD-H. Chae氏との共同研究をめざす、研究交流を行った分担者の川島は一般の双曲・楕円型連立系のある種の特異極限を論じた。この特異極限で双曲・楕円型連立系の解が対応する双曲・放物型連立系の解に収束することを、その収束の速さも込めて証明した。また、輻射気体の方程式系ではこの特異極限は、Boltzmann数とBouguer数の積を一定にしたままBoltzmann数を零に近づける極限に対応していることを明らかにした。分担者の隠居はVlasov-Poisson-Fokker-Planck方程式(VPFP方程式)の初期値問題に対して,重み付きソボレフ空間において不変多様体を構成し、解の時間無限大での漸近形を導出した

This objective of this project is to pursue the behavior of solutions of nonlinear partial differential equations of parabolic type.In collaboration with Wei-Ming Ni (University of Minnesota) and Kanako Suzuki (Tohoku University), Takagi studied the behavior of solutions of a reaction-diffusion system of activator-inhibitor type proposed by Gierer and Meinhardt and obtained the following results : (i)In the case where the initial data are constant functions, there exist solutions blowing up in finite time if the activator activates the its production stronger than that of the inhibitor. There are two types of blow-up solutions. Either (a) only the activator blows up or (b) both the activator and the inhibitor blow up. In the former case, we can choose the initial value so that the inhibitor converges to any specified positive number. (ii)In the case where the equation for the activator does not contain the source term, no solution blows up in finite time if the activator produces the inhibitor more than itself. Moreover, in this case the collapse of patterns can occur. Here, by the collapse of patterns we mean that the solution converges to the origin as the time variable tends to infinity.Nishiura studied scattering phenomena of pulse solutions and spot solutions. He found that various input/output relationships can be formed depending on the local dynamics in the neighborhood of unstable steady-states or periodic solutions and on the location of solution orbits.Yanagida considered a certain quasilinear parabolic equation and showed that the solution is either (a) globally increasing, (b) a traveling wave, or (c) extinct in finite time, depending on the initial data. The asymptotic behavior of the solution is also investigated.Kozono proved that in three dimensional exterior domains one can construct weak solutions of the Navier-Stokes equations which satisfy the strong energy inequality for all square-integrable initial data

1.Navier-Stokes方程式に関するミレニアム問題の解説3次元Navier-Stokes方程式の大きな初期値に対する時間大域的滑らかな解の存在問題は,2000年にクレイ研究所からミレニアムにおける数学の7つの難題のひとつとして提唱された.本研究では,Lerayによる時間大域的な弱解の存在からはじめて,Serrinによる一意正則な弱解のクラスL^s(0,T;L^r(R^n)),2/s+n/γ【less than or equal】-1を中心に総合的な解説を行った.とくに,スケール不変則に対する藤田-加藤の原理を紹介し,同方程式の時間局所的な強解C([0,T);L^n(R^n))の果たした重要性を指摘した.解の特異点集合のHausdorff次元の評価,除去可能孤立特異点の特徴付け,後進自己相似解による爆発解の非存在についても触れた.調和解析学における最近の研究成果が,Navier-Stokes方程式の考察に寄与した例を2,3挙げ,今後の研究の指針を与えた.2.Navier-Stokes方程式の適切性と流体力学との関連流体力学サイドにおいては,今日,計算機能力の飛躍的な進歩に伴ってNavier-Stokes方程式の解を数値実験によって求め,乱流をも含む様々な流れの場を矛盾なく説明しているようである.一方,コンピューターを用いる以前に,まずは解析計算によって解の性質を調べようと試みる古典的な純粋数学の立場もある.本研究では,Navier-Stokes方程式の数学サイドから研究を紹介し,ミレニアム問題を中心とした同方程式に関する課題を解説した.とくに,「乱流の発生が解の正則性の崩壊と対応している」との数学者の見解と,多くの流体力学者によって指示されている「乱流の発生にはの解の特異性の議論は必要ない」との知見との比較を行った.渦度が有限でとどまる限り,解の正則性が保証されることに注目し,乱流発生が渦度の挙動と密接な関係にあることを偏微分方程式の適切性に関する研究から解き明かした.3.Navier-Stokes方程式の軟解とエネルギー等式Navier-Stokes方程式から導かれる積分方程式の解を"軟解"(mild solution)という.Katoは,初期値α∈L^n(R^n)であれば,あるT>0とC([0,T);L^n(R^n))に属する一意的な軟解uが存在することを示した.本研究では,L^2(R^n)∩L^n(R^n)に属する初期値をもつ"すべての軟解"uはLeray-HopfクラスL^∞(0,T;L^2(R^n))∩L^2(0,T;H^1(R^n))に属し,かつエネルギー等式を満たすことを証明した

