Convergence to equilibria for compressible Navier-Stokes equations with large data
This paper studies the large-time behavior of solutions to an initial-boundary value problem with periodic boundary conditions for compressible barotropic Navier-Stokes equations in \(\mathbb{R}^n\) (\(n=2,3\)). Assuming that the external force is potential and that the pressure satisfies certain conditions (e.g., in the isentropic case, the specific heat ratio \(\gamma >6/5\) for \(n=2\) and \(\gamma >3/2\) for \(n=3\)), the authors prove that for any sequence \(t_n\to\infty\) and any global (renormalized) weak solution \((\rho ,u)\) there is a subsequence \(\{s_n\}\) such that \(\rho (s_n)\to\rho_\infty\) in the space \(L^r_\omega (\Omega)\) of spatially periodic functions, \(r>1\) suitable, where \(\rho_\infty\) is an equilibrium density. Moreover, if the equilibrium is unique, then \(\rho (t)\to\rho_\infty\) in \(L^r_\omega (\Omega)\) as \(t\to\infty\). The main ingredients in the proof are careful and tricky a priori estimates and the exploitation of the energy estimate established for weak solutions.
scientific article; zbMATH DE number 1993715