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

Volume 39 / 2012

Jolanta Socała, Wojciech M. Zajączkowski Applicationes Mathematicae 39 (2012), 231-242 MSC: Primary 35D05; Secondary 35D10, 35K05, 35K20, 35Q30, 76D03, 76D05. DOI: 10.4064/am39-2-8


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.


  • 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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