Global existence of solutions to 2-D Navier-Stokes flow with non-decaying initial data in half-plane

TL;DR

This paper proves the global existence of solutions to the 2-D Navier-Stokes equations in a half-plane with non-decaying initial data, introducing new techniques applicable to 3-D cases and weak solutions.

Contribution

It introduces a novel method for solving 2-D Navier-Stokes with non-decaying data and extends to weak solutions in 3-D, a first in the field.

Findings

Established global solutions for 2-D Navier-Stokes with non-decaying initial data.

Developed a technique applicable to 3-D Navier-Stokes weak solutions.

First demonstration of weak solutions for 3-D Navier-Stokes with such initial data.

Abstract

We investigate the Navier-Stokes initial boundary value problem in the half-plane $R^2_+$ with initial data $u_0 \in L^\infty(R^2_+)\cap J_0^2(R^2_+)$ or with non decaying initial data $u_0\in L^\infty(R^2_+) \cap J_0^p(R^2_+), p > 2$ . We introduce a technique that allows to solve the two-dimesional problem, further, but not least, it can be also employed to obtain weak solutions, as regards the non decaying initial data, to the three-dimensional Navier-Stokes IBVP. This last result is the first of its kind.