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Grant amount：\160680000 （ Direct Cost: \123600000 、 Indirect Cost：\37080000 ）

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Grant amount：\17680000 （ Direct Cost: \13600000 、 Indirect Cost：\4080000 ）

The spectra of linearized operators around spatio-temporal periodic states of the compressible Navier-Stokes system were investigated in detail to obtain a precise description of the large time behavior of solutions around such periodic states. The structure of the spectrum of the linearized operator of the artificial compressible system was studied around the bifurcation point of stationary solutions and it was proved that if the artificial Mach number is sufficiently small, then the spectrum is decomposed into two parts, one is given by a perturbation of the spectrum for the incompressible system and the other one arises from the compressible aspect of the equations. This analysis was extended to the case of the linearized operator at the Couette flow of the compressible Navier-Stokes equations.

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Grant amount：\3120000 （ Direct Cost: \2400000 、 Indirect Cost：\720000 ）

The purpose of this project is to study the structure of boundary layers in limiting process of the large time behavior of solutions and the zero Mach number limit for the compressible Navier-Stokes equations. We derived the second order term of the asymptotic expansion of solutions in large time for the Cauchy problem on the whole space. We also study the spectral relations of the linearized operators around stationary solutions between the artificial compressible system and the incompressible Navier-Stokes system. It was shown that the spectrum for the artificial compressible system near the imaginary axis is decomposed into a part given by a perturbation of the spectrum for the incompressible system and a part arising from the compressible aspect of the system. We then established a sufficient condition so that a stable stationary solution of the incompressible system is also stable as a solution of the artificial compressible system.

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Grant amount：\4940000 （ Direct Cost: \3800000 、 Indirect Cost：\1140000 ）

We consider the initial boundary value problems for incompressible Navier-Stokes equations in half space and perturbed half space. We proved the time decay estimates for the solutions in weighted Lp space. We also consider the semilinear heat equations and dissipative wave equations in two dimension exterior domains. We showed the L2 boundedness with respect to the space and time valuables of the solutions for initial data in the Hardy space.

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Grant amount：\191100000 （ Direct Cost: \147000000 、 Indirect Cost：\44100000 ）

The challenging problem on global well-posedness of the Navier-Stokes equations had been so fully investigated that several remarkable results are obtained. Furthermore, our DNS of the uniformly isotropic turbulence is still by far the larger computational performance so that we could deal with the turbulent fluid with the high Reynolds number without any error of the experiment and indeterminacy. Our study has been based on the DNS of such a world highest standard and we could succeed to overcome difficulty of turbulence with the high Reynolds number. In this way, our research projects have developed the modern mathematical analysis, the applied mathematics, computational science and hydrodynamics and hopefully will lead the relevant subjects to the world-wide level.

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Grant amount：\9360000 （ Direct Cost: \7200000 、 Indirect Cost：\2160000 ）

To establish mathematical theory for bifurcation and stability in the compressible Navier-Stokes equation, we studied the stability of stationary and time-periodic parallel flows. We proved that the asymptotic behavior of parallel flow is described by a linear heat equation when the space dimension n is greater than or equal to 3, and by a one-dimensional viscous Burgers equation when n=2. In the case of the Poiseuille flow, we derived a sufficient condition for the instability in terms of the Reynolds and Mach numbers. Furhtermore, we proved the bifurcation of a familiy of space-time-peirodic traveling waves when the Poiseuille flow is getting unstable. As a first step of the stability analysis of space-periodic patterns, we investigate the stablity of the motionless state on periodic infinte layer, and derived the asymptotic leading part of the perturbation by using the Bloch transformation.

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Grant amount：\5460000 （ Direct Cost: \4200000 、 Indirect Cost：\1260000 ）

We considered the shallow water and long wave approximations for water waves over a periodically oscillating bottom, whose horizontally spatial scale is ε. We analyzed the homogenized limit (ε→０) of the solution to the shallow water equations by the method of multiple scales and determined explicitly the behavior of the solution. We also analyzed the homogenized limit and the long wave limit at the same time to a Boussinesq type equation. Moreover, we clarified the structure of a model for water waves which is obtained by the use of the variational structure.

