Global well-posedness of unsteady motion of viscous incompressible capillary liquid bounded by a free surface

Yoshihiro Shibata

doi:10.3934/eect.2018007

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2018, Volume 7, Issue 1: 117-152. Doi: 10.3934/eect.2018007

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Global well-posedness of unsteady motion of viscous incompressible capillary liquid bounded by a free surface

Yoshihiro Shibata

Department of Mathematics and Research Instituteof Science and Engineering, Waseda University, Ohkubo 3-4-1, Shinjuku-ku, Tokyo 169-8555, Japan

Received: April 30, 2016

Revised: July 31, 2017

Published: February 2018

Partially supported by JSPS Grant-in-aid for Scientific Research (A) 17H0109 and Top Global University Project. Adjunct faculty member in the Department of Mechanical Engineering and Materials Science, University of Pittsburgh

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In this paper, we prove the global well-posedness of free boundary problems of the Navier-Stokes equations in a bounded domain with surface tension. The velocity field is obtained in the $L_p$ in time $L_q$ in space maximal regularity class, ($2 < p < ∞$, $N < q < ∞$, and $2/p + N/q < 1$), under the assumption that the initial domain is close to a ball and initial data are sufficiently small. The essential point of our approach is to drive the exponential decay theorem in the $L_p$-$L_q$ framework for the linearized equations with the help of maximal $L_p$-$L_q$ regularity theory for the Stokes equations with free boundary conditions and spectral analysis of the Stokes operator and the Laplace-Beltrami operator.

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Mathematics Subject Classification: Primary: 35R35; Secondary: 35Q30, 76D05, 76D03.

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