Existence of global strong solution for the compressible Navier-Stokes system and the Korteweg system in two-dimension

This paper is dedicated to the study of viscous compressible barotropic fluids in dimension $N = 2$. We address the question of the global existence of strong solutions with large initial data for compressible Navier-Stokes system and Korteweg system with friction. In the first case we are interested by slightly extending a famous result due to V. A. Vaigant and A. V. Kazhikhov in [34] concerning the existence of global strong solution in dimension two for a suitable choice of viscosity coefficient $(\mu(\rho) = \mu > 0$ and $\lambda(\rho) = \lambda\rho^{\beta}$ with $\beta > 3)$ in the torus. We are going to weaken the condition on by assuming only $\beta > 2$ essentially by taking profit of commutator estimates introduced by Coifman et al in [7] and using a notion of effective velocity as in [34]. In the second case we study the existence of global strong solution with large initial data in the sense of the scaling of the equations for Korteweg system with degenerate viscosity coefficient and with friction term. It allows us in particular to prove the existence of global strong solution with large initial data in energy space when $N = 2$. Let us point out that these results depend in an essential way on the structure of the viscosity coefficients.

Keywords

fluids mechanics, Navier Stokes equations, Korteweg system, harmonic analysis

2010 Mathematics Subject Classification

35A01, 35D35, 35Q40, 76N10

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Published 25 September 2013