## Differential and Integral Equations

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

### On a free boundary problem for the Navier-Stokes equations

Yoshihiro Shibata and Senjo Shimizu

#### 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