Construction of a Lyapunov functional for 1D-viscous compressible barotropic fluid equations admitting vacua

0352455 - MÚ 2011 RIV FR eng J - Journal Article
Penel, P. - Straškraba, Ivan
Construction of a Lyapunov functional for 1D-viscous compressible barotropic fluid equations admitting vacua.
Bulletin des Sciences Mathématiques. Roč. 134, č. 3 (2010), s. 278-294. ISSN 0007-4497. E-ISSN 1952-4773
R&D Projects: GA ČR GA201/08/0012
Institutional research plan: CEZ:AV0Z10190503
Keywords : compressible fluid * Navier-Stokes equations * asymptotic behavior
Subject RIV: BA - General Mathematics
Impact factor: 0.722, year: 2010
http://www.sciencedirect.com/science/article/pii/S0007449709000153

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 Hi 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. Straskraba, 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 rho(rho) = a rho(gamma) with any gamma > 0 (a > 0 a constant).
Permanent Link: http://hdl.handle.net/11104/0191960