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

Volume 36 / 2009

Jolanta Socała, Wojciech M. Zaj/aczkowski Applicationes Mathematicae 36 (2009), 453-463 MSC: 35Q30, 35Q35, 76D03. DOI: 10.4064/am36-4-5


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 $v,\theta \in W_s^{2,1}({\Omega }^T)$, $\nabla p\in L_s({ \Omega }^T)$, $s>2$.


  • Jolanta SocałaState Higher Vocational School in Racibórz
    Słowacki St. 55
    47-400 Racibórz, Poland
  • Wojciech M. Zaj/aczkowskiInstitute of Mathematics
    Polish Academy of Sciences
    Śniadeckich 8, 00-956 Warszawa, Poland
    Institute of Mathematics and Cryptology
    Cybernetics Faculty
    Military University of Technology
    Kaliskiego 2, 00-908 Warszawa, Poland

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