TVD Fields and Isentropic Gas Flow

TL;DR

This paper investigates the existence of TVD fields in isentropic gas dynamics, showing that such fields do not exist for the system with a gamma-law pressure, highlighting limitations in controlling solution variation.

Contribution

It provides a representation result for scalar TVD fields in hyperbolic systems and proves their nonexistence in isentropic gas dynamics with gamma-law pressure.

Findings

No nontrivial TVD field exists for isentropic gas dynamics with gamma-law pressure.

The paper extends understanding of variation bounds in hyperbolic conservation laws.

Results highlight limitations in applying TVD methods to certain gas dynamics systems.

Abstract

Little is known about global existence of large-variation solutions to Cauchy problems for systems of conservation laws in one space dimension. Besides results for $L^\infty$ data via compensated compactness, the existence of global BV solutions for arbitrary BV data remains an outstanding open problem. In particular, it is not known if isentropic gas dynamics admits an a priori variation bound which applies to all BV data. In a few cases such results are available: scalar equations, Temple class systems, $2\times 2$-systems satisfying Bakhvalov's condition, and, in particular, isothermal gas dynamics. In each of these cases the equations admit a TVD (Total Variation Diminishing) field: a scalar function defined on state space whose spatial variation along entropic solutions does not increase in time. In this paper we consider strictly hyperbolic $2\times 2$-systems and derive a representation result for scalar fields that are TVD across all pairwise wave interactions, when the latter are resolved as in the Glimm scheme. We then use this to show that isentropic gas dynamics with a $\gamma$-law pressure function does not admit any nontrivial TVD field of this type.