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Compactness of bounded quasientropy solutions to the system of equations of an isothermal gas. (Russian, English) Zbl 1034.76050
Sib. Mat. Zh. 44, No. 2, 459-472 (2003); translation in Sib. Math. J. 44, No. 2, 366-377 (2003).
The author introduces a new notion of solution (which is called a quasientropy solution) to the system of equations of an isothermal gas with planar waves \[ u_t = -p_x,\quad v_t = u_x,\quad p = \frac{k}{v},\quad (t,x) \in Q = (0,T)\times\mathbb R, \] in the Lagrange variables \((x,t)\) under the initial value conditions \[ \left.u\right| _{t=0} = u_0(x),\quad \left.v\right| _{t=0} = v_0(x)\geq 0. \] The main aim is to prove compactness of the set of quasientropy solutions satisfying the inequalities \[ | u| \leq c_0,\quad 0 < c_0^{-1} \leq v \leq c_0, \] where \(c_0\) is a fixed positive number. The author uses the method of compensated compactness based on the well-known div-curl-Lemma by F. Murat.
The rest of the article is devoted to the reduction of the Yang measures to the Dirac measures which appear in the technique developed for proving compactness of bounded quasientropy solutions.

MSC:
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35L45 Initial value problems for first-order hyperbolic systems
35B45 A priori estimates in context of PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
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