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Grant amount：\35230000 （ Direct Cost: \27100000 、 Indirect Cost：\8130000 ）

We studied systems of nonlinear partial differential equations in the fields of gas dynamics, elasto-dynamics and plasma physics. We investigated the dissipative structure and decay property of the systems and proved the asymptotic stability of various nonlinear phenomena of vibration and wave propagation. Also we developed a general theory on nonlinear stability analysis for hyperbolic systems of conservation equations with relaxation and observed that the time-weighted energy method, semigroup approach and the technique of harmonic analysis are useful in the stability analysis.

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Grant amount：\4160000 （ Direct Cost: \3200000 、 Indirect Cost：\960000 ）

We consider the initial boundary value problems for the semilinear damped wave equations and the semilinear heat equations in two dimension exterior domains. We give the L2 boundedness with respect to the space and time valuables of thesolutions. Also,we consider the zero relaxation time approximation to the compressible Navier-Stokes-Poisson system to derive the mono-polar drift-diffusion system of degenerated type in a framework of the theory of weak solution.

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Grant amount：\4420000 （ Direct Cost: \3400000 、 Indirect Cost：\1020000 ）

In numerical computations of tsunamis due to submarine earthquakes, it is frequently assumed that the initial displacement of the water surface is equal to the permanent shift of the seabed and that the initial velocity field is equal to zero and the shallow water equations are often used to simulate the propagation of tsunamis. Under appropriate assumptions, we give a mathematically rigorous justification of this tsunami model starting from the full water-wave problem.

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Grant amount：\146120000 （ Direct Cost: \112400000 、 Indirect Cost：\33720000 ）

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.

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Grant amount：\6760000 （ Direct Cost: \5200000 、 Indirect Cost：\1560000 ）

We studied the asymptotic behavior of solutions of the compressible Navier-Stokes equation which describes motion of viscous fluids. We analyzed the stability properties of stationary solutions such as the motionless state and parallel flows in detail. It was proved that these stationary solutions are asymptotically stable if they are small enough in some sense. Furthermore, it was shown that the disturbances behave like solutions of convective heat equations in large time.

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Grant amount：\4550000 （ Direct Cost: \3500000 、 Indirect Cost：\1050000 ）

We consider the Navier-Stokes-Poisson equation describing the motion of compressible viscous isentropic gas flow under the self-gravitational force. We proved the existence of finite energy weak solutions in three dimensional bounded domain and discussed the stability of equilibrium. Also, we consider the equations to the Magneto-Hydrodynamics, and proved the existence of weak solutions.

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Grant amount：\13530000 （ Direct Cost: \11100000 、 Indirect Cost：\2430000 ）

We studied nonlinear partial differential equations in the field of gas dynamics, fluid dynamics and elasto-dynamics. We investigated the dissipative properties of the systems and proved the asymptotic stability of various nonlinear phenomena.

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Grant amount：\3870000 （ Direct Cost: \3300000 、 Indirect Cost：\570000 ）

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Grant amount：\3400000 （ Direct Cost: \3400000 ）

We consider the initial boundary value problem of the Compressible Navier-Stokes equations and show the asymptotic behavior to the solutions for the small initial data near the equilibrium states. Concerning the initial boundary value problem of the Compressible Navier-Stokes-Poisson equations in a bounded domain, we prove the existence theorem of weak solutions globally in time. Also, we consider the component-wise regularity of the solution to stationary Maxwell or Stokes systems.
Asymptotic behavior of solutions to the compressible Navier-Stokes equation on the half is considered around a given constant equilibrium. A solution formula for the linearized problem is derived, and some estimates for solutions of the linearized problem are obtained. It is shown that, as in the case of the Cauchy problem, the leading part of the solution of the linearized problem is decomposed into two parts, one behaves like diffusion waves and the other one behaves like purely diffusively. There, however, appear some aspects different from the Cauchy problem, especially in considering spatial derivatives. It is also shown that the solution of the linearized problem approaches in large times to the solution of the nonstationary Stokes problem in some spaces; and, as a result, a solution formula for the nonstationary Stokes problem is obtained Large time behavior of solutions of the nonlinear problem is then investigated in some by applying the results on the linearized analysis and the weighted energy method. The results indicate that there may be some nonlinear interaction phenomena not appearing in the Cauchy problem.
We consider the Navier-Stokes-Poisson equation describing the motion of compressible viscous isentropic gas flow under the self-gravitational force. We prove the existence of finite energy weak solutions in three dimensional bounded domain and discuss the stability of equilibrium.
We consider the component-wise regularity of the solution to stationary Maxwell or Stokes systems. We assume that there is a surface, regarded as an interface, and the solution to one of those systems is smooth except for this interface. Then, only under those assumptions, we can show that some components of solution are smooth across the interface. Namely, in the Maxwell system, the normal component of solution is always regular across the interface. In the Stokes system, on the other hand, the singularity of solution across the interface can arise only to the normal derivatives of its tangential components. Furthermore, those results are shown to be optimal.

