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 -