On the global well-posedness and decay of a free boundary problem of the Navier-Stokes equation in unbounded domains

arXiv2202.11304MaRDI QIDQ6391910

Kenta Oishi, Yoshihiro Shibata

Publication date: 22 February 2022

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 $L_p$ in time and $L_q$ in space framework in a uniformly $H_i^{2}nfty$ domain $OmegasubsetB{R}^N$ for $Ngeq4$. We assume the unique solvability of the weak Dirichlet problem for the Poisson equation and the $L_q$-$L_r$ 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 cite{Shiba17CIME}. The restriction $Ngeq4$ is required to deduce an estimate for the nonlinear term $$ arising from $$. However, we establish the results in the half space $hsp$ for $Ngeq3$ by reducing the linearized problem to the problem with $$, where $$ is the right member corresponding to $$.

Mathematics Subject Classification ID

Navier-Stokes equations for incompressible viscous fluids (76D05) Navier-Stokes equations (35Q30)

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