Compensated Compactness Method on Non-isentropic Polytropic Gas Flow

TL;DR

This paper develops a new compensated compactness approach to prove the global existence of bounded entropy solutions for non-isentropic polytropic gas flow, incorporating a novel relation between Riemann invariants and entropy.

Contribution

It introduces a new technique linking Riemann invariants and entropy, extending compensated compactness methods to non-isentropic gas dynamics.

Findings

Proves global existence of bounded entropy solutions.

Establishes a relation between Riemann invariants and entropy variable.

Constructs a non-classical generalized solution for the system.

Abstract

In this paper, we are concerned with a model of polytropic gas flow, which consists the mass equation, the momentum equation and a varying entropy equation. First, a new technique, to set up a relation between the Riemann invariants of the isentropic system and the entropy variable $s$, coupled with the maximum principle, is introduced to obtain the a-priori $L^{\infty}$ estimates for the viscosity-flux approximation solutions. Second, the convergence framework from the compensated compactness theory on the system of isentropic gas dynamics is applied to prove the pointwise convergence of the approximation solutions and the global existence of bounded entropy solutions for the Cauchy problem of the system with bounded initial data. Finally, as a by-product, we obtain a non-classical bounded generalized solution $(\rho,u,s),$ of the original non-isentropic polytropic gas flow, which satisfies the mass equation and the momentum equation, the entropy equation with an extra nonnegative measure in the sense of distributions.