Solvability ?in the large? of a system of equations of the one-dimensional motion of an inhomogeneous viscous heat-conducting gas

Some Uniform Estimates and Large-Time Behavior of Solutions to One-Dimensional Compressible Navier–Stokes System in Unbounded Domains with Large Data

“…It follows from (2.1) and (1.11) that for any α ∈ [2,3], 15) which together with Holder's inequality yields that 16) where in the second inequality we have used (2.1) and the following simple fact that for any w ∈ H 1 (Ω),…”

“…For initial boundary value problems in bounded domains the existence and uniqueness of global (generalized) solutions and the regularity have been known. Moreover, the global solution is asymptotically stable as time tends to infinity; see [1][2][3][18][19][20][21]23] among others. For the Cauchy problem (1.1)-(1.6) and the initial boundary value problems (1.1)-(1.5) (1.7) and (1.1)-(1.5) (1.8) (in unbounded domains), Kazhikhov [15] (also cf.…”

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

“…Comparing with the related known results in the case of n = 1, the mathematical difficulties for problem (7)- (12) with n 2 lie in the coefficients that depend on solutions themselves through (7). To overcome such difficulties, we adapt the arguments from [7,[11][12][13].…”

“…For initial data that are only integrable or/and in L ∞ , the existence of global weak solutions is known, see, e.g. [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.…”

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

“…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.…”