Abstract

In this paper, we establish the unique existence and some decay properties of a global solution of a free boundary problem of the incompressible Navier–Stokes equations in Lp in time and Lq in space framework in a uniformly H∞2 domain Ω⊂RN for N≥4. We assume the unique solvability of the weak Dirichlet problem for the Poisson equation and the Lq-Lr estimates for the Stokes semigroup. The novelty of this paper is that we do not assume the compactness of the boundary, which is essentially used in the case of exterior domains proved by Shibata. The restriction N≥4 is required to deduce an estimate for the nonlinear term G(u) arising from divv=0. However, we establish the results in the half space R+N for N≥3 by reducing the linearized problem to the problem with G=0, where G is the right member corresponding to G(u).

Highlights

  • Navier–Stokes Equation inA free boundary problem for the viscous incompressible Navier–Stokes equations describes the motion of a fluid in time-dependent domains, such as a drop of water, an ocean of infinite extent and finite or infinite depth, or liquid around a bubble

  • The present paper is concerned with the unique existence and decay of a global solution to these problems without taking account of surface tension

  • We develop the global well-posedness and decay properties of the solution of the transformed problem (3) in general domains stated in Theorem 2

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Summary

Introduction

A free boundary problem for the viscous incompressible Navier–Stokes equations describes the motion of a fluid in time-dependent domains, such as a drop of water, an ocean of infinite extent and finite or infinite depth, or liquid around a bubble. When surface tension is taken into account, the same result was proved by Solonnikov [4] in L2 framework under the same assumptions and the additional assumption that the domain Ω is close to a ball In this case, the boundary condition should be. In the half-space and without taking account of surface tension, similar results in the L p -Lq framework have not been shown because of the non-compactness of the boundary ∂Ω. In L p -Lq framework, we establish the global well-posedness and decay properties of (1) in the half space for a sufficiently small initial velocity v0. We state our main results on the global well-posedness of (1) and decay properties of the solution in general domains with N ≥ 4 and in the half-space with N ≥ 3.

Main Results
Proof of Theorem 2
Estimate for the Stokes Problem in the General Domain
Estimate for the Nonlinear Terms in General Domain
Proof of Theorem 3
The Lq -Lr Estimates
Estimate for the Stokes Problem in the Half Space
Estimate for the Nonlinear Terms in the Half Space
Conclusions
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