Mathematical analysis of interface problems in mathematical physics

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In this research, we consider the Stokes equation with Neumann boundary condition which is obtained as a linearized equation of the free boundary problem for the Navier-Stokes equation. We analyzed this problem by the following procedure : (1) Analysis of the resolvent problem (2) Generation of Analytic semigroups (3) L_p-L_q estimates(1)Obtained is the L_p estimate of solutions to the resolvent problem for Stokes system with Neumann type boundary condition in a bounded or exterior domain in R^n. The result has been obtained by Grubb and Solonnikov by the systematic use of theory of pseudo-differential operators. In this paper, we give an essentially different proof from theirs. The core of my approach is to estimate the solutions in the whole space and half-space case. We apply the Fourier multiplier theorem to solution of the model problems.(2)First we introduce the Helmholtz decomposition. Then we delete pressure trem and reduce to the problem only including velocity vector. Then we generated analytic semigroup to this reduced Stokes equation.(3)We obtained local energy decay estimates and L_p-L_q estimates of the solutions to the Stokes equation with Neumann boudary condition. Comparing with the non-slip (Dirichlet) boundary condition case, we have a better decay estimate for the gradient of the semigroup because of the null net force at the boundary.