On global solutions to the inhomogeneous, incompressible Navier–Stokes equations with temperature-dependent coefficients

In this paper, we study the initial-boundary value problem for the full inhomogeneous, incompressible Navier–Stokes equations with temperature-dependent viscosity and heat conductivity coefficients. The viscosity coefficient may be degenerate in the sense that it may vanish in the region of absolutely zero temperature. Our main result is to prove the global existence of large weak solutions to such a system. The proof is based on a three-level approximate scheme, the Galerkin method, De Giorgi’s method, and appropriate compactness arguments.

Keywords

global existence, Galerkin method, De Giorgi’s method

2010 Mathematics Subject Classification

35D30, 35Q35, 76D03, 76D05

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This work was supported by the National Natural Science Foundation of China (Grant No. 12101154), and by the Natural Science Foundation of Heilongjiang Province (Grant No. LH2022A005).

Received 9 October 2022

Accepted 11 July 2023

Published 1 February 2024