The Navier-Stokes problem modified by an absorption term
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In this work we consider the Navier-Stokes problem modified by the absorption term $|\textbf{u}|^{σ-2}\textbf{u}$, where $σ>1$, 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 $1<σ<2$, exponentially decay in time if $σ=2$ and decay with a power-time rate if $σ>2$. % We prove also that for a general non-zero body forces, the weak solutions exponentially decay in time for any $σ>1$. 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 $1<σ<2$.
22 pages
22 pages