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Grant amount：\12900000 （ Direct Cost: \12900000 ）

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.

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Grant amount：\3500000 （ Direct Cost: \3500000 ）

The KdV equation and the Kawahara equation were derived formally from the basic equations for surface waves as long wave approximations. After rewriting the equations in an appropriate non-dimensional form, we have two non-dimensional parameters δ and ε the ratio of the water depth to the wave length and the ratio of the amplitude of the free surface to the water depth, respectively. The limit δ→0 corresponds to the long wave approximation. More precisely, the limit δ=ε^2→0 corresponds to the KdV limit and the limit δ=ε^4→0 corresponds to the Kawahara limit. T.Iguchi gave mathematically rigorous justifications for the KdV and the Kawahara limits and proved that the solutions of these approximate equations in fact approximate the solutions of the original basic equations for an appropriate long time interval. He also analyzed an effect of the presence of an uneven bottom to these long wave approximations.
T.Miyakawa investigated the Navier-Stokes equations for a two-dimensional incompressible viscous fluid in the cases where the fluid is occupied the entire space or outside of the unit disc. He discovered the relation between the group symmetries of the solution and the space-time decay properties of the solution. He also investigated the Euler equation for a two-dimensional incompressible ideal fluid occupied an exterior domain and discovered a relation between the square integrability of the pressure and the effect of the flow to the obstacle.
S.Nishibata investigated the asymptotic behavior in time of the spherically symmetric solution for compressible Navier-Stokes equations in the exterior domain of the sphere. He proved that a stationary solution is asymptotically stable under suitable assumptions for the initial data and the external forces. He did not suppose any smallness conditions for the data.
Y.Kagei investigated the asymptotic stability of a stationary solution for the compressible Navier-Stokes equations in the half-space. He discovered a nice solution formula to the linearized problem. By using the formula and carrying out the analysis very carefully for the oscillatory integrals, he obtained the best possible decay estimates and proved the asymptotic stability.

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Grant amount：\8100000 （ Direct Cost: \8100000 ）

We studied many kinds of objects in the function theory of several complex variables from the viewpoints of the theory of singularities. In particular, we are interested in the boundary behavior of the holomorphic functions which are square integrabel. Concretely the Bergman kernel and Szegoe kernel are very important integral kernel and they have many important information of the boundary behavior of holomotphic functions. As is very well known, the case of strictly pseudoconvex domains has many strong results about the Bergman kernel and Szegoe kernel. For example, the asymptotic expansion due to C.Fefferman reveals completely their boundary behaviors. We are interested in the weakly pseudoconvex domains case. The general case is very difficult to analyze and so we restrict ourselves to the objects in the case of finite type in the sense of D'Angelo. From the definition of finite type, the argument from algebraic geometry and singularity theory are valuable. We introduced the concepts of"Newton polyhedra"into the analysis of the Bergman kernel and showed that its singularity can be expressed in terms of the topological information of the Newton polyhedra. Moreover, we analyzed the construction of peak functions of any finite type domains.

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Grant amount：\11400000 （ Direct Cost: \11400000 ）

We studied asymptotic behavior of solutions and stability of nonlinear waves for equations of gas motion with dissipative structure.
1.We developed the energy method in the Sobolev space W^{1,p} for n-dimensional scalar viscous conservation law and derived the optimal decay estimates in W^{1,p}. The method was also applied to the stability problem for rarefaction waves and stationary waves.
2.We introduced the notion of entropy for n-dimensional hyperbolic conservation laws with relaxation and developed the Chapman-Enskog theory. Moreover, we proved the global existence and optimal decay of solutions in a L^2 type Sobolev space.
3.For the compressible Navier-Stokes equation in the n-dimensional half space, we proved the asymptotic stability of planar stationary waves. To develop the theory in the Sobolev space of order [n/2]+1, we need additional considerations for local existence results.
4.For the dissipative Timoshenko system, we derived qualitative decay estimates of solutions by applying the energy method in Fourier space. We found that the dissipative structure is so weak in high frequency region and it causes the regularity loss in the decay estimates.
5.For dissipative wave equation with a nonlinear convection term, we proved the global existence and optimal decay of solutions in L^p. Moreover, we showed that the solution approaches the nonlinear diffusion waves given in terms of the self similar solutions of the Burgers equation. Derivation of detailed pointwise estimates of the fundamental solutions is crucial in the proof.

