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Discontinuous solutions of the Navier-Stokes equations for multidimensional flows of heat-conducting fluids. (English) Zbl 0904.76074
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.

##### MSC:
 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 35Q30 Navier-Stokes equations 80A20 Heat and mass transfer, heat flow (MSC2010)
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