Long time existence of solutions to 2d Navier–Stokes equations with heat convection
Tom 36 / 2009
Applicationes Mathematicae 36 (2009), 453-463
MSC: 35Q30, 35Q35, 76D03.
DOI: 10.4064/am36-4-5
Streszczenie
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 $v,\theta \in W_s^{2,1}({\Omega }^T)$, $\nabla p\in L_s({ \Omega }^T)$, $s>2$.
