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

热对流方程空间周期分岔稳态解附近解的动力学

基本信息

批准号：
11640208
负责人：
KAGEI Yoshiyuki
金额：
$ 2.43万
依托单位：
KYUSHU UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640208/
关键词：
Oberbeck-Boussinesq equation Bifurcation Asympotic behavior 解の安定性解の慚正挙動

项目摘要

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.

Y.Kagei 证明 Obebeck-Boussinesq 方程的一些平稳解即使处于线性稳定性的临界点也是无条件稳定的。随后，Kagei 推导出了一个热对流模型方程，其中考虑了粘性耗散加热的影响。结果表明，该模型方程的对流一阈值大于通常的Oberbeck-Boussinesq方程的阈值，并且各种空间周期稳态解在该阈值处发生跨临界分叉。 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。结果，导出了小解的长时间渐近。 S.Kawashima研究了一般双曲-椭圆系统的奇异极限问题，证明了在奇异极限下双曲-椭圆系统的解收敛于相应的双曲-抛物线系统的解。川岛还研究了半空间中离散玻尔兹曼方程的初始边值问题，并证明了几种边界条件下平稳解的存在性及其渐近稳定性。 T.Okawa 证明，对于一类半线性色散方程，像狄拉克 δ 函数那样，初始值具有一个奇异点的解在空间和时间变量上变为实解析，除了初始时间。小川还研究了三维欧拉方程的爆炸问题，并根据广义贝索夫空间的某些半范数给出了爆炸的充分条件。 T.Iguchi 研究了稳态表面波的分岔问题并对可能的分岔模式进行了分类。