Applied Mathematical Analysis of Fluid Mechanics

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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.