第9回 解析学賞受賞 特別講演

I studied various mathematical problems on Schroedinger equations and obtained following results during the period 2016-18: 1) I obtained a sufficient condition for the existence and uniqueness of the propagator for quantum many body systems which are in an external electro-magnetic field whose potentials increase quadratically-linearly at spatial infinity and which interact each others by potentials which carry very strong local singularities. The propagator preserves a large subspace where energy observable may be defined and computed in a natural way; 2) I decided exponents of Lebesgue spaces in which wave operators of scattering theory for Schroedinger operators are continuous when they are regular or singular at threshold; 3) I studied spectral and scattering theory for Schroeinger operators with point interactions in 2 and 3 dimensions. I obtained the asymptotic expansion of resolvent at threshold and proved wave operators are bounded in Lebesgue spaces of certain order.

We extract a dispersive and dissipative effect from the typical example in the nonlinear dispersive equations such as the nonlinear Schroedinger equation and nonlinear dissipative equations such as the Navier-Stokes system or the drift diffusion system and research the critical problems that arose from a balanced situation between the stabilize effects from dispersive and dissipative and the instability caused from nonlinear interaction. In particular, we establish the maximal regularity for the nonlinear dissipative system and applied for the critical problems and singular limit problems in Keller-Segel system or ill-posedness problem of mathematical fluid mechanics.

研究代表者の小川は研究協力者の岩渕 司と共同で, 2乗のべき型非線形項を持つ非線形シュレディンガー方程式の適切性と非適切性の臨界を研究し, 空間1次元においては非斉次Sobolev空間のもつ非斉次構造が, 不変スケールから予想される臨界スケールに至ることを阻害することを示し, さらに臨界性を実補間空間であるベソフ空間で分類した場合の臨界補間指数を同定した. また2次元に対しては予想される臨界スケールに至ることを示した. 4次元以上においては堤誉志雄による最良の結果が知られており, 2次の非線型性に対して残る問題は3次元のみとなった. また同様の事実は非線形熱方程式に対しても成立することを述べた. これらの結果は解の形式的な漸近展開を, モデュレーション空間において正当化し, 解の2次近似が臨界空間よりも広いクラスで解の不安定性を引き起こすことに起因する. 漸近展開を正当化することにより, 従来あった背理法による議論を経由せずに証明が可能となる. 一方, 半導体モデルに現れる, 移流拡散方程式には双極性のモデルと単極性モデルが存在する. 双方の初期値問題に対しても同様な臨界適切性を研究し, 双極性のモデルは単極モデルよりも適切な函数空間のクラスが狭いことを, 非線形干渉の対称性に着目して示した. また副産物として, 2次元渦度のNavier-Stokes方程式の可解性について既存の結果が双極型移流拡散方程式の非線形項と類似であるにもかかわらず, 単極型と同等の函数空間まで適切性が示されることについて, 非圧縮条件が非線形構造に対して対称性を与えることに起因することを突き止めた.

We study the basic model for incompressible two-phase flows with phase transitions consistent with thermodynamics in the case of constant but non-equal densities of the phases. We employ the direct mapping approach to transform the problem locally in time to a fixed domain. The proof of local well-posedness is based on maximal regularity of the underlying principal linearization and the contraction mapping principle. We extend our well-posedness result to general geometries, study the stability of the equilibria of the problem, and show that a solution which does not develop singularities exist globally, and if its limit set conatins a stable equilibrium it converge to this equilibrium as time goes to infinity.

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 obtained comparison principle for unbounded viscosity solutions of degenerate elliptic PDE with superlinear gradient terms. We presented a representation formula for viscosity solutions of integro-differential equations of Isaacs type. We established the local maximum principle fro Lp-viscosity solutions of fully nonlinear uniformly elliptic PDE with unbounded coefficients to the first derivatives. We discussed regularity and large time behavior of viscosity solutions of integro-differential equations with coercive first derivative terms. We obtained existence and uniqueness of entire solutions of fully nonlinear elliptic equations with superlinear growth in the first derivatives. We showed the Lipschitz continuity of viscosity solutions of mean curvature flow equations with bilateral obstacles.

Analyzing simultaneous rational approximations of irrational p-adic numbers by using multi-dimensional p-adic approximation lattices, we investigate some recurrent properties of discrete orbits given by quasi-periodic dynamical systems, the frequencies of which are weak Liouville p-adic numbers and we show some unpredictability properties of the orbits. For the symbolic dynamical systems given by the coefficient sequences of expansions of p-adic numbers we give some inequality relations between the recurrent dimensions and the topological entropy of these systems.
For the shortest vector problems of p-adic approximation lattices we compare the theoretical solutions given by the simultaneous approximation problems (SAP) and the numerical solutions estimated by the LLL algorithm. By using these results we propose a new lattice based cryptosystem, the private keys of which are the SAP solutions of p-adic lattices.

