Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data.

*(English)*Zbl 0836.35120Considered 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.

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.

Reviewer: W.Velte (Würzburg)

##### 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 |