A blow-up criterion of strong solutions to the compressible viscous heat-conductive flows with zero heat conductivity
In this article is considered the compressible viscous heat-conductive Navier-Stokes (HCNS) system with zero heat conductivity on the torus \({\mathbb T}^3\) for the density \(\rho\), the velocity \(\vec{v}\), and the temperature \(\theta\), given by the equations \[ \partial_t\rho+ \text{div}\;\rho\vec{v}=0, \] \[ \partial_t(\rho \vec{v})+ \text{div}\;(\rho\vec{v}\otimes \vec{v})= -\nabla p+\mu \Delta\vec{v}+(\lambda+\mu)\nabla\text{div}\;\vec{v}, \] \[ \partial_t(\rho e)+ \text{div}\;(\rho \vec{v}e)+p\;\text{div}\;\vec{v} =\frac{\mu}{2}|\nabla \vec u+\nabla^t\vec u|^2+\lambda (\text{div}\;\vec{v})^2, \] with initial conditions \[ \left. (\rho,\vec v,\theta)\right|_{t=0}=(\rho_0,\vec v_0,\theta_0)\;\;\text{in}\;{\mathbb T}^3, \] where the pressure is \(p=R\rho\theta\) and the internal energy is \(e=c_V\theta\). The authors prove the following blow-up result \textbf{Theorem:} Suppose that the initial data satisfy \[ \inf \rho_0>0,\;\;\rho\in W^{1,q}\;\text{for some}\;\;3<q<6,\;\vec v_0\in H^2,\;\theta_0\geq 0,\;\theta_0\in W^{1,q}. \] Then there exist a positive constant \(T_0\) and a unique strong solution \((\rho,\vec v,\theta)\) of (HCNS) such that \[ \inf \rho>0,\;\;\rho\in C([0,T_0];W^{1,q}),\;\;\partial_t\rho\in C([0,T_0];L^q), \] \[ \vec u\in C([0,T_0];H^2)\cap L^2(0,T_0;H^1), \] \[ \inf \theta>0,\;\;\theta\in C([0,T_0];W^{1,q}),\;\;\partial_t\theta\in C([0,T_0];L^q). \] Let \(T_*\) the maximal time of existence of \((\rho,\vec v,\theta)\). If \(T_*<\infty\), then we have \[ \lim_{T\rightarrow T_*} \left\| (\rho,\rho^{-1},\theta)\right\|_{L^{\infty}(0,T;L^{\infty})}=\infty, \] provided that \(25\mu-3\lambda>0\). This result extends to the zero heat conductivity case previous results involving non-zero Fourier fluxes.
scientific article; zbMATH DE number 5988276
scientific article