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Grant amount：\5200000 （ Direct Cost: \5200000 ）

Y.Kagei and T.Kobayashi investigated the stability of the motionless equilibrium with constant density of the compressible Navier-Stokes equation on the half space and gave a solution formula for the linearized problem to derive decay estimates for solutions to the linearized problem. Combining these results with the energy method, they obtained decay estimates for perturbations. The results also indicate that there may be some nonlinear interaction phenomena not appearing in the Cauchy problem. Kagei studied a nonhomogeneous Navier-Stokes equations for thermal convection motions. He showed the existence of global weak solutions and investigated the Oberbeck-Boussinesq limit of the equation under consideration. Kobayashi investigated local interface regularity of solutions of the Maxwell equation, Stokes equation and Navier-Stokes equation. S. Kawashima proved that the solution of a general hyperbolic-elliptic system are approximated in large times by the ones of the corresponding hyperbolic-parabolic system. Kawashima also established the $W^{1.p}$-energy method for multi-dimensional viscous conservation laws and obtained the sharp $W^{1.p}$ decay estimates. Kawashima gave a notion of an entropy for hyperbolic systems of balance laws, which enables to understand the dissipative structure of the systems. T.Ogawa extended the logarithmic Sobolev inequalities to homogenous and inhornogeneous Bosev spaces. Using these inequalities he improved the Serrin-type condition for regularity of solutions to the incompressible Navier-Stokes equation, Euler equation and Harmonic flows. Ogawa also proved the finite-time blow up of solutions to the drift-diffusion equations. T.Iguchi studied the bifurcation problem of water waves and classified the bifurcation patters in terms of the Fourier coefficients which represent the bottom of the domain. Iguchi also investigated conservation laws with a general flux. He introduced a notion of "piecewise genuinely nonlinear" and constructed the entropy solutions for the small initial values.

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Grant amount：\2800000 （ Direct Cost: \2800000 ）

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。

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Grant amount：\13900000 （ Direct Cost: \13900000 ）

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.

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Grant amount：\10200000 （ Direct Cost: \10200000 ）

We study the stability of nonlinear waves for hyperbolic-elliptic coupled systems in radiation hydrodynamics and related equations.
1. By using the Fourier transform, we give a representation formula for the fundamental solutions to the linearized systems of hyperbolic-elliptic coupled systems and verify that the principal part of the fundamental solutions is given explicitly in terms of the heat kernel. Also, we obtain the sharp pointwise estimates for the error terms.
2. We obtain the pointwise decay estimate of solutions to the hyperbolic-elliptic coupled systems by using the representation formula for the fundamental solution and the corresponding estimates. Furthermore, we prove that the solution is asymptotic to the superposition of diffusion waves which propagate with the corresponding characteristic speeds.
3. We discuss a singular limit of the hyperbolic-elliptic coupled systems. We prove that at this limit, the solution of the hyperbolic-elliptic coupled system converges to that of the corresponding hyperbolic-parabolic coupled system.
4. We show the existence of stationary solutions to the discrete Boltzmann equation in the half space. It is proved that the stationary solution approaches the far field exponentially and is asymptotically stable for large time.
5. We study the asymptotic behavior of nonlinear waves for the isentropic Navier-Stokes equation in the half space. For the out-flow problem, we prove the asymptotic stability of nonlinear waves such as (1)stationary wave, (2)rarefaction wave, and (3)superposition of stationary wave and rarefaction wave.

