Wave-front tracking for the equations of isentropic gas dynamics

Abstract

Abstract. We study the 2 × 2 system of conservation laws of the form, which are the model equations of isentropic gas dynamics. Weak global in time solutions are obtained by Nishida-Smoller (CPAM 1973) provided (γ − 1) times the total variation of the initial data is sufficiently small. The aim of this paper is to give an alternative proof by using the Dafermos-Bressan-Risebro wavefront tracking scheme. We obtain new estimates of the total amount of interactions, which also imply the asymptotic decay of the solution. The main idea is to define appropriate amplitude to the path that is a continuation of shock fronts.

Dependence of entropy solutions in the large for the Euler equations on nonlinear flux functions

“…Consider two shock curves of first family, which start from the points (r 0 , s 1 ) = (r(U 0 ), s(U 1 )) and (r 0 , s 0 ) = (r(U 0 ), s(U 0 )) and are continuous to the points (r, s 2 ) and (r, s), respectively. Then there exists a constant c 0 > 0 such that, for c ≥ c 0 , Again, following [1] (see also [24]), we obtain the wave interaction estimates corresponding to Lemma 3.4 and conclude the existence of a global entropy solution U µ to (4.1) by the front-tracking method.…”

“…This lemma enables us to introduce the Glimm functional for the approximate solutions which are constructed by the front-tracking method with the same notations as in [1]. Let J be the space-like curve and denote S j (J) as the set of j−shock waves crossing J, and S(J) = S 1 (J) ∩ S 2 (J).…”

“…Following [1] (see also [24]), the δ−approximate solutions U δ,µ are globally defined and converge to the entropy solution U µ to (3.1) as δ ↓ 0. See Lemmas 2.2 and 2.3.…”

Front tracking for a model of immiscible gas flow with large data

“…Asakura shows the convergence of front tracking for the p-system [3] and for the Euler equations [2] with large initial data. The conditions on the initial data are the same as obtained in [21] and [17].…”

The Solution of the Cauchy Problem With Large Data for a Model of a Mixture of Gases

“…Asakura applies front tracking to show existence of a solution for the p-system [3] and for the Euler equations [2] with large initial data. The conditions on the initial data are the same as obtained in [16] and [12].…”