Well-posedness theory for non-homogeneous incompressible fluids with odd viscosity

TL;DR

This paper develops a mathematical framework proving local well-posedness for non-homogeneous incompressible fluids with odd viscosity, overcoming derivative loss by revealing a hidden hyperbolic structure.

Contribution

Introduces a novel set of good unknowns that uncover a hyperbolic structure, enabling well-posedness analysis for fluids with non-dissipative odd viscosity.

Findings

Established local well-posedness in Sobolev spaces.

Identified continuation criteria for solutions.

Demonstrated the hyperbolic structure underlying the system.

Abstract

Several fluid systems are characterised by time reversal and parity breaking. Examples of such phenomena arise both in quantum and classical hydrodynamics. In these situations, the viscosity tensor, often dubbed ``odd viscosity'', becomes non-dissipative. At the mathematical level, this fact translates into a loss of derivatives at the level of \textsl{a priori} estimates: while the odd viscosity term depends on derivatives of the velocity field, no parabolic smoothing effect can be expected. In the present paper, we establish a well-posedness theory in Sobolev spaces for a system of incompressible non-homogeneous fluids with odd viscosity. The crucial point of the analysis is the introduction of a set of \emph{good unknowns}, which allow for the emerging of a hidden hyperbolic structure underlying the system of equations. It is exactly this hyperbolic structure which makes it possible to circumvent the derivative loss and propagate high enough Sobolev norms of the solution. The well-posedness result is local in time; two continuation criteria are also established.