Mathematical analysis of thermal convection equations

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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 hype … More rbolic-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. Less