On a free boundary problem for the Navier-Stokes equations

We consider a free boundary problem for the Navier-Stokes equation in Rⁿ (n 2). We prove a local in time unique existence theorem for any initial data and a global in time unique existence theorem for some small initial data. The problem we consider in this paper was already treated by V. ~Solonnikov 15. But, recently in 10 we proved an Lₚ-Lq maximal regularity theorem for the Stokes equation with Neumann boundary condition which is a linearized version of the free boundary problem for the Navier-Stokes equation treated in this paper. Our proof is based on this theorem. Therefore our solution is obtained in the space W^2, 1ₐ, ₏ (2 < p < and n < q <) while a solution in 15 is in W^2, 1q = W^2, 1ₐ, ₐ (n < q <). Moreover, our proof of the global in time existence theorem is much simpler than 15, because in 10 we established a maximal regularity theorem on the whole time interval (0, ) with exponential stability. The results obtained in this paper were already announced in Shibata-Shimizu 11.