# Wydawnictwa / Banach Center Publications / Wszystkie tomy

## On an existence theorem for the Navier-Stokes equations with free slip boundary condition in exterior domain

### Tom 81 / 2008

Banach Center Publications 81 (2008), 457-470 MSC: Primary 35Q30; Secondary 76D05. DOI: 10.4064/bc81-0-28

#### Streszczenie

This paper deals with a nonstationary problem for the Navier-Stokes equations with a free slip boundary condition in an exterior domain. We obtain a global in time unique solvability theorem and temporal asymptotic behavior of the global strong solution when the initial velocity is sufficiently small in the sense of $L^n$ ($n$ is dimension). The proof is based on the contraction mapping principle with the aid of $L^p$-$L^q$ estimates for the Stokes semigroup associated with a linearized problem, which is also discussed. In particular, we mainly discuss the local energy decay property of the semigroup which is a key estimate to prove the $L^p$-$L^q$ estimates in an exterior domain.

#### Autorzy

School of Science and Engineering
Waseda University
3-4-1 Ōkubo, Shinjuku-ku
Tokyo 169-8555, Japan
e-mail
• Norikazu YamaguchiDepartment of Mathematical Sciences
School of Science and Engineering
Waseda University
3-4-1 Ōkubo, Shinjuku-ku
Tokyo 169-8555, Japan
e-mail

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