TY - JOUR

T1 - On the Global Well-Posedness and Decay of a Free Boundary Problem of the Navier–Stokes Equation in Unbounded Domains

AU - Oishi, Kenta

AU - Shibata, Yoshihiro

N1 - Funding Information:
† Partially supported by Top Global University Project, JSPS Grant-in-aid for Scientific Research (A) 17H0109.
Publisher Copyright:
© 2022 by the authors. Licensee MDPI, Basel, Switzerland.

PY - 2022/3/1

Y1 - 2022/3/1

N2 - 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 (Formula presented). 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 div v = 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).

AB - 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 (Formula presented). 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 div v = 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).

KW - Free boundary problem

KW - General domain

KW - Global well-posedness

KW - Navier–Stokes equation

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U2 - 10.3390/math10050774

DO - 10.3390/math10050774

M3 - Article

AN - SCOPUS:85126309388

SN - 2227-7390

VL - 10

JO - Mathematics

JF - Mathematics

IS - 5

M1 - 774

ER -