(1)古典的直交級数について.実数直線の開区間上の重み付きハーディ空間において,ヤコビ級数に対する移植定理と乗子定理を示した.これは,重み付きのL^P空間におけるMuckenhouptの結果を,重み付きハーディ空間にまで拡張したものである.ヤコビ級数に対して,古典的な場合を含むハーディの不等式を示した.ハンケル変換に関する移植作用素がハーディ空間において有界であることを示した.(2)重み付きHardy空間の函数論的な特徴付け.実数直線の開区間上の重み付きハーディ空間について,特殊の場合に、正則関数のなす古典的なハーディ空間に対するBurkholder-Gundy-Silversteinの定理と同様の定理が成り立つことを,B.MuckenhouptとE.M.Steinが超球多項式による関数展開に関する考察から導入した一般化正則関数を用いて,示した.(3) Littlewood-Paley関数とMarcinkiewicz積分について.積分核に対する弱い仮定の下で,これらが,重み付きの空間やCampanato空間での評価を持つことを示した.(4)関数方程式への調和解析と実解析の応用.Waveletを応用して,Sobolev-Lieb-Thirringの一般化を示し,Schrodinger作用素の負の固有値の精密な評価を得た.Navier-Stokes方程式の研究に種々の関数空間や不等式などを応用し,Besov空間における解の性質の解明,弱解の内部正則性,孤立特異点の除去可能性の特徴付け,などの結果を得た.また,斉次Triebel-Lizorkin空間における双線形不等式を確立し,その応用としてNavier-Stokes方程式の局所古典解の延長可能性を論じた

(1)Heat Convection Pronlem : To extend the bifurcation curves obtained by the local bifurcation theory into the analytically unknown region in the solution space, to investigate the change of stability of the solution on the extended bifurcation curves and to know the global bifurcation structure, we use new computer assisted analysis for Boussinesq equation. Especially, we showed by computer assisted proofs the existence of extended bifurcation curves of the roll-type solutions. We also formulate a method to determine the point of secondary bifurcation on the extended bifurcation curves.(2)The cavity flows of Navier-Stokes equation are proved to exist by a revised numerical verification method for the higher Reynolds number. We reformulated the Newton method for the fixed point equation in the infinite dimensional space.(3)The blow-up of the solution of Navier-Stokes equation is proved to be characterized by the two components of vorticity, which means that its three components are not necessary to protect the blow-up.(4)Forced nonlinear wave equations are investigated by the Newton method in the infinite dimensional Banach space. The inverse operator of linearized equation at the approximate (constructed by computers) solution can be approximated in the norm by a pseudo diagonal operator.(5)In the Lorenz model equation we proved the existence of singularly degenerate heteroclinic cycle, which is an invariant set. We suppose that it will give the chaotic attractor by a perturbation

1. Constructions of very weak solutions of the Navier-Stokes equations in exterior domains.We show the unique existence of local very weak solutions to the prescribed non-homogeneous boundary data which belong to the larger class than the usual trace class. Our solutions satisfy the Serrin condition which implies the scaling invariant class.2. New regularity criterion on weak solutions of the Navier-Stokes equations.We prove that every turbulent solution which is α-Hoelder continuous in the kinetic energy in the time interval with α>1/2 necessarily regular.3. Helmholtz-Weyl de composition in unbounded domains with non-compact boundaries of uniformly C^2-class.Despite of a counter example of valiclity of the Helmholtz-Weyl decomposition in L^r, we introduce the space of sum and intersection of L^r and prove the Helmholtz-Weyl decomposition in such spaces. As an application, we can define the Stokes operator

