Global well-posedness for the 2D incompressible heat conducting Navier-Stokes equations with temperature-dependent coefficients and vacuum

TL;DR

This paper proves the global existence of strong solutions for 2D incompressible heat conducting Navier-Stokes equations with temperature-dependent coefficients and vacuum, showing exponential decay of velocity and temperature over time.

Contribution

It establishes the first global well-posedness results for this class of nonisentropic systems with vacuum and temperature-dependent viscosity and heat conductivity.

Findings

Global existence of strong solutions under smallness conditions on parameters

Solutions exhibit exponential decay in velocity and temperature

Initial data can be arbitrarily large with vacuum presence

Abstract

We consider the initial boundary problem of 2D non-homogeneous incompressible heat conducting Navier-Stokes equations with vacuum, where the viscosity and heat conductivity depend on temperature in a power law of Chapman-Enskog. We derive the global existence of strong solution to the initial-boundary value problem, which is not trivial, especially for the nonisentropic system with vacuum. Significantly, our existence result holds for the cases that the viscosity and heat conductivity depend on $\theta$ with possibly different power laws (i.e., $\mu=\theta^{\alpha}, \kappa=\theta^{\beta}$ with constants $ \alpha,\beta\geq0$) with smallness assumptions only on $\alpha$ and the measure of initial vacuum domain. In particular, the initial data can be arbitrarily large. Moreover, it is obtained that both velocity and temperature decay exponentially as time tends to infinity.