A unified duality framework for barotropic, quantum and Korteweg fluids

TL;DR

This paper develops a unified dual variational framework for multiple fluid models, proving existence and consistency of solutions, and establishing a principle that constrains entropy dissipation rates.

Contribution

It introduces a novel abstract duality framework applicable to various fluid models, ensuring solution existence and a new entropy dissipation principle.

Findings

Unified duality framework for multiple fluid models

Existence of dual solutions for initial data in Radon measures

No duality gap and a Dafermos entropy dissipation principle

Abstract

We investigate a dual variational formulation, in the spirit of Brenier, for several compressible fluid models: the compressible barotropic Euler system, the quantum Euler system, and the Euler-Korteweg system. We identify a unified abstract framework encompassing all three systems, which enables a simultaneous analysis. By introducing time-adaptive weights, we establish the consistency of the duality scheme on large time intervals. We prove the existence of variational dual solutions to the corresponding Cauchy problems for continuous, vacuum-free initial data in spaces of finite Radon measures, and establish the absence of a duality gap. As an application, we formulate and prove a 'Dafermos principle' for these models: no subsolution can dissipate the total entropy earlier or at a faster rate than the corresponding strong solution on its interval of existence. We also discuss connections between our abstract consistency result and Brenier's shock-free substitutes for entropy solutions of Burgers' equation.