Applied Mathematical Analysis of Fluid Mechanics

一般情况下，定常的Navier-Stokes方程的解的存在性是已知的，只有在严格的外流条件下或在较小的雷诺数时才是已知的。在一般出流条件下，我们研究了定常的Navier-Stokes方程和Boussinesq方程解的存在性。我们得到了两种结果。对于任意空间维度和调和函数的常p倍梯度边值问题，除了p的至多离散可数情形外，证明了上述方程解的存在性.值得注意的是，p可以是任意大的.对于二维情形，在对称假设下，Amick证明了解的存在性.Fujita得到了这种情况下边值问题的螺线对称延拓的具体构造。利用这种方法，Morimoto和Fujita得到了具有流入和流出的管状区域中定常的Navier-Stokes方程解的存在性。

The existence of solutions to the stationary Navier-Stokes equations is known, in general context, only under the stringent outflow condition or for small Raynolds number. We studied the existence of solutions to the stationary Navier-Stokes equations and of the Boussinesq equations under general outflow condition. We obtained two kinds of results. For arbitrary space dimension and for the boundary value of constant p times gradient of harmonic function, the existence of solution for the above equations is shown except for at most discrete countable case of p. It is to be noted that p can be arbitrary large.For two dimensional case, under the assumption of symmetry, Amick showed the exis tence of solutions. Fujita obtained the concrete construction of the solenoidal symmetric extension of the boundary value in this case. Using this method, Morimoto and Fujita obtained the existence of solutions to the stationary Navier-Stokes equations in a tube-like domain with inflow and outflow.