Applied Mathematical Analysis of Fluid Mechanics

一般流出条件下的Navier-Stokes方程的求解问题是具有多重连通边界的区域无法解决的问题。它以小雷诺数或在严格的流出条件下而闻名。1996年，H.Morimoto和s.u ukai获得了二维环形区域的一些结果。1997年，H.Fujita和H.Morimoto研究了边值为调和函数梯度的n维区域情况，发现即使在大雷诺数下也存在解，但有一些例外情况。此后，1998年，H.Fujita在二维对称区域得到了满足Leray型不等式的对称边值的螺线形推广，并成功地获得了Navier-Stokes方程解的先验估计。证明了解的存在性。这一结果早在1984年就已由Ch.Amich给出，但Fujita的方法更为实际和有用，并正在用于严格流出条件的情况。将此方法应用于一般流出条件下的二维无限对称通道，得到如下结果：对于对称且边界分量有限的半无限通道、V形通道和Y形通道，证明了在泊泽尔流不是很强的情况下，存在一个满足边界条件且在无限范围内趋向泊泽维尔流的解。

The problem to find a solution to the Navier-Stokes equations under the general outflow condition is unsolved problem for the domain having multiply connected boundary. It was known for small Reynolds number or under stringent outflow condition. In 1996, H.Morimoto and S.Ukai obtained some results for 2-dimensional annular domain. In 1997, H.Fujita and H.Morimoto studied the n-dimensional domain case with the boundary value which is gradient of a harmonic function and found the existence of solution even for large Reynolds number with some exceptional case, After that, in 1998, for 2-dimensionl symmetric domain, H.Fujita obtained the solenoidal extension of the symmetric boundary value satisfying Leray type inequality and succeeded to obtain an a priori estimate for solutions of Navier-Stokes equations, which proves the existence of solutions. The result was already shown by Ch.Amich in 1984, but the method of Fujita is more practical and useful and is on the way used for stringent outflow condition case. Applying this method for 2-dimensional infinite symmetric channel under general outflow condition, we obtained the follwings. For semi-infinte channel, V shaped channel and Y shaped channel, symmetric and having some finite boundary components, it is shown the existence of a solution satisfying the boundary condition and tending to Poiseuille flows in the infity if the Poiseulle flow is not so strong.