The Navier-Stokes problem modified by an absorption term

TL;DR

This paper studies a modified Navier-Stokes problem with an absorption term, proving existence, uniqueness, and various decay and extinction properties of weak solutions across different conditions and parameters.

Contribution

It introduces and analyzes the Navier-Stokes problem with an absorption term, establishing existence, uniqueness, and decay behaviors of solutions for various  and force conditions.

Findings

Existence of weak solutions for all dimensions  

Finite-time extinction for 1<<2 with zero forces

Exponential decay for =2 and non-zero forces

Abstract

In this work we consider the Navier-Stokes problem modified by the absorption term $|\textbf{u}|^{\sigma-2}\textbf{u}$, where $\sigma>1$, which is introduced in the momentum equation. % For this new problem, we prove the existence of weak solutions for any dimension $N\geq 2$ and its uniqueness for N=2. % Then we prove that, for zero body forces, the weak solutions extinct in a finite time if $1<\sigma<2$, exponentially decay in time if $\sigma=2$ and decay with a power-time rate if $\sigma>2$. % We prove also that for a general non-zero body forces, the weak solutions exponentially decay in time for any $\sigma>1$. In the special case of a suitable forces field which vanishes at some instant, we prove that the weak solutions extinct at the same instant provided $1<\sigma<2$.