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Grant amount：\2100000 （ Direct Cost: \2100000 ）

研究実績は以下のとおり.
小川は研究分担者の後藤、松本とともに、平均曲率流方程式に対するBMOアルゴリズムに対して解析を加え、特に研究協力者の三沢正史(電気通信大)の協力のもと、類似の手法のp-Laplace型作用素を持つ退化放物型の偏微分方程式の解に対するlevel set functionによる数値解析の手法を試みた。そこでは、p-Laplace型の方程式にはBMPアルゴリズムの直接の適用が不可能であることが判明した。次に、小川は共同研究者である、石井克幸(神戸商船大)とともに、BMOアルゴリズムとAllen-Charn方程式の特異極限による平均曲率流方程式への近似理論の類似性に着目して、BMOアルゴリズムとAllen-Charn方程式の特異極限を統一的に扱う理論の構築を構想し、証明を試みた。現在その漸近展開における解析で部分的な結果を得ている。この手法は、Charn-Hiliard方程式のような、さらに高階の放物型方程式における特異極限問題に、適用が可能であると予想され、さらに複雑な界面運動を記述する、Eguchi方程式系への応用が見込まれる。
さらに小川は研究協力者の山田想(ヴィジュアルテクノロジー)と共同でモンテカルロ法による非線形楕円型方程式の境界値問題についての数値解析に関するシュミレーションを行い、並列高速への可能性を探った。ことに並列化において有利な領域における特定の部分のみに対する解の高速計算処理に力点を置いて、研究を行った。
杉田は複雑な関数の数値積分におけるさまざまな現象を確率数値解析の視点から考察した.とくにランダム性が小さくて複雑な関数に対しても安定した数値積分を可能にする離散的ランダムワイルサンプリングを提唱した.
伊藤は表面拡散による3相境界運動を記述する幾何学的偏微分方程式に対して,3相が含まれる領域の境界が長方形的である場合に,初期値がエネルギーのミニマイザーに近ければ時間大域解が存在すること,解は時間無限大でエネルギーのミニマイザーになることを考察した.

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Grant amount：\3800000 （ Direct Cost: \3800000 ）

Y.Kagei showed that some stationary solutions of the Obebeck-Boussinesq equation is unconditionally stable even when they are at criticality of the linearized stability. Kagei then derived a model equation of thermal convection in which the effect of viscous dissipative heating is taken into account. It was shown that the threshold of the onest of convection for this model equation is larger than that for the usual Oberbeck-Boussinesq equation and various space-periodic stationary solutions bifurcate at the threshold transcritically. Kagei also studied the Cauchy problem for the Vlasov-Poisson-Fokker-Planck equation and constructed invariant manifolds in some weighted Sobolev spaces. As a result, long-time asymptotics of small solutions were derived. S.Kawashima studied a singular limit problem for a general hyperbolic-elliptic system and proved that in the singular limit the solution of the hyperbolic-elliptic system converges to the solution of the corresponding hyperbolic-parabolic system. Kawashima also studied initial boundary value problems for discrete Boltzmann equations in the half-space and showed the existence of stationary solutions under several boundary conditions and their asymptotic stability. T.Ogawa showed that for a class of semilinear dispersive equations, solutions with initial values having one singular point like the Dirac delta function become real analytic in space and time variables except at the initial time. Ogawa also studied blow-up problem for the three dimensional Euler equation and gave a sufficient condition for blow-up in terms of some semi-norm of a generalized Besov space. T.Iguchi studied bifurcation problem of stationary surface waves and classified possible bifurcation patterns.

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Grant amount：\3200000 （ Direct Cost: \3200000 ）

Under this Grant-in-Aid, I made joint research with Professor Victor G.Kac, and obtained the following results :
1. "Integrable representations" for affine superalgebras are never easy concept, but should be treated carefully. We found that integrable representations consist of two kinds, namely principal-integrable representations and subprincipal-integrable representations, and gave the explicit and complete list of all highest weights for principal and subprincipal integrable modules.
2. We gave an explicit construction of fundamental sl(m|n)^- and osp(m|n)^-modules by using free bosons and free fermions. Using this explicit construction, we calculated the characters and obtained three kinds of character formulas --- Weyl-Kac type, theta-function type and quasi-particle type. From these character formulas, we found that the characters of fundamental sl(m|1)^-modules are Appell's elliptic functions which were discovered by Appell in 1880's but have been forgotten over one hundred years. These functions are not modular functions, but we succeeded to compute their asymptotics.
3. The trivial representation of an affine superalgebra sl(2|2)^ is a representation of critical level, since its dual Coxeter is equal to 0. So there was no known denominator identity for such superalgebras. We obtained explicitly the denominator formula for sl(2|2)^ by using Riemann's theta relations.
4. It is known by the theory of Frenkel-Kac-Wakimoto (1994) that the W-algebra of an usual affine Lie algebra and its representations are constructed in terms of the quantized Drinfeld-Sokolov reduction. But, for affine superalgebras, an immediate extension of this method fails to give a right W-algebra, and the construction of the W-algebra associated to affine superalgebras has long been a problem. We succeeded to resolve the difficulty by tensoring the factor, which arises from the algebraic variety, with the usual BRST-complex. The W-algebra of an affine superalgebra sl (2|1)^ obtained by this method is the direct sum of the centerless Virasoro algebra and the N=2 superconformal algebra. This theory enables us to make a detail investigation on representations of the N=2 superconformal algebra by means of admissible representations of sl(2|1)^. Actually we found that, other than the usual minimal series representations, there exist curious series of N=2 representations whose characters are half-modular functions. This research is now in progress very intensively.

