Stability of generalized solutions to equations of one-dimensional motion of viscous heat-conducting gases

Abstract

ABSTRACT. Nonhomogeneous initial boundary value problems for a specific quasilinear system of equations of composite type are studied. The system describes the one-dimensional motion of a viscous perfect polytropic gas. We assume that the initial data belong to the spaces Loo(fl) or L2(fl) and the problems under consideration have generalized solutions only. For such solutions, a theorem on strong stability is proved i i.e., estimates for the norm of the difference of two solutions are expressed in terms of the sums of the norms of the differences of the corresponding data. Uniqueness of generalized solutions is a simple consequence of this theorem.KEY WORDS: strong stability of solutions, generalized solution, equation of motion of a gas, viscous heatconducting polytropic gas, quasilinear system of composite type, initial boundary value problem.Systems of equations governing the motion of a viscous gas form an interesting and important class of quasilinear systems of equations of composite type [1,2]. Results on the "global" existence and uniqueness of the regular generalized solutions of initial boundary value problems for equations of motion of a viscous heat-conducting gas with initial data from Wl(fl) is outlined in [2, Chap. 2]. A theorem on stability of such solutions in the strong norm is given in [3]. The stability of regular generalized solutions was studied in detail in [4,5] (in the finite difference version, but the same techniques apply in the differential case).The "global" existence of (weak) generalized solutions for these problems with initial data from Lq(~) with some q was obtained in [6,7] Under somewhat more rigid conditions on the initial data, uniqueness and stability of generalized solutions was also established in [6]. An earlier result on stability is contained in [8 I. It should be noted that the weaker the class of generalized solutions, the more complicated is the proof c r uniqueness and stability in it.Earlier, in [9], the authors proved the strong stability of generalized solutions for systems of equations of motion of a viscous barotropic gas. (Strong stability implies the possibility of expressing an estimate for the norm of the difference of two solutions in terms of the sums of the norms of the differences of the corresponding data.) The present paper can be regarded as a natural continuation and extension of these studies. In it we establish the strong stability of generalized solutions from [6] for systems of equations of motion of a viscous heat-conducting gas without imposing any additional conditions on the data. Three initial boundary value problems with inhomogeneous boundary conditions are studied. Uniqueness of generalized solutions from [6] is shown to be a simple consequence of stability. Moreover, the proof of one of the two results on (strong) stability announced in [6] is given. The proof essentially uses the techniques and auxiliary results outlined in [9], as well as one result from [7]. Only the new features of the proofs bearing on the specific character of t...

Stability of weak solutions to the compressible Navier–Stokes equations in bounded annular domains

“…be the weak solutions corresponding to the initial data (r 0i , 0i , v 0i , 0i ) (i = 1, 2) satisfying conditions (13). Then, the following bound…”

“…In this section, we prove Theorem 1.1 by adapting and modifying the arguments in [12][13][14]. The proof is broken up into several lemmas.…”

“…[6][7][8][9][10][11]. Recently, Zlotnik and Amosov [6,12,13] proved the uniqueness and the Lipschitz continuous dependence on data of weak solutions to initial boundary-value problems in a bounded interval, while Jiang and Zlotnik [14] obtained a similar result for the Cauchy problem.…”

“…The aim of this paper is to prove the uniqueness and Lipschitz continuous dependence on initial data of weak solutions to problem (1)- (5) with Lebesgue initial data, extending therefore the corresponding one-dimensional results in [6,[12][13][14] to the multidimensional problems with spherical symmetry in annular domains.…”

“…The calculations can be ensured by the below assumptions (13). The Eulerian coordinates (r, t) are connected to the Lagrangian coordinates (y, t) by the relation Equations (1)- (3) in the new variable (y, t) read…”