A unified duality framework for barotropic, quantum and Korteweg fluids
Dmitry Vorotnikov
TL;DR
The paper develops a unified dual variational framework for several compressible fluid models (barotropic Euler, quantum Euler, Euler–Korteweg) by introducing time-adaptive weights and a convex entropy structure. It proves the existence of variational dual solutions in spaces of finite Radon measures and shows there is no duality gap, enabling a Dafermos-type principle that precludes premature entropy dissipation by subsolutions relative to strong solutions. The framework also connects to Brenier’s shock-free substitutes for Burgers’ equation, unifying them with the fluid models in a variational setting. Overall, it provides a robust tool for selecting physically meaningful weak solutions across multiple fluid models and reveals links to optimal transport structures.
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.