Differential and Integral Equations

On a free boundary problem for the Navier-Stokes equations

Yoshihiro Shibata and Senjo Shimizu

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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].

Article information

Source
Differential Integral Equations Volume 20, Number 3 (2007), 241-276.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356039501

Mathematical Reviews number (MathSciNet)
MR2293985

Zentralblatt MATH identifier
1212.35353

Subjects
Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10]
Secondary: 35R35: Free boundary problems 76D03: Existence, uniqueness, and regularity theory [See also 35Q30] 76D07: Stokes and related (Oseen, etc.) flows

Citation

Shibata, Yoshihiro; Shimizu, Senjo. On a free boundary problem for the Navier-Stokes equations. Differential Integral Equations 20 (2007), no. 3, 241--276. https://projecteuclid.org/euclid.die/1356039501.


Export citation