Long time existence of regular solutions to 3d Navier–Stokes equations coupled with heat convection

Jolanta Socała¹ ; Wojciech M. Zajączkowski²

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 39 (2012) no. 2, pp. 231-242.

Résumé

We prove long time existence of regular solutions to the Navier–Stokes equations coupled with the heat equation. We consider the system in a non-axially symmetric cylinder, with the slip boundary conditions for the Navier–Stokes equations, and the Neumann condition for the heat equation. The long time existence is possible because the derivatives, with respect to the variable along the axis of the cylinder, of the initial velocity, initial temperature and external force are assumed to be sufficiently small in the $L_2$ norms. We prove the existence of solutions such that the velocity and temperature belong to $W_\sigma ^{2,1}(\varOmega \times (0,T))$, where $\sigma >{5/3}$. The existence is proved by using the Leray–Schauder fixed point theorem.

Détail

DOI : 10.4064/am39-2-8

Keywords: prove long time existence regular solutions navier stokes equations coupled heat equation consider system non axially symmetric cylinder slip boundary conditions navier stokes equations neumann condition heat equation long time existence possible because derivatives respect variable along axis cylinder initial velocity initial temperature external force assumed sufficiently small norms prove existence solutions velocity temperature belong sigma varomega times where sigma existence proved using leray schauder fixed point theorem