Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data. (English) Zbl 0836.76082

Summary: We extend to general polytropic pressures \(P (\rho) = K \rho^\gamma\), \( \gamma > 1\), the existence theory for isothermal \((\gamma = 1)\) flows of Navier-Stokes fluids in two and three space dimensions, with fairly general initial data. Specifically, we require that the initial density is close to a constant in \(L^2\) and \(L^\infty\), and that the initial velocity is small in \(L^2\) and bounded in \(L^{2^n}\) (in two dimensions the \(L^2\) norms must be weighted slightly). Solutions are obtained as limits of approximate solutions corresponding to mollified initial data. The key point is that the approximate densities are shown to converge strongly, so that nonlinear pressures can be accommodated, even in the absence of any uniform regularity information for the approximate densities.


76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35Q30 Navier-Stokes equations
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