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

Abstract

Abstract. In this paper we study mixture of gases, each governed by a gamma law. The system is modeled by the p-system with variable gamma. We use this model to study immiscible gas flow. The main result is that the Cauchy problem with large data is shown to have a solution. We use the Glimm scheme for the proof. The result is illustrated by numerical examples.

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

“…This solution is found as a limit of the Glimm scheme [13] or of the front tracking method [15,4]. In [16] we extended the existence result to large initial data for (1.1) by using the Glimm scheme. In this paper we prove that a front tracking algorithm converges to a weak solution, thereby giving an alternative existence argument.…”

“…This is the same functional as the Glimm functional defined in [16], and the two first terms are similar to the Glimm functional used in [21]. The linear functional F and the two quadratic functionals Q 1 and Q 2 are defined by…”

“…The only difference between the functionals above and the functionals used for the Glimm scheme in [16] is that the above ones are defined on time lines, while the functionals in [16] are defined on mesh curves.…”

“…Note also that the outgoing fronts are not altered in this process. This method corresponds to the use of inner diamonds for the Glimm scheme in [16]. DiPerna [10] constructs the outgoing solution by resolving the interaction into a composition of binary interactions.…”

“…Furthermore, only the case with two interacting fronts or waves is the same in front tracking and in the Glimm scheme. Still, the following, simple symmetry property for system (1.1) proved in [16], is useful also for the interactions in front tracking. Lemma 3.3.…”