Construction of a Lyapunov functional for 1D-viscous compressible barotropic fluid equations admitting vacua |
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Authors: | Patrick Penel Ivan Straškraba |
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Institution: | a Université du Sud Toulon-Var, Mathématique & Laboratoire S.N.C., BP 20132, 83957 La Garde, France b Mathematical Institute, Academy of Sciences of the Czech Republic, ?itná 25, 115 67 Praha 1, Czech Republic |
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Abstract: | The Navier-Stokes equations for a compressible barotropic fluid in 1D with zero velocity boundary conditions are considered. We study the case of large initial data in H1 as well as the mass force such that the stationary density is uniquely determined but admits vacua. Missing uniform lower bound for the density is compensated by a careful modification of the construction procedure for a Lyapunov functional known for the case of solutions which are globally away from zero I. Straškraba, A.A. Zlotnik, On a decay rate for 1D-viscous compressible barotropic fluid equations, J. Evol. Equ. 2 (2002) 69-96]. An immediate consequence of this construction is a decay rate estimate for this highly singular problem. The results are proved in the Eulerian coordinates for a large class of increasing state functions including p(ρ)=aργ with any γ>0 (a>0 a constant). |
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Keywords: | 35Q30 35B40 76N15 |
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