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

点击翻译按钮获取中文摘要

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.