Dynamics of solutions near space-periodic bifurcating steady solutions of thermal convection equations

Y.Kagei证明了Obebeck-Boussinesq方程的某些定常解是无条件稳定的，即使它们处于线性化稳定性的临界点。然后，Kagei导出了考虑粘性耗散加热影响的热对流模型方程。结果表明，该模型方程的最小对流阈值比通常的Oberbeck-Boussinesq方程的最大对流阈值大，并且各种空间周期定常解在该阈值处发生转录分叉。Kagei还研究了Vlasov-Poisson-Fokker-Planck方程的柯西问题，并在一些加权Sobolev空间中构造了不变流形。由此，得到了小解的长时间渐近性。S.Kawashima研究了一般双曲-椭圆组的奇异极限问题，证明了在奇异极限下，双曲-椭圆组的解收敛于相应的双曲-抛物组的解。Kawashima还研究了半空间中离散Boltzmann方程的初边值问题，证明了在几种边界条件下定常解的存在性和渐近稳定性。T.Ogawa证明了对于一类半线性色散方程，初值具有一个奇点的解，如Dirac Delta函数，除了在初始时刻外，在空间和时间变量上都是实解析的。Ogawa还研究了三维Euler方程的爆破问题，并利用广义Besov空间的半范数给出了爆破的一个充分条件。井口研究了定常表面波的分叉问题，并对可能的分叉模式进行了分类。

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.