Global existence and asymptotic behavior of solutions to a free boundary problem for the 1D viscous radiative and reactive gas
This paper is concerning a free-boundary problem for the compressible Navier-Stokes system which describes the 1D-flows of a viscous heat-conducting radiative and reactive gas, with no restrictions on the size of initial data. The Stefan-Boltzmann radiative law is considered for pressure and internal energy. The temperature appear both in a ``perfect polytropic contribution and a fourth-order radiative part'' in the state equation (1.3), quite similar with the relation (1.16) given in [\textit{B. Ducomet} and \textit{A. Zlotnik}, Arch. Ration. Mech. Anal. 177, No. 2, 185--229 (2005; Zbl 1070.76044)], where is specified that `` this model can be shown to model correctly some astrophysical situations of interest like stellar evolution or interstellar medium dynamics''. The new element of the paper is to remove some restrictions on the specific volume, imposed in previous works. With some assumptions on the heat conductivity, the existence of a unique global solution is proved, in Sobolev spaces \(H^2\) and \(H^4\) (for reactant mass fraction). The point is to obtain uniform positive lower and upper bounds for the specific volume. For this, some important estimates given in [\textit{J. Jiang} and \textit{S. Zheng}, J. Math. Phys. 53, No. 12, 123704, 33 p. (2012; Zbl 1331.35401)] are used or adapted for the present assumptions. The time-asymptotic limits of the global solutions are also obtained. The main tools are Sobolev's, Holder's and Gronwall's inequalities, embedding theorems, interpolation inequality and Poincaré's inequality. A large list of references is given in the last part.
scientific article; zbMATH DE number 6881939
scientific article