The Navier-Stokes problem modified by an absorption term

Abstract

In this work we consider the Navier-Stokes problem modified by the absorption term $|\text{u}{|}^{\sigma -2}\text{u}$, where $\sigma >1$, which is introduced in the momentum equation. % For this new problem, we prove the existence of weak solutions for any dimension $N\ge 2$ 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 $1<\sigma <2$.

22 pages