Dynamics of the incompressible Euler equations at critical regularity

Abstract

The purpose of this work is to study the incompressible Euler equations in scaling critical spaces, the function spaces which are left invariant under the natural scaling transformations for the Euler equations. We demonstrate that under appropriate symmetry assumptions, which depends on the dimension of the physical space, the Euler equations are well-posed in certain critical spaces.

It turns out that such critical spaces contain solutions which are radially homogeneous, whose dynamics is necessarily described by a system of one less dimension. In the case of 2D Euler equation, we obtain from this procedure a new 1D fluid model, and it turns out that the long-time dynamics of this model can be analyzed under some mild assumptions on the initial data.

We proceed to show well-posedness results of ``hybrid type'', by which we mean that if the initial data consists of a radially homogeneous piece and a smooth piece vanishing at the origin, the solution continues to have this decomposition, with the homogeneous part solving the lower dimensional system. As an immediate consequence, we obtain a conditional blow-up result, which states that if there is a finite time singularity for the 2D model arising from the 3D Euler equations, then there is a finite time blow-up for a Lipschitz continuous solution of the 3D Euler equation with compact support and hence of finite energy.

As an application of the critical spaces introduced in this work, we obtain a global well-posedness result for a certain class of singular vortex patches in two dimensions.