From the highly compressible Navier-Stokes equations to the porous medium equation -- rate of convergence

Ewelina Zatorska, Boris Haspot
2015 Discrete and Continuous Dynamical Systems. Series A  
We consider the one-dimensional Cauchy problem for the Navier-Stokes equations with degenerate viscosity coefficient in highly compressible regime. It corresponds to the compressible Navier-Stokes system with large Mach number equal to ε −1/2 for ε going to 0. When the initial velocity is related to the gradient of the initial density, the densities solving the compressible Navier-Stokes equations -ρε converge to the unique solution to the porous medium equation [14, 13] . For viscosity
more » ... r viscosity coefficient µ(ρε) = ρ α ε with α > 1, we obtain a rate of convergence of ρε in L ∞ (0, T ; H −1 (R)); for 1 < α ≤ 3 2 the solution ρε converges in L ∞ (0, T ; L 2 (R)). For compactly supported initial data, we prove that most of the mass corresponding to solution ρε is located in the support of the solution to the porous medium equation. The mass outside this support is small in terms of ε. ∇uε+∇ t uε 2 denotes the symmetric part of the gradient of u ε , µ( ε ) and λ( ε ) denote the two Lamé viscosity coefficients satisfying µ( ε ) > 0, 2µ( ε ) + N λ( ε ) ≥ 0, where N is the space dimension.
doi:10.3934/dcds.2016.36.3107 fatcat:kgulekigvvgjdm6zb3cbf42ysu