Global smooth solutions for the compressible viscous and heat-conductive gas

Abstract

Abstract. This paper is concerned with the global existence of smooth solutions to a system of equations describing one-dimensional motion of a self-gravitating, radiative and chemically reactive gas. We have proved that for any arbitrary large smooth initial data, the problem under consideration admits a unique globally smooth (classical) solution. Our results have improved those results by Umehara and Tani ([J. Differential Equations,

Global Existence and Uniqueness of Nonlinear Evolutionary Fluid Equations

“…Umehara and Tani [129], Qin et al [103] and Qin, Hu and Wang [104] proved the global existence of smooth solutions for a self-gravitating radiative and reactive gas.…”

“…Our methods are mainly based on the techniques in Qin [101,[103][104][105], that is, we carefully estimate the solution and its higher derivatives in terms of functions A, X, Y and Z (see their definitions below) and resort to delicate interpolation techniques. …”

A regularizing effect of radiation in one-dimensional compressible MHD equations

“…It turns out that the key steps are to bound the L 2 -norm of the gradient of the density and to estimate the upper bound of the temperature. To achieve these, the growth condition (1.12) with suitably large q > 0 in the previous works [25][26][27][28][30][31][32] is technically needed for their analysis. So, in the case that the heat conductivity is only a positive constant, the situation becomes somewhat different and some new ideas have to be developed.…”

“…. for the studies of other 1D models for radiative gases, among all of which the growth condition (1.12) with different exponent q > 0 was required; see, for instance, q ≥ 4 in [27,28,31], q ≥ 2 in [30,32], and q ≥ 1 in [29]. For radiative gas-dynamics, it was pointed out in [27,28] that the growth condition (1.12) with suitably large q > 0 plays a key role in the proof of the global a priori estimates.…”

“…θ 4 ) in (1.8) will cause some new difficulties in the study of radiative gasdynamics, and hence, the growth condition (1.12) with suitably large q > 0 is technically needed in the previous works to obtain some better estimates of the temperature (see, e.g. [25][26][27][28][29][30][31][32]). Motivated by the results achieved, it is therefore mathematically interesting to improve or remove such growth restriction on the heat conductivity, although (1.12) with q ≥ 3 is physically valid for radiative gases in LTE (see [16]).…”