Discontinuous solutions of the Navier-Stokes equations for multidimensional flows of heat-conducting fluids
The author considers the Navier-Stokes equations for compressible, heat-conducting fluids in \(\mathbb{R}^n (n=2,3)\), i.e. \(\rho_t+ \text{div} (\rho u)=0\), \(\rho u_t+ \rho (u\cdot \nabla) u=\text{div} \sigma\), \(\rho e_t +\rho u\cdot \nabla e= \Delta T+ \sigma^{jk} u^j_{x_k}\), where the unknowns \(\rho,u\) and \(e\) are the fluid density, velocity and specific internal energy respectively. Furthermore, \(\sigma^{jk} =\varepsilon (u^k_{x_j} +u^j_{x_k}) +((\lambda-\varepsilon) \text{div} u-P) \delta_{jk}\) denotes the stress tensor and \(P=P (\rho,e)\), \(T=T(e)\) are the pressure and the temperature. Under suitable conditions on the parameters \(\varepsilon, \lambda,P\) and \(T\), the author proves that the system has a global weak solution with initial data \((\rho_0, u_0, e_0)\) (essentially) provided that \(\| \rho_0-\widetilde \rho \|_{L^2 \cap L^\infty} +\| u_0 \|_{H^s \cap L^4} +\| e_0- \widetilde e\|_{L^2}\) is sufficiently small and \(\text{essinf} e_0>0\). Here, \(\widetilde \rho, \widetilde e\) are positive constants and \(s=0\) when \(n=2\), \(s> {1\over 3}\) when \(n=3\). In particular, the initial data are allowed to be discontinuous across a hypersurface of \(\mathbb{R}^n\). The main part of the proof consists in establishing a priori estimates for solutions of the above problem. Since the usual energy arguments are not sufficient, a lot of work is spent on the derivation of pointwise bounds for the solution \((\rho,u,e)\). In this context, the ``effective viscous flux'' \(F= (\varepsilon +\lambda) \text{div} u- P(\rho, \varepsilon)+ P(\widetilde \rho,\widetilde e)\) plays a crucial role at several steps in the analysis.
scientific article; zbMATH DE number 1096003