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Global C ∞ ‐solutions to 1D compressible Navier–Stokes equations with density‐dependent viscosity
Author(s) -
Ding Shijin,
Huang Jinrui,
Liu Xiaoe,
Wen Huanyao
Publication year - 2011
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1461
Subject(s) - uniqueness , mathematics , isentropic process , compressibility , bounded function , viscosity , mathematical analysis , navier–stokes equations , primitive equations , zero (linguistics) , differential equation , physics , thermodynamics , simultaneous equations , linguistics , philosophy
In this paper, we consider the existence of global smooth solutions to 1D compressible isentropic Navier–Stokes equations with density‐dependent viscosity and free boundaries. The initial density ρ 0 ∈ W 1,2 n is bounded below away from zero and the initial velocity u 0 ∈ L 2 n . The viscosity coefficient µ is proportional to ρ θ with 0<θ⩽1, where ρis the density. The existence and uniqueness of global solutions in H i ([0,1])( i = 1,2,4) have been established in ( J. Math. Phys. 2009; 50 :023101; Meth. Appl. Anal. 2005; 12 :239–252; J. Differ. Equations 2008; 245:3956–3973; Commun. Pure Appl. Anal. 2008; 7 :373–381). By mathematical induction method, we will establish the existence of global smooth solutions to 1D compressible isentropic Navier–Stokes equations with density‐dependent viscosity and free boundaries when the initial data ρ 0 and u 0 are smooth. Copyright © 2011 John Wiley & Sons, Ltd.