Premium
Global Existence of Smooth Solutions and Convergence to Barenblatt Solutions for the Physical Vacuum Free Boundary Problem of Compressible Euler Equations with Damping
Author(s) -
Luo Tao,
Zeng Huihui
Publication year - 2016
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21562
Subject(s) - mathematics , mathematical analysis , pointwise , euler equations , ansatz , rate of convergence , compressibility , physics , mathematical physics , channel (broadcasting) , electrical engineering , thermodynamics , engineering
Abstract For the physical vacuum free boundary problem with the sound speed being C 1/2 ‐Hölder continuous near vacuum boundaries of the one‐dimensional compressible Euler equations with damping, the global existence of the smooth solution is proved, which is shown to converge to the Barenblatt self‐similar solution for the porous media equation with the same total mass when the initial datum is a small perturbation of the Barenblatt solution. The pointwise convergence with a rate of density, the convergence rate of velocity in the supremum norm, and the precise expanding rate of the physical vacuum boundaries are also given. The proof is based on a construction of higher‐order weighted functionals with both space and time weights capturing the behavior of solutions both near vacuum states and in large time, an introduction of a new ansatz, higher‐order nonlinear energy estimates, and elliptic estimates.© 2016 Wiley Periodicals, Inc.