Long time existence of solutions to 2d Navier–Stokes equations with heat convection

Jolanta Socała¹ ; Wojciech M. Zaj/aczkowski²

¹ State Higher Vocational School in Racibórz Słowacki St. 55 47-400 Racibórz, Poland
² Institute of Mathematics Polish Academy of Sciences Śniadeckich 8, 00-956 Warszawa, Poland and Institute of Mathematics and Cryptology Cybernetics Faculty Military University of Technology Kaliskiego 2, 00-908 Warszawa, Poland

Applicationes Mathematicae, Tome 36 (2009) no. 4, pp. 453-463

Résumé

Global existence of regular solutions to the Navier–Stokes equations for $(v,p)$ coupled with the heat convection equation for $\theta $ is proved in the two-dimensional case in a bounded domain. We assume the slip boundary conditions for velocity and the Neumann condition for temperature. First an appropriate estimate is shown and next the existence is proved by the Leray–Schauder fixed point theorem. We prove the existence of solutions such that $v,\theta \in W_s^{2,1}({\Omega }^T)$, $\nabla p\in L_s({ \Omega }^T)$, $s>2$.

Détail

DOI : 10.4064/am36-4-5

Keywords: global existence regular solutions navier stokes equations coupled heat convection equation theta proved two dimensional bounded domain assume slip boundary conditions velocity neumann condition temperature first appropriate estimate shown existence proved leray schauder fixed point theorem prove existence solutions theta omega nabla omega