We studied the location of the blow-up set for the solutions for a semilinear heat equation with large diffusion, under the homogeneous Neumann boundary condition, in a bounded smooth domain of the Euclidean space. This was a joint work with Professors Noriko Mizoguchi and Hiroki Yagisita. We proved that, if the diffusive coefficient is sufficiently large, for almost all initial data, the solution blows-up in a finite time only near the maximum points of the projection of the initial data onto the second Neumann eigenspace. This is the first result that explains the relation between the eigenfunctions and the location of the blow-up set.On the other hand, we studied the movement of the maximum points (hot spots) of the solutions of the heat equations. In particular, we considered the solution for the Cauchy-Neumann problem and the Cauchy-Dirichlet problem to the heat equation in the exterior domain of a ball. This exterior domain is very simple, but it is difficult to study the movement of hot spots. By using harmonic functions, we obtained some good asymptotic behavior of the hot spots as the time tends to infinity. After that, we studied the decay rate of derivatives of the solution and the movement of hot spots for the solution of the heat equation, with Professor Yoshitsugu Kabeya. By this study, we can understand the mechanism how to decide the decay rate of the derivatives of the solutions and the movement of hot spots

In what follows, the main research results of this project are described.In the academic years of 2003-4, we studied the time local well-posedness of the Cauchy problem for the modified KdV equation with periodic boundary condition. In 1993, Bourgain proved that this Cauchy problem is time locally well-posed in $H^s$, s ≧1/2. Furthermore, in 1996, Bourgain also proved that the solution map fails to belong to $C^3$ in $H^s$, s<1/2, though it is analytic in $H^s$, s≧ 1/2. Takaoka and Tsutsumi made a close investigation into the problem of what is the difference between the cases $H^s$, s ≧1/2 and $H^s$, s<1/2. It was showed that the Cauchy problem is still locally well-posed in $H^s$, s>1/3, but the solution map is not uniformly continuous because of the occurrence of nonlinear oscillation.In the academic year of 2005, Tsutsumi studied the asymptotic behavior of solution for the quadratic nonlinear Schrodinger equation in two space dimensions with Akihiro Shimomura. In two space dimensions, the quadratic nonlinearity is a boader between the short renge case and the long range case and the quadratic nonlinearity has a special interest from a viewpoint of nonlinear scattering theory. It was showed that for a nonlinearity of squared modulous of the unkown, the solution does not approach a free solution, while it is already known that the rest of other quadratic nonlinearities belong to the shourt range interaction. casse.In the academic year of 2006, Tsutsumi studied the unconditional uniqueness of solution for the Cauchy problem of the nonlinear Schrodinger equation. When the solution is constructed, one usually impose a condition that the solution belongs to an auxiliary space associated with the Stricahrtz estimate. The unconditional uniqueness means that the solution is unique even though it is not in this kind of auxiliary space. It was proved that the unconditional uniqueness holds for the solution in the critical Sobolev space associated with the scaling invariance of nonlinear SchrOdinger equations. This is a substantial improvement over the results by Furioli and Terraneo

The main researcher, Prof.Ogawa obtained the following results. He researched for the Sobolev type inequality of the critical type, especially for the real interpolation spaces such as Besov and Triebel-Lirzorkin spaces and generalized it for the abstract Besov and Lorentz space. Those inqualities involving the logarithmic interpolation order can be applied for the regularity and uniqueness criterion of the seimilinear partial differential equation. In a series of collaboration with the research colabolators, he shows that the reguarlity and uniquness criterion for the weak solution of the 3 dimensional Navier-Stokes equations and break down condition for the Euer equation. In a similar method, he also showed the regularity criterion for the smooth solution of the 2 dimensional harmonic heat flow into a sphere. In particular, for the weak solution of the harmonic heat flow, the similar regularity criterion is also holds. The result is obtained by establishing the "monotonicity formula" for the mean oscillation of the energy density of the solutions.He also consider the asymptotic behavior of the solution for the semi-lineear parabolic equation of the non-local type. Those system appeared in a various Physical scaling such as semi-conductor simulation model, Chemotaxis model and the birth of star in Astronomy. The system is involving Poisson equation as the field generated by the dencity of the charge or mucous ameba and the non-local effect is essential for the analysis of the solution. He particulariy investigated to the critical situation, 2-dimensional case, and showed that there exists a time local solution in the critical Hardy space, time global solution upto the threshold initial density and finite time blow-up for the system of forcusing drift-diffusion case. Besides, the asymootitic behavior of the solution for small data is characterized by the heat kernel. Moreover if the field equation is purterbed in a certain nonlinear way, then there exist two solutions for the same initial data in a radially symmetric case.He also studied for the asymptotic behavior of the solution for the semi-linear damped wave equation in whole and half spaces and exterior domains and show the small solution is going to be decomposed into the solutions of the linear heat equation, some combination of linear wave equation with nonlinear effect. This was shown for 1 and 3 dimensional cases before, however the mothod there could not be applicable for the 2dimensional case

