Mathematical analysis of thermal convection equations

Y.Kagei和T.Kobayashi研究了可压缩Navier-Stokes方程在半空间上的等密度不动平衡的稳定性，并给出了线性化问题的解公式，从而导出了线性化问题解的衰减估计。将这些结果与能量法相结合，他们得到了扰动的衰减估计。结果还表明，柯西问题中可能存在一些未出现的非线性相互作用现象。Kagei研究了热对流运动的非齐次Navier-Stokes方程。他证明了整体弱解的存在性，并研究了所考虑方程的Oberbeck-Boussinesq极限。Kobayashi研究了Maxwell方程、Stokes方程和Navier-Stokes方程解的局部界面正则性。S. Kawashima证明了一般双曲-椭圆型方程组的解在大时间内可由相应的曲线解近似。川岛还建立了$W^{1。p}$-能量法求解多维粘性守恒定律，得到了尖锐的$W^{1。P}$衰变估计。Kawashima给出了平衡定律双曲系统的熵的概念，使我们能够理解系统的耗散结构。T.Ogawa将对数Sobolev不等式推广到齐次和非齐次Bosev空间。利用这些不等式，他改进了不可压缩Navier-Stokes方程、Euler方程和调和流解的正则性的serrin型条件。Ogawa还证明了漂移扩散方程解的有限时间爆破性。T.Iguchi研究了水波的分岔问题，并用表示域底部的傅里叶系数对分岔模式进行了分类。井口还研究了一般通量的守恒定律。他引入了“分段真正非线性”的概念，并构造了小初始值的熵解。少

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