Models described by drift-diffusion equations having a typical mass transport structure, it is understood and classified as one of the non-local interaction system. In this research, we consider the global behavior of the solution to a semi-linear drift-diffusion system in two and three-dimensional cases. In addition, we also study a system of degenerate drift-diffusion equations and make it clear the global behavior
of the weak solution. Furthermore, we find the global structure of the solutions are quite similar to the one of the nonlinear disspersive equations and between the critical exponents, we classified the global behavior of the solution and find the threshold value of the global existence.

The purpose of this project is to build mathematical theories on reaction-diffusion systems and on the deformation of curves and surfaces, which are necessary to understand, through mathematical models, the dynamic process of morphogenesis in embryonic stages.
In particular, we have succeeded in building a theory on pattern formation in strongly heterogeneous environments, and this will help us devise more biologically realistic models of pattern formation.

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.

The main reseacher T. Ogawa researched several nonlinear partial differential equations with critical structure and find various criticality in each problems. He studied the following topics with colabolators. Two dimensional drift-diffusion system in the critical Besov spaces and established maximal regularity for the heat equation in non-reflexivie Banach spaces. Higher order expansion of the solution for the drift-diffusion system in higher space dimensions, the global existence for the nonlinear damped wave system, the critical Sobolev inequality with logarithmic type and generalization to abstract Besov spaces, WKB approximation for nonlinear Schrodinger equations with Poisson equations, the scaling critical solvabilityfor quadratic nonlinear Schrodinger equation and critical well-posedeness in lower space dimensions.

Various types of nonlinear PDEs (nonlinear elliptic equations, nonlinear diffusion equations, nonlinear wave equations, nonlinear Schrodinger equations) arising in physics and engineering were synthetically studied from the viewpoint of the theory of nonlinear evolution equations by using the techniques from the theory of nonlinear functional analysis, the theory of functions of a real variable, the theory of ordinary differential equations and the calculus of variations.

Partial differential equations describing diffusion phenomena have been widely considered. To know the relationship between the behavior of solutions and the geometry of domain, we showed both the relationship between the initial behavior and the curvatures of the boundary and that between the existence of a stationary level surface with time and the symmetry of domain. In particular, we obtained characterizations of the sphere, the hyperplane, and the circular cylinder involving a stationary level surface. These yielded a new development of inverse problems determining the geometry of domain. Also, as a by-product, we obtained Liouville-type theorems for viscosity solutions of fully nonlinear elliptic partial differential equations describing an important class of Weingarten hypersurfaces.

We develop the method to prove maximal regularity by proving R-bounded of an solution operator in view of operator-valued Fourier-multiplier theorem. As an application of the maximal regularity, we prove local solvability of free boundary problems for the Navier-Stokes equations with surface tension in a scale invariant Sobolev space. Moreover we prove maximal regularity of the Cauchy problem for the heat equation in homogeneous Besov space that is not a UMD (unconditional martingale differences) Banach space.

We study the shapes of the solutions of diffusive equations, and reveal the relationship between the movement of hot spots and the harmonic functions. Furthermore we studied the asymptotics of the solutions for the semilinear heat equations. In particular, we characterized the location of blow-up set for blow-up solutions, and also gave the decay estimates of the deference between global in time solutions and their asymptotics.

We carried out the investigation about the structure of solutions of nonlinear parabolic and elliptic equations. Our main results are as follows : Next, we studied the existence and uniqueness of solutions with moving singularities for a nonlinear parabolic partial differential equation. We also showed that there exists a solution with a moving singularity that changes its type suddenly., and made clear the asymptotic behavior of singular solutions that converges to a singular steady state. We also studied a chemotaxis system, and made clear the structure of self-similar solutions that blows up by concentrating to a point in finite time.
For a reaction-diffusion system, which is called a Gierer-Meinhardt system, we studied the mathematical structure of pattern formation, and also made clear the behavior of time-dependent solutions.

