## Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data.(English)Zbl 0836.35120

Considered are the time dependent Navier-Stokes equations for compressible, isothermal flow in the whole space (dimension $$n = 2$$ and $$n = 3)$$, when the initial density $$\rho_0$$ is close to a constant in $$L^2$$ and $$L^\infty$$, and the initial velocity is small in $$L^2$$ and bounded in $$L^{2^n}$$. (When $$n = 2$$, the norm in $$L^2$$ must be weighted slightly.) For the special equation of state $$P (\rho) = \text{const} \rho$$, and under a slight restriction for the coefficients of viscosity, the author proves existence of weak solutions to the Cauchy problem by first mollifying the initial data, obtaining thus smooth solutions $$u^\delta$$, $$\rho^\delta$$, and then passing to the limit as $$\delta \to 0$$.
Via pointwise bounds for the density $$\rho$$, the author finds that the asymmetric part of the velocity gradient is relatively smooth while the symmetric part is not. The trace of the symmetric part plus a pressure term, however, is in fact continuous. This quantity, called “effective viscous flux”, plays a crucial role in the entire analysis. For more general equations of state, also a great deal of technical and qualitative information is obtained.

### MSC:

 35Q30 Navier-Stokes equations 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 35D05 Existence of generalized solutions of PDE (MSC2000) 35R05 PDEs with low regular coefficients and/or low regular data
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