On the well-possedness of time-dependent three-dimensional Euler fluid flows

TL;DR

This paper proves the global well-posedness of a modified 3D Euler fluid model where dissipation activates when the velocity gradient exceeds a threshold, ensuring unique solutions for regular initial data.

Contribution

It introduces a novel threshold-based dissipation mechanism in the 3D Euler equations and proves global well-posedness for this modified model.

Findings

Existence of unique weak solutions for the modified Euler model.

Global-in-time well-posedness under the threshold dissipation mechanism.

The model maintains regularity for arbitrary regular initial velocities.

Abstract

We study the mathematical properties of time-dependent flows of incompressible fluids that respond as an Euler fluid until the modulus of the symmetric part of the velocity gradient exceeds a certain, a-priori given but arbitrarily large, critical value. Once the velocity gradient exceeds this threshold, a dissipation mechanism is activated. Assuming that the fluid, after such an activation, dissipates the energy in a specific manner, we prove that the corresponding initial-boundary-value problem is globally-in-time well-posed in the sense of Hadamard. In particular, we show that for an arbitrary, sufficiently regular, initial velocity there is a unique weak solution to the spatially periodic problem.