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2007 On a free boundary problem for the Navier-Stokes equations
Yoshihiro Shibata, Senjo Shimizu
Differential Integral Equations 20(3): 241-276 (2007).

Abstract

We consider a free boundary problem for the Navier-Stokes equation in $\mathbb R^n$ ($n \geqq 2$). We prove a local in time unique existence theorem for any initial data and a global in time unique existence theorem for some small initial data. The problem we consider in this paper was already treated by V.~Solonnikov [15]. But, recently in [10] we proved an $L_p$-$L_q$ maximal regularity theorem for the Stokes equation with Neumann boundary condition which is a linearized version of the free boundary problem for the Navier-Stokes equation treated in this paper. Our proof is based on this theorem. Therefore our solution is obtained in the space $W^{2,1}_{q,p}$ ($2 < p < \infty$ and $n < q < \infty$) while a solution in [15] is in $W^{2,1}_q = W^{2,1}_{q,q}$ ($n < q < \infty$). Moreover, our proof of the global in time existence theorem is much simpler than [15], because in [10] we established a maximal regularity theorem on the whole time interval $(0, \infty)$ with exponential stability. The results obtained in this paper were already announced in Shibata-Shimizu [11].

Citation

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Yoshihiro Shibata. Senjo Shimizu. "On a free boundary problem for the Navier-Stokes equations." Differential Integral Equations 20 (3) 241 - 276, 2007.

Information

Published: 2007
First available in Project Euclid: 20 December 2012

zbMATH: 1212.35353
MathSciNet: MR2293985

Subjects:
Primary: 35Q30
Secondary: 35R35 , 76D03 , 76D07

Rights: Copyright © 2007 Khayyam Publishing, Inc.

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Vol.20 • No. 3 • 2007
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