Head investigator, F. Takahashi, in cooperation with other investigators, has made studies on the construction, asymptotic behavior, and blow-up analysis of solutions to various nonlinear elliptic equations arising from Gauge theories which were proposed by physicists.Also we have studied the long time asymptotics of time global solutions, or blow up behavior of time local solutions to heat flows associated with the above nonlinear elliptic equations. Such nonlinear elliptic equations have variational structures, and have strong relations to the critical inequalities such as Sobolev, or Trudinger-Moser inequality on compact manifolds or on bounded domains in Euclidean spaces. They also have quantized blow-up mechanisms and exhibit mass-concentration phenomena commonly.In the former half of the term of our project, we have established the existence of solutions to some mean filed equations which come from the statistical mechanics of many vortices with a neutral orientation in a perfect fluid. Our study has become a trigger of other studies of the equilibrium mean field equations, and now, this is one of the most active area in the fields.In the latter half, head investigator has begun to study the blow up analysis and some qualitative properties of blowing-up solutions to nonlinear elliptic equations with the critical Sobolev exponents. These studies lead to the current research project of my own.In summary, we have clarified the relations between various properties of blowing-up solutions and those of singular limits, pictured typical mass or energy quantization phenomena, and established many analytical tools which will be useful to the future studies of this kind, through our research project

1. 時間方向にLorentz 空間を採用した最大正則性定理によるNavier-Stoes方程式の強解最大正則性定理において，基礎となるBanach空間として斉次Besov空間$B^s_{p, γ}と取り，時間区(0, T) における可積分空間としてはLorentz 空間L^{α,q}(0, T) を採用した．すなわち，L^{α, q}(0,T; B^s_{p, γ})なるBochner時空間の最大正則定理の基礎空間として，熱方程式初期値問題を考察した．初期値属する斉次Besov空間 B^k_{r, q} としては，k= 2+n/r -(2/α+ n/p - s), n/p &lt;n/r &lt;α/2 + n/pなる関係式が自然であることを証明した．2. 境界が時間に依存する外部領域におけるStokes 方程式に関する最大正則性定理とそのNavier-Stokes方程式への応用3次元空間内のおけるコンパクトな曲面が時間に依存して動く時，その外部領域であるを非柱状時空間領域において，Stokes方程式に対する時間大域的なL^p-最大正則性定理を証明した．ただし，$1&lt; q &lt; 3/2$ である．応用として，非柱状時空間領域がある固定された柱状領域に十分近いとき，小さなデータに対するNavier-Stokes 方程式の古典解の一意的存在を証明した．３．尺度不変な斉次Besov空間における定常Navier-Stokes 方程式の解の存在と正則性ｎ次元空間において，与えられた外力が斉次ベゾフ空間 B^{-3+ n/p}_{p, q}$ で十分小さければ，B^{-1+n/p}_{p,q}に属する定常Navier-Stokes 方程式の解が一意的に存在することを証明した．ただし，$1 &lt;p &lt; n,&nbsp; 1 &lt; q&lt; ∞ である．応用として，定常Navier-Stokes 方程式に対する自己相似解が得られる．

Caffarelli-Kohn-Nirenbergによって提唱されたNavier-Stoes 方程式の適切な弱解を，より広い局所的なエネルギー不等式を満たすものに拡張し，一般化された適切な弱解と名付け，無限遠方で弱い増大度を仮定するならば．初期値のエネルギー有限性が，時間発展後も運動エネルギーとその散逸が有限に留まること証明した．更にエネルギー等式が成り立たしめ得ることも示した．特に2 次元平面においては，一般の非有界領域においても，渦度の遠方での減衰度と，領域の境界におけるある種の積分量の符号を仮定するならば，時間発展後も解の渦とその一階偏導関数は領域全体で自乗可積分であることを証明示した．応用として，Navier-Stoes 方程式に対するLiouvile型定理を確立した．&nbsp;

