A bound from below for the temperature in compressible Navier-Stokes equations
The paper considers the following full system of compressible Navier-Stokes equations for heat conducting fluid, filling a bounded three-dimensional domain \(\Omega\) in a time interval \([0,T]\): \[ \partial_t\rho + \text{div}(\rho u)=0, \] \[ \partial_t(\rho\theta) + \text{div}(\rho u \theta) - \text{div}(\kappa \nabla\theta)+R\rho\theta \operatorname{div}u=2\mu|Du|^2 + \lambda |\operatorname{div}u|^2, \] \[ \partial_t(\rho u) + \text{div}(\rho u u) - \text{div}(2\mu Du + \lambda \operatorname{div}u Id)+\nabla P(\rho\theta)=0, \] representing respectively the continuity, the temperature and the momentum equation, where \(\rho\geq 0\), \(\theta\geq 0\) and \(u(t,x)\in\mathbb R^3\) denotes respectively the density, temperature and velocity field of the fluid, and the pressure law is \[ P(\rho\theta)=p_e(\rho) + R\rho\theta, \] with \(\mu,\lambda\) the Lamé coefficients (depending on \(\rho\) and \(\theta\)). This PDE system is ended with the non-slip condition for the velocity and homogeneous Neumann condition for the temperature. Finally, initial conditions are denoted by \(\rho_0\), \(\theta_0\) and \(u_0\) respectively. Assuming the existence of the so-called admisible solutions, which satisfying the following energy (or entropy) inequality for such equations: \[ \frac{d}{dt}\int_\Omega \rho \varphi(\theta) - \int_\Omega \mu\varphi'(\theta)|Du|^2 - \int_\Omega \lambda \varphi'(\theta)|\operatorname{div}u|^2 + \int_\Omega \kappa\varphi ''(\theta)|\nabla\theta|^2 \leq -R \int_\Omega \rho\theta \varphi'(\theta)\operatorname{div}u \] for some appropiate functions \(\varphi\), under reasonable assumptions on the coefficients \[ \mu>0,\quad 2\mu+3\lambda\geq \nu(\theta)>0 \quad \forall \theta, \] \[ \nu(\theta)\geq C\theta, \quad \kappa(\theta)\geq \kappa_{\min}>0, \quad \text{for small }\theta, \] and on the initial data \[ \int_\Omega \rho_0 [\ln (1/\rho_0)]_+ < \infty \] (which imply in particular that the initial temperature may not vanish on an open set), the paper proves that the temperature cannot vanish for positive times and is uniformly bounded away from zero on any time interval \([t_0,T]\) for any \(t_0:0<t_0<T\). The proof is inspired by De Giorgi's proof of Hölder regularity for the solutions of the elliptic equation with discontinuous coefficients.
scientific article; zbMATH DE number 5571615