On the existence of weak solutions to the equations of non-stationary motion of heat-conducting incompressible viscous fluids

Abstract

SUMMARYThis paper is concerned with the equations of non-stationary motion in 3D of heat-conducting incompressible viscous fluids with temperature-dependent viscosity. The conservation of internal energy includes the usual dissipation term. We prove the existence of a 'weak solution with defect measure' to the system of PDEs under consideration. Our method of proof is based on a regularization of the equations of conservation of momentum.

On Stability of Solutions to Equations Describing Incompressible Heat-Conducting Motions Under Navier’s Boundary Conditions

“…The Navier-Stokes system with temperature dependent viscosity coupled with the equation of conservation of internal energy has been studied in [3,5,7,17].…”

“…Moreover, Feireisl and Málek [7] studied the existence of weak solutions to the Navier-Stokes-Fourier system with temperature dependent viscosity. Naumann [17] proved the existence of weak solutions to system (1) 1 -(1) 3 with the Dirichlet boundary conditions and with temperature-dependent viscosity and heat conductivity coefficients. Moreover, it is assumed that the r.h.s.…”

Two-component viscous flows with density-dependent transition layer

“…In the subsequent considerations the transformation to Lagrangian coordinates is essential. The free boundary problem (12), (13), (17), (18)- (23) is transformed into a phase field problem with fixed boundary. Then we can show that regions with zero mass density in the interior of the domain are impossible.…”

Stability of two-dimensional heat-conducting incompressible motions in a cylinder

“…Equations describing nonstationary heat-conducting incompressible viscous motions have been analyzed in papers [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Equations describing nonstationary heat-conducting incompressible viscous motions have been analyzed in papers [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18].…”

“…The Navier-Stokes system with temperature-dependent viscosity coupled with the equation of conservation of internal energy has been examined in papers [3,5,7,16]. Paper [5] is concerned with the two-dimensional problem with the periodic boundary conditions, while the other mentioned papers are devoted to the three-dimensional case.…”