# A remark on a Liouville problem with boundary for the Stokes and the Navier-Stokes equations

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A remark on a Liouville problem with boundary for the Stokes and the Navier-Stokes equations Yoshikazu Giga ∗ Graduate School of Mathematical Sciences University of Tokyo 3-8-1 Komaba Meguro-ku Tokyo 153-8914, Japan and Department of Mathematics, Faculty of Sciences King Abdulaziz University P. O. Box 80203 Jeddah 21589, Saudi Arabia Abstract We construct a Poiseuille type flow which is a bounded entire solution of the nonstationary Navier-Stokes and the Stokes equations in a half space with non-slip boundary condition. Our result in particular implies that there is a nontrivial solution for the Liouville problem under the non-slip boundary condition. A review for cases of the whole space and a slip boundary condition is included. 2000 Mathematics Subject Classification. Primary: 35Q30; Secondary: 35B40, 76D05, 76D07. Key words and phrases. Liouville problem, Navier-Stokes equations, non-slip boundary condition, Poiseuille type flow. ∗Partly supported by the Japan Society for the Promotion of Science through grant Kiban (S) 21224001. 1 1 Introduction A classical Liouville theorem says that there is no bounded harmonic function in Rn other than a constant function. Non-existence of nontrivial entire solutions for a partial differential equation is often called a Liouville problem. Such a problem is important not only for classification of entire solutions but also for applications to a blow-up argument. A blow-up argument is a powerful tool to obtain a bound for solutions and their regularity. It was first introduced by De Giorgi [8] (see also [14, Theorem 8.1]) for the study of minimal surfaces. To obtain an a priori bound for a semilinear elliptic equation Gidas and Spruck [9] first introduced a blow-up argument. Among other results they obtained a bound for solutions of ∆u+ up = f, u ≥ 0, 1 < p < (n+ 2)/(n− 2) (1.1) in a smoothly bounded domain in Rn (n ≥ 2) with homogeneous Dirichlet condition, i.e. zero boundary condition, where f is a given data. The a

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