Lagrangian-Eulerian Methods for Uniqueness in Hydrodynamic Systems

TL;DR

This paper introduces a Lagrangian-Eulerian approach to establish uniqueness and local existence of solutions in limited smoothness spaces for various incompressible hydrodynamic models, including complex fluids and magneto-hydrodynamics.

Contribution

It develops a novel Lagrangian-Eulerian framework to prove mathematical properties of hydrodynamic systems with limited regularity.

Findings

Proves uniqueness of solutions in path spaces.

Establishes local existence for complex fluid models.

Applies to magneto-hydrodynamics equations.

Abstract

We present a Lagrangian-Eulerian strategy for proving uniqueness and local existence of solutions in path spaces of limited smoothness for a class of incompressible hydrodynamic models including Oldroyd-B type complex fluid models and zero magnetic resistivity magneto-hydrodynamics equations.