On existence of solutions for the nonstationary Stokes system with boundary slip conditions

Wisam Alame¹

¹ Faculty of Mathematics and Information Sciences Warsaw University of Technology Pl. Politechniki 1 00-661 Warszawa, Poland

Applicationes Mathematicae, Tome 32 (2005) no. 2, pp. 195-223.

Résumé

Existence of solutions for equations of the nonstationary Stokes system in a bounded domain $ \Omega \subset {{\Bbb R}}^3$ is proved in a class such that velocity belongs to $W_p^{2,1}({ \Omega }\times (0,T))$, and pressure belongs to $W_p^{1,0}({ \Omega }\times (0,T))$ for $p>3$. The proof is divided into three steps. First, the existence of solutions with vanishing initial data is proved in a half-space by applying the Marcinkiewicz multiplier theorem. Next, we prove the existence of weak solutions in a bounded domain and then we regularize them. Finally, the problem with nonvanishing initial data is considered.

Détail

DOI : 10.4064/am32-2-7

Keywords: existence solutions equations nonstationary stokes system bounded domain omega subset bbb proved class velocity belongs omega times pressure belongs omega times proof divided three steps first existence solutions vanishing initial proved half space applying marcinkiewicz multiplier theorem prove existence weak solutions bounded domain regularize finally problem nonvanishing initial considered