Variational Dual Solutions for Incompressible Fluids

TL;DR

The paper develops a variational dual framework for incompressible fluids by recasting Euler and Navier–Stokes as concave dual optimization problems built from weak formulations and designable base states $\bar{V}$. Through a sequence of constructions—first for Euler, then extended to NS with additional variables and even a pressure as an independent field—the authors establish existence of dual extremizers and show that classical weak solutions are recoverable via a dual-to-primal mapping, while also analyzing regularity and limitations. A central result is a $\Gamma$-convergence analysis showing the inviscid NS$\to$Euler limit as viscosity $\nu$ vanishes, within this variational framework, thereby connecting the dual formulation to the classical PDE limit. Collectively, the work provides a non-standard, variational lens on fluid dynamics that supports existence, approximation, and convergence results beyond traditional weak formulations, with potential implications for computational methods and the understanding of regularity in incompressible flows.

Abstract

We consider a construction proposed in \cite{acharyaQAM} that builds on the notion of weak solutions for incompressible fluids to provide a scheme that generates variationally a certain type of dual solutions. If these dual solutions are regular enough one can use them to recover standard solutions. The scheme provides a generalisation of a construction of Y. Brenier for the Euler equations. We rigorously analyze the scheme, extending the work of Y.Brenier for Euler, and also provide an extension of it to the case of the Navier-Stokes equations. Furthermore we obtain the inviscid limit of Navier-Stokes to Euler as a $Γ$-limit.

Theorems & Definitions (5)

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