Global weak solutions of the Navier-Stokes equations for intermittent initial data in half-space

TL;DR

This paper establishes the existence and regularity of global weak solutions to the Navier-Stokes equations in a half-space with initial data that can grow intermittently at infinity, extending the theory to more general initial conditions.

Contribution

It introduces new a priori estimates and pressure bounds for weak solutions with intermittent initial data in the half-space, and constructs global discretely self-similar solutions.

Findings

Existence of global weak solutions for intermittent initial data.

Eventual regularity of solutions under certain conditions.

Construction of global discretely self-similar solutions.

Abstract

We prove existence of global-in-time weak solutions of the incompressible Navier-Stokes equations in the half-space $\mathbb{R}^3_+$ with initial data in a weighted space that allow non-uniformly locally square integrable functions that grow at spatial infinity in an intermittent sense. The space for initial data is built on cubes whose sides $R$ are proportional to the distance to the origin and the square integral of the data is allowed to grow as a power of $R$. The existence is obtained via a new a priori estimate and stability result in the weighted space, as well as new pressure estimates. Also, we prove eventual regularity of such weak solutions, up to the boundary, for $(x,t)$ satisfying $t>c_1|x|^2 + c_2$, where $c_1,c_2>0$, for a large class of initial data $u_0$, with $c_1$ arbitrarily small. As an application of the existence theorem, we construct global discretely self-similar solutions, thus extending the theory on the half-space to the same generality as the whole space.