研究実績は以下のとおり。
1.研究代表者の小川は、連携研究者の石渡通徳(室蘭工大・工)、研究協力者の高橋太(阪市立大・理)、黒木場正城(福岡大・理)らとともに2次元上で楕円型部分が非線形に摂動されたKeller-Segel系の時間適切性について研究し、初期値が十分小さく解に一定の変分法的特性を満たす場合には解が一意的に存在すること、またそれ以外の場合には一般に解の一意性が崩れ、適切性が成立しないことを変分的手法により示した。特に初期値問題が非適切であって少なくとも球対称の初期値からは少なくとも解が2つ存在することを示した。方法はヒルベルト-シュミット法と楕円型方程式の臨界点解からの分岐解をとらえることにより示される。
2.研究代表者の小川は研究協力者の清水扇丈(静岡大・理)と共同で2次元Drift-diffusion方程式を臨界Besov空間で考え、局所解の存在定理と時間大域的可解性を示した。その際に非回帰的Banach空間における最大正則性定理を証明し、L^1に近い空間における擬似的なエネルギー不等式が成り立つこと、またL^1空間では同様の不等式が一般には成立しないことを示した。
3.研究代表者の小川は研究協力者の山本征法(東北大大学院博士3年)と共同で、高次元drift-diffusion方程式の解の減衰について研究し、時間大域解の解の漸近挙動を高次の項まで展開した。特にこの問題に固有のキャンセル効果により高次漸近展開項がより簡潔に表せることと、高次の展開項が一般には消えないことを示し、高次項の誤差項に対する下からの減衰評価を与えた。

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.

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.

Collapse of patterns is a newly found phenomenon characteristic to some reaction-diffusion systems possessing singular nonlinearities, where patterns are formed at first but eventually converge to a nonregular steady state. We have given sufficient conditions for patterns to collapse and also for solutions to blow-up in finite time. In addition, qualitative properties of solutions such as the dynamics of maximum points and/or asymptotic forms of solutions have been studied in detail.
Moreover, movement of planar closed curves driven by bending energy is considered as a lower dimensional analogue for the geometric variational problem which determines the shape of red blood cells. All the critical points of the energy functional under some constraints are found and the gradient flow of the constraint minimization problem has been constructed.

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.

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

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.

今年度得られた主たる結果は以下のものである.
1.破壊力学における亀裂進展力となるエネルギー解放率について,Lipschitz連続な領域写像を用いてその数学的定式化を行った.更に,Frechet微分・Banach空間における抽象的パラメータ付き変分問題の一般論とLax-Milgramの定理の応用としてその領域積分表現を得た.それにより見通しのよい数学的枠組みが構築され,より弱い条件への精密化・高階エネルギー微分への拡張が得られた.(木村)
2.いくつかの自由境界問題及びパターン形成問題に対して,その数値シミュレーションと数学解析の手法の開発を行った.取り扱った問題は,流れ下における最短時間経路問題に対するマーカー粒子法の基礎付け,反応拡散系に現れるパターンダイナミクスに対するアダプティブメッシュ有限要素法の応用,などである.(木村)
3.鉄磁性体の2次元ising型spinモデル(シグマ模型)に対する連続体近似を考え,特異性の発生について考察した.(小川)
4.重力自己崩壊に関連する半線形熱方程式系の解の時間大域的存在と解の一様有界性について,球対称の解に限定して,初期条件を閾値となる8πより小さい初期値から時間大域的な有界な解が存在してなめらかに成ることを示した.(小川)
5.多孔質媒体中の流体に現れる自由境界問題の離散化として,特異極限を使った方法があり,自由境界を自然に扱うことができる.スケール変換の普遍性の手法をもとに,数値実験で絞られた候補の妥当性について検討した.(中木)
6.流体のある種の自由境界を離散的に記述する方法として,多数の渦点による近似が知られている.この渦点問題に対して,まず少数の渦点の挙動について研究した.その結果,7個の場合に緩和振動を起こすことを見つけ,数学的な証明を行った.(中木)

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.

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.