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Grant amount：\3100000 （ Direct Cost: \3100000 ）

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.

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Grant amount：\3100000 （ Direct Cost: \3100000 ）

(1) As for solutions to Navier-Stokes equations, which describe motions of incompressible fluids in unbounded domains, decays at infinity of stationary solutions and time-decays of non-stationary measured in several LP-like norms were studied, and an influence of non- lineality of the equations on the decay was found out. (2) Sufficient conditions for classical and Sobolev type global solutions to Burgers-Helmholtz system were established, and asymptotic behavior of the solutions were obtained. The existence of traveling waves and theire asymptotic stability were seen. (3) A mathematical definition of entropy functions for compressible Euler-Helmholtz system was established. (4) The asymptotic stability of steady flows in infinite layers of viscous incompressible fluids in critical cases of stability has been verified. (5) For the initial value problems associated with Korteweg-de Vires equations, an analytic smoothing effect was found out. (6) A class where one can handle formal asymptotic expansions of solutions to quasilinear positive symmetric systems of hyperbolic equations was introduced, and its basic properties were studied. (7) A new complexification of an abstract Wiener space was proposed. A complex change of variable based on the complexification was established, and appled to study asymptotic behaviors of stochastic oscillatry integrals (methods of stationary phase, saddle point methods, and so on).

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Grant amount：\6800000 （ Direct Cost: \6800000 ）

We study the initial value problem for a class of hyperbollic-elliptic coupled systems including the equation of a radating gas. For the simplest model system of a radiating gas :
1.We give sufficient conditions for the non-existence and the existence of classical global-solutions.
2.We prove the global existence and asymptotic decay of smooth solutions in Sobolev spaces. The solution approaches the diffusion wave which is defined in terms of the self-similar solution of the viscous Burgers equation.
3.We prove the asymptotic stability of rarefaction waves which are defined in terms of the centered rarefaction wave of the inviscid Brugers equation.
4.We show the existence of traveling wave solutions of shock profile. These shock waves have a discontinuity only when the shock strength is greater than a critical value. Moreover, we prove the asymptotic stability of smooth shock waves.
5.We show the glogal existence of weak solutions to the Riemann problem. The jump contained in the weak solution decays exponentially, and the solution approaches the corresponding smooth shock wave as time tends to infinity.
6.We give a mathematical definition of the entropy function and prove the equivalence of the existence of an entropy function and the symmetrization of the system. Next, we formulate the stability condition, and under that condition we show the global existence and asymptotic decay of smooth solutions in Sobolev spaces. Furthermore, we observe that the solution is time-asymptotically approximated by the solution to the eorresponding hyperbolic-parabolic coupled system. These results are applicable to the equation of a radiating gas.

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Grant amount：\1000000 （ Direct Cost: \1000000 ）

2枚の水平平行板間に流体をいれ、下から一様に加熱していくと静止状態が不安定化を起こして対流が発生し、さまざまな対流パターンが見られる。この対流はブシネスク方程式で記述され、現象に対応するさまざまな定常解が得られている。これらの定常解の中でもロール型対流解については、ブシネスク方程式から形式的に導かれた簡単なモデル方程式を用いて、その安定性の解析が行われてきた。しかしながら、ブシネスク方程式を用いてのロール型対流解の安定性の数学的に厳密な解析はあまり行われていない。本研究では、ロール型対流解の安定性をブシネスク方程式を用いて明らかにし、また、モデル方程式としてよく知られたギンツブルグ-ランダウ方程式やスウィフト-ホ-ヘンバーグ方程式の数学的に厳密な導出を行うことを目標とした。本年度の研究では、ブシネスク方程式のロール型対流解のまわりでの線形化作用素のスペクトルを調べ、ロール型対流解の2次元攪乱に対する線形化安定性および不安定性に関するエックハウスの判定条件の証明をWolf von Wahlと共同で与えた。この結果はInternational Journal of Non-Linear Mechanicsに発表予定である。今後は3次元攪乱に対する安定性、不安定性の判定条件を与え、ギンツブルグ-ランダウ方程式やスウィフト-ホ-ヘンバーグ方程式の数学的に厳密な導出を行いたい。また、ロール型対流解が不安定な場合、ロール型対流解の近傍に初期値をとる初期値問題の解がどのような振る舞いをするのかについて力学系の手法を用いて考察してみたい。