(i) 一般領域におけるStoeks 作用素の最大正則性Stokes 作用素のq-乗可積分空間理論はq=2 の場合を除き、一般の領域では定義が出来ないことが知られている。そこで、反例が構成されている非コンパクトな境界をもつn 次元空間内の非有界領域を取り扱った。通常のq-乗可積分空間に代わるものとして、２乗可積分空間とq-乗可積分指数の和および共通部分からなる関数空間を導入した。これらの関数空間はともに、関数自身の無限遠方では減衰の速度が２乗可積分関数と同程度であることを要請したものである。その結果、領域の境界が一様にC1-級であれば、非コンパクト領域においてもStokes 作用素はこれらの関数空間において定義可能であり、正則半群を生成するとともに最大正則性定理を満たすことが明らかにされた。(ii) Navier-Stokes 方程式の弱解の正則性に関する新たな指標３次元有界領域におけるNavier-Stokes 方程式の弱解で強エネルギー不等式満たすクラスの正則性を考察した。従来はSerrin によって提唱された時空間におけるスケール不変な可積分空間において正則性の指標が確立されていたが、本研究では運動エネルギーとエネルギー散逸量に着目した。すなわち、前者に対しては指数が1/2 より大きな時間変数のヘルダー連続関数であり、また後者に対しては積分量の時間爆発レートが-1/2 より遅ければ、弱解が滑らかであることを証明した。これら２つの指標は、時空間の関数のセミノルムと見なすとき、スケール変換則に関して不変であることに注意が必要である。

1. 回転する障害物の周りの定常Navier-Stokes 方程式の解の存在と一意性3次元空間において障害物が回転し，かつ回転軸と同じ方向に並進運動する場合に，その外部領域 $\Omega$ において非圧縮性粘性流体のNavier-Stokes 方程式の定常解の存在と一意性を考察した．実際，回転の角速度を$\omega$,並進速度を$u_{\infty}$かつ外力$f = \dive F$ が条件$|\omega| + |u_{\infty}| + \|F\|_{L^{\frac32, \infty}} << 1$ であれば，$\nabla u \in L^{\frac32, \infty}(\Omega)$ であって，$u\in L^{3,\infty}(\Omega)$ である小さい解 $u$ が一意的に存在することを証明した．より一般的な一意性定理として，与えられデータ$\omega\in \re^3$, $u_{\infty}\in \re^3$, $F \in L^{\frac32,\infty}(\Omega)$ が十分小さく，かつ$F\in L^{\frac32,\infty}(\Omega) \cap L^{q,\infty}(\Omega)$, $3/2 < r < 3$ であれば，我々の構成した解 $u$ は$\nabla u \in L^{\frac32,\infty}(\Omega) \cap L^{q,\infty}(\Omega)$ なるクラスで一意的であることを証明した．さらに，これらのデータが小さい限りにおいては，データーに関する解の連続依存性が成立する．2. 外部領域における定常Navier-Stokes 方程式の弱解の一意性とエネルギー不等式の関係3次元外部領域$\Omega$においては，Leray により任意の外力$\dive F$, $F\in L^2(\Omega)$ に対して，$\nabla u\in L^2(\Omega)$ でエネルギー不等式 $\|\nabla u\|^2_{L^2(\Omega)} \le \dis{-\int_{\Omega}F\cdot\nabla u}dx$を満たす弱解 $u$ の存在が示されている．しかし，そのような弱解については，空間 $L^{3, \infty}(\Omega)$ における小ささを仮定する必要があった．本研究では，弱解そのものに対する小ささではなく，与えられた外力$F\in L^2(\Omega)\cap L^{\frac32,\infty}(\Omega)$ が空間$L^{\frac32,\infty}(\Omega)$ において十分小さければ，$\nabla u \in L^2(\Omega)$ であってエネルギー不等式を満たす弱解$u$ は一意的に存在することを証明した．この結果は期待できる定常Navier-Stokes 方程式の弱解の存在と一意性に関しては，最良の結果と言える．