Global weak solution of 3D-NSE with exponential damping

TL;DR

This paper establishes the global existence of solutions for 3D incompressible Navier-Stokes equations with an exponential damping term, using advanced approximation and interpolation techniques to handle nonlinearities.

Contribution

It introduces a novel approach to prove global solutions for Navier-Stokes with exponential damping, overcoming challenges in nonlinear term limits.

Findings

Proved global existence of solutions with exponential damping.

Developed polynomial approximation for nonlinear damping term.

Utilized new interpolation methods between function spaces.

Abstract

In this paper we prove the global existence of incompressible Navier-Stokes equations with damping $\alpha (e^{\beta |u|^2}-1)u$, where we use Friedrich method and some new tools. The delicate problem in the construction of a global solution, is the passage to the limit in exponential nonlinear term. To solve this problem, we use a polynomial approximation of the damping part and a new type of interpolation between $L^\infty(\mathbb{R}^+,L^2(\mathbb{R}^3))$ and the space of functions $f$ such that $(e^{\beta|f|^2}-1)|f|^2\in L^1(\mathbb{R}^3)$. Fourier analysis and standard techniques are used.