This paper concerns the free boundary problem for the Navier-Stokes equations without surface tension in the Lp in time and Lq in space setting with 2<p<∞ and N<q<∞. A local in time existence theorem is proved in a uniform Wq2-1/q domain in the N-dimensional Euclidean space RN (N≥2) under the assumption that the weak Dirichlet-Neumann problem is uniquely solvable. Moreover, a global in time existence theorem is proved for small initial data under the additional assumption that Ω is bounded. This was already proved by Solonnikov  by using the continuation argument of local in time solutions which are exponentially stable in the energy level under the assumption that the initial data is orthogonal to the rigid motion. We also use the continuation argument and the same orthogonality condition for the initial data. But, our argument about the continuation of local in time solutions is based on some decay theorem for the linearized problem, which is a different point than .
- Bounded domain
- Free boundary problem
- Global in time unique existence theorem
- Local in time unique existence theorem
- Navier-Stokes equations
- Uniform Wq2-1/q domain
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