Lyapunov Functional Method for 1D Radiative and Reactive Viscous Gas Dynamics

We consider the compressible Navier–Stokes system for 1D-flows of a viscous heat-conducting gas, with the pressure law and a one-order kinetics to include radiative effects and reactive processes. The mass force and the ignition phenomenon are also taken into account. For large data and under general assumptions on the heat conductivity, we establish global-in-time bounds and exponential stabilization for solutions in Lq and H1 norms. To this end, we construct new global Lyapunov functionals and show that they describe the dynamics of solutions for any t≧0. A short proof of the corresponding global existence is also included for completeness.

[1]  Jerrold Bebernes,et al.  Mathematical Problems from Combustion Theory , 1989 .

[2]  A. Zlotnik.  On equations for one-dimensional motion of a viscous barotropic gas in the presence of a body force , 1992 .

[3]  Song Jiang.  On the asymptotic behavior of the motion of a viscous, heat-conducting, one-dimensional real gas , 1994 .

[4]  B. Ducomet.  SOME ASYMPTOTICS FOR A REACTIVE NAVIER–STOKES–POISSON SYSTEM , 1999 .

[5]  A. Bressan,et al.  Global a priori estimates for a viscous reactive gas , 1985, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[6]  P. Secchi.  On the uniqueness of motion of viscous gaseous stars , 1990 .

[7]  A. Amosov,et al.  Semidiscrete method of solving the quasiaveraged equations of one-dimensional motion of a viscous heat-conducting gas , 1997 .

[8]  Gui-Qiang G. Chen.  Global solutions to the compressible Navier-Stokes equations for a reacting mixture , 1992 .

[9]  S. N. Antont︠s︡ev,et al.  Boundary Value Problems in Mechanics of Nonhomogeneous Fluids , 1990 .

[10]  Yuming Qin.  Global existence and asymptotic behavior for a viscous, heat-conductive, one-dimensional real gas with fixed and constant temperature boundary conditions , 2002 .

[11]  P. Lions.  Mathematical topics in fluid mechanics , 1996 .

[12]  B. Kawohl.  Global existence of large solutions to initial boundary value problems for a viscous, heat-conducting, one-dimensional real gas , 1985 .

[13]  Stabilization for equations of one-dimensional viscous compressible heat-conducting media with nonmonotone equation of state , 2002, math-ph/0206005.

[14]  Pierre-Louis Lions,et al.  Mathematical Topics in Fluid Mechanics: Volume 2: Compressible Models , 1998 .

[15]  A. Amosov,et al.  Solvability “in the large” of a system of equations of the one-dimensional motion of an inhomogeneous viscous heat-conducting gas , 1992 .

[16]  L. Hsiao,et al.  Large-time behaviour of solutions for the outer pressure problem of a viscous heat-conductive one-dimensional real gas , 1996 .

[17]  B. Ducomet.  A model of thermal dissipation for a one-dimensional viscous reactive and radiative gas , 1999 .

[18]  Yuming Qin.  Global existence and asymptotic behavior for a viscous, heat-conductive, one-dimensional real gas with fixed and thermally insulated endpoints , 2001 .

[19]  A. Zlotnik,et al.  Properties and asymptotic behavior of solutions of some problems of one-dimensional motion of a viscous barotropic gas , 1994 .

[20]  Sergei F. Shandarin,et al.  The large-scale structure of the universe: Turbulence, intermittency, structures in a self-gravitating medium , 1989 .

[21]  A. Zlotnik.  Global behaviour of 1‐D viscous compressible barotropic flows with free boundary and self‐gravitation , 2003 .

[22]  On the Navier-Stokes Equations for Exothermically Reacting Compressible Fluids , 2002 .

[23]  Paolo Secchi,et al.  On the motion of gaseous stars in the presence of radiation , 1990 .

[24]  O. Ladyženskaja.  Linear and Quasilinear Equations of Parabolic Type , 1968 .

[25]  A. A. Zlotnik.  Stress and heat flux stabilization for viscous compressible medium equations with a nonmonotone state function , 2003, Appl. Math. Lett..

[26]  B. Guo,et al.  Asymptotic Behavior of the Solution to the System for a Viscous Reactive Gas , 1999 .

[27]  Shigenori Yanagi.  Asymptotic stability of the solutions to a full one-dimensional system of heat-conductive, reactive, compressible viscous gas , 1998 .

[28]  B. Ducomet,et al.  Stabilization for viscous compressible heat-conducting media equations with nonmonotone state functions , 2002 .

[29]  Achim Weiss,et al.  Stellar Structure and Evolution , 1990 .

[30]  D. Mihalas,et al.  Foundations of Radiation Hydrodynamics , 1985 .

[31]  David Hoff,et al.  Global Solutions to a Model for Exothermically Reacting, Compressible Flows with Large Discontinuous Initial Data , 2003 .

[32]  Song Jiang.  On initial boundary value problems for a viscous, heat-conducting, one-dimensional real gas , 1994 .