Applied Mathematical Analysis of Fluid Mechanics

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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.