On the Lp-Lq decay estimate for the Stokes equations with free boundary conditions in an exterior domain

Article type: Research Article

Authors: Shibata, Yoshihiro^{a; b; c}

Affiliations: [a] Department of Mathematics, Waseda Univeristy, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan. E-mail: yshibata@waseda.jp | [b] Research Institute of Sciences and Engineering, Waseda Univeristy, Japan | [c] Department of Mechanical Engineering and Materials Science, University of Pittsburgh, USA

Abstract: This paper deals with the Lp-Lq decay estimate of the C0 analytic semigroup {T(t)}t⩾0 associated with the perturbed Stokes equations with free boundary conditions in an exterior domain. The problem arises in the study of free boundary problem for the Navier–Stokes equations in an exterior domain. We proved that ‖∇jT(t)f‖Lp⩽Cp,qt−j2−N2(1q−1p)‖f‖Lq(j=0,1) provided that 1<q⩽p⩽∞ and q≠∞. Compared with the non-slip boundary condition case, the gradient estimate is better, which is important for the application to proving global well-posedness of free boundary problem for the Navier–Stokes equations. In our proof, it is crucial to prove the uniform estimate of the resolvent operator, the resolvent parameter ranging near zero.

Keywords: Exterior domains, Stokes equations, free boundary problem, without surface tension, L_p-L_q decay estimate

DOI: 10.3233/ASY-171449

Journal: Asymptotic Analysis, vol. 107, no. 1-2, pp. 33-72, 2018