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

Volume 39 / 2012

Jolanta Socała, Wojciech M. Zajączkowski Applicationes Mathematicae 39 (2012), 23-41 MSC: Primary 35D05; Secondary 35D10, 35K05, 35K20, 35Q30, 76D03, 76D05. DOI: 10.4064/am39-1-2


We examine the Navier–Stokes equations with homogeneous slip boundary conditions coupled with the heat equation with homogeneous Neumann conditions in a bounded domain in $\mathbb R^3$. The domain is a cylinder along the $x_3$ axis. The aim of this paper is to show long time estimates without assuming smallness of the initial velocity, the initial temperature and the external force. To prove the estimate we need however smallness of the $L_2$ norms of the $x_3$-derivatives of these three quantities.


  • Jolanta SocałaState Higher Vocational School in Racibórz
    Słowacki St. 55
    47-400 Racibórz, Poland
  • Wojciech M. ZajączkowskiInstitute 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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