Bifurcation and structure of solutions for fluid dynamical systems

我们主要关注的是描述可压缩热对流运动的流体动力学系统，在重力作用下沿倾斜平面或垂直平面向下的粘性不可压缩流。通过构造预解算子并估计其范数，证明了完全非线性问题解的存在性，研究了解的分歧结构和解的稳定性。要证明分支解的存在性和渐近性，关键是这些预解算子在适当的函数空间中生成一个解析半群。证明了可压缩热对流方程定常解的存在性，并从热传导状态得到了分支分支。

We are concerned mainly with fluid dynamical systems describing motions for compressible heat convection, viscous incompressible flow down an inclined plane or down a vertical plane under the gravity. After constructing resolvent operators and estimating their norms we proved existence of solutions to full nonlinear problems and studied bifurcation structure and stability of solutions. To show existence proof and asymptotic property of bifurcating solutions, it is the key point that those resolvent operator generate an analytic semigroup in a suitable function space. We proved the existence of stationary solution to the compressible heat convection and obtained the bifurcating branch from the heat conducting state.