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石井克幸と協力者の後藤陽子と共同で平均曲率流方程式を等高面の方法で考え、そのBence-Merrimen-Osher型の数値解析アルゴリズムの解への収束を、半線形熱方程式の解に対する特異摂動の観点から考え、粘性解の方法により証明した。平均曲率流方程式は特に画像処理の際のノイズ消去に有効に用いられるがその場合のBMOアルゴリズムの有効性が示せた.
また,小川は単独で,鉄磁性体の2次元ising型spinモデル(シグマ模型)に対する連続体近似を考え、その半線形化方程式のエネルギー空間における可解性を新しいゲージ変換を考えることにより与えた。また関連して、粘性が入る場合に鉄磁性体モデルとSchrodinger写像、調和写像熱流との相関を議論し、それぞれ係数が極限と成る場合の状況について、ゲージ変換による議論、単調性公式による議論により特異性の発生について考察した。
分担者の服部はプレシルピンスキーガウケットと呼ばれる無限フラクタル格子上の単純ランダムウォークと自己回避確率連鎖を連続的に内挿する自己抑制・吸引的確率連鎖の族を構成し,変位の指数を与えた.
分担者の木村はパラメータを含む移流項を持つ楕円型方程式の第一固有値の特異摂動問題を考察した.移流の代表速度を表すパラメータが無限大に近づくとき起こる固有値の指数減衰現象について,空間1次元の場合に精密な漸近挙動評価を得た.
分担者の松本は生成作用素の定義域が稠密でないanalytic semigroupおよび、integrated semigroupの時間に依存しない非線形摂動を考察し、汎関数を用いた一般的な増大条件の下で、弱解を与える発展作用素が存在するための必要十分条件を、方程式に対する陰的差分近似の存在によって与えた。

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.

1.The head investigator and co-investigator T.Ogawa has studied on the sufficient condition for solutions to the Benjamin-Ono equation to be analytic in space and time variables for t>0 with E.Kaikina and P.I.Naumkin. Under the condition that initial data have some singularity at the origin, we show that the solution is analytic for t>0.
2.We have investigated the existence of solutions for the initial value problem of the Benjamin-Ono equation. However it is impossible to apply Picard's iteration method to the Benjamin-Ono equation in usual Sobolev space, we show that Picard's iteration method is applicable for some Sobolev space with homogeneous and inhomogeneous multipliers.
3.We have investigate the propagation of singularities for the nonlinear wave equations with nonlinear term satisfying null condition. Regularity estimate can be improved under the null condition for nonlinear term.

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.

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

1. The H-2-construction to treat the singular perturbation for the self-adjoint operator by the operator theoretical method has been studied by K. Watanabe. The author obtains some results for the operator, for example, necessary and sufficient condition of the existence of embedded eigenvalues, representation of the scattering matrix and etc.
2. The regularity of the solutions for the Maxwell, Stokes and Navier-Stokes equation with the interface has been investigated by K. Watanabe. Especially the following results is remarkable : if the tangential component does not have the singularity, then the regularity of the solution gains rank one.
3. The partial differential equation with the dissipative term has been studied by K. Watanabe. The relationship of this spectrum type and the behavior of the time decay of the solutions has been studied.
4. Krein's formula (which is a generalization of the second resolvent equation) was studied by S.T. Kuroda and published
5. The finite elements method for the bi-harmonic Dirichlet problems on the polygon in the plane (not necessary convex) has been studied by A. Mizutani.
6. The behaviors of the solution at the time infinity for the system of the nonlineat partial differential equations (for example, the coupled Schrodinger and Klein-Goldon) have been studied by A. Shimura and published.
7. The regularity and the uniqueness for the initial date problems of the Euler equation have been studied by T. Ogawa and published.
8. The scattering theory for the dissipative system has been studied by M. Kadowaki and published.

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.

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。

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.

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.

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.

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).

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 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.

本研究の成果は多岐に渡るので、研究代表者の三宅正武の研究成果を中心に研究実績の報告をする。解析的偏微分方程式のグルサ-問題は従来では優級数の方法による研究がほとんどであったが、三宅はテプリッツ作用素のフレドホルム性の議論を数列空間に適用することに依って、各種のジュブレイ族空間における可解性のみならず、初めてグルサ-問題においてフレドホルム性の概念が自然に導入される事を明らかにすると共に、指数公式を与えた。これは、従来の優級数に依る方法は単に可解性の証明にのみ有効であったのに対して、関数解析的手法及び結果が自然に適用されることを与える優れた方法である事を示している。
また、一般の常微分方程式系に対して、各種のジュブレイ族空間での指数公式をジュブレイ位数に付随して定義される行列式から定まる表象を用いて与えた。これらの結果は、常微分作用素及び偏微分作用素の何れに対してもフレドホルム性がテプリッツ作用素によって統一的に議論することが出来る事を示している。
この成果のもとに、不確定特異点型偏微分作用素に対してもジュブレイ族空間における指数公式を証明し、形式べき級数解の収束性を特徴付る事に成功した。これは、柏原・河合・ショストランドによる結果の類似で、テプリッツ作用素の立場からの特徴付である。
また、形式べき級数解に対する漸近解析の手始めとして、熱方程式をモデルとして形式解のボレル総和可能性を特徴付ると共に、熱核が発散級数解のボレル和の解析接続から得られる事や佐藤超関数としての表現で与えられる事を明らかにした。