Premium
Existence of global strong solution for the compressible Navier–Stokes equations with degenerate viscosity coefficients in 1D
Author(s) -
Haspot Boris
Publication year - 2018
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201700050
Subject(s) - mathematics , compressibility , degenerate energy levels , norm (philosophy) , mathematical analysis , viscosity , navier–stokes equations , compressible flow , physics , mechanics , thermodynamics , law , quantum mechanics , political science
We consider Navier–Stokes equations for compressible viscous fluids in the one‐dimensional case. We prove the existence of global strong solution with large initial data for compressible Navier–Stokes equation with viscosity coefficients of the form∂ x ( ρ α ∂ x u )with1 2 < α ≤ 1 (it includes in particular the important physical case of the viscous shallow water system when α = 1 ). The key ingredient of the proof relies to a new formulation of the compressible equations involving a new effective velocity v (see [13][B. Haspot, 2014], [14][B. Haspot, 2016], [16][B. Haspot, 2015], [17][B. Haspot, 2016]) such that the density verifies a parabolic equation. We estimate v in L t , x ∞ norm which enables us to control the L t , x ∞ norm of 1 ρ by using the maximum principle.