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Grant amount：\900000 （ Direct Cost: \900000 ）

2枚の水平平行平板間に流体をいれ、下から一様に加熱していくと静止状態が不安定化を起こして対流が発生し、さまざまな対流パターンが見られる。この対流はブシネスク方程式で記述され、現象に対応するさまざまな定常解が得られている。これらの定常解の中でもロール型対流解については、ブシネスク方程式から形式的に導かれた簡単なモデル方程式を用いて、その安定性の解析が行われてきた。しかしながら、ブシネスク方程式を用いてのロール型対流解の安定性の数学的に厳密な解析は行われていない。本研究では、ロール型対流解の安定性をブシネスク方程式を用いて明らかにし、また、モデル方程式としてよく知られたギンツブルグ-ランダウ方程式やスウィフト-ホ-ヘンバーグ方程式の数学的に厳密な導出を行うことを目標とした。本年度の研究では、ブシネスク方程式のロール型対流解の2次元攪乱に対する安定性を考え、ロール型対流解のまわりでの線形化作用素のスペクトルを調べ、物理学者によって得られたロール型対流解の2次元安定性に関する結果(エックハウス不安定性)を数学的に厳密に証明した。今後は3次元攪乱に対する安定性を考え、ロール型対流解のジグザグ不安定性やクロスロール不安定性などの証明を行いたい。また、3次元攪乱に対する安定性が明らかになれば、ロール型対流解のまわりの攪乱の挙動を記述するモデル方程式であるギンツブルグ-ランダウ方程式とスウィフト-ホ-ヘンバーグ方程式のブシネスク方程式からの導出を証明したい。

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Grant amount：\2100000 （ Direct Cost: \2100000 ）

非圧縮粘性流の挙動に関し、宮川は外部領域における定常解のエネルギー安定性を論じ、摂動の減衰率についてほぼ最良と思われる結果を得た。この研究はドイツ連邦共和国のWolfgang Borchersと共同で行われた。さらに宮川は、全空間における流れの減衰の問題に対し、Hardy空間の理論を用いて上記の結果を拡張した。
川島は一次元の気体モデルの方程式の衝撃波解と拡大波解の安定性を論じ、これまでに知られた結果を改良した。特に衝撃波解の安定性については、与えられた摂動の減衰率を導くことに初めて成功した。
隠居はロール型渦対流の摂動論を研究し、摂動の周期を変えると安定解が一度不安定になった後さらにまた安定になることを示した。この結果は現象とよく一致し、この問題に対する数理モデルの信頼度をさらに高めた。

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Grant amount：\2000000 （ Direct Cost: \2000000 ）

1.非圧縮粘性流の挙動に関し、宮川は、外部領域における定常問題を扱い、解の空間減衰度について最良の結果を得た。また解の一意性についての判定規準を改良した。さらに、平行平板間の流れ及び通路で結ばれた流れの数学的研究の出発点となるベクトル場の分解定理を証明した。
2.川島は、一次元粘弾性体の方程式に対し、一般的な構成則の下で衝撃波の存在を示し、その安定性を論じた。また、ボルツマン方程式の離散モデルの拡大波解の存在を示し、その安定性を論じた。
3.隠居は、散逸効果を伴う一般の熱対流方程式の二次元問題について、対応する力学系がアトラクターを持つことを示し、その次元の評価を行った。さらに、ロール型渦対流の臨界状態での安定性を詳しく調べた。
4.準線形双曲型方程式、曲面の発展方程式、流れの統計理論については、従来の理論では扱えない新しいモデルが提起され、現在研究が進行中である。

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Grant type：Competitive

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Grant type：Competitive

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