Energy-localization in Navier-Stokes models with reaction terms
Radu Precup
Babeş-Bolyai University, Cluj-Napoca, Romania
Abstract:
We discuss the localization of velocity for a problem of the type
{−div\ (A(x)∇u)+η0(x)u+κ0(x)(u⋅∇)u+∇p=Φ(x,u) in Ωdiv\ u=0 in\ Ωu=0 on ∂Ω,
where Φ is a reaction term dependent on velocity. First we obtain the
localization of the enstrophy, namely r≤|u|H10(Ω)≤R, and then, the localization of
the kinetic energy, that is r≤|u|L2(Ω)≤R. The bounds r and R are estimated in terms of
the reaction force Φ and of system coefficients. The proofs are based
on the fixed point formulation of the problem and on the fixed point index.
The results come from a joint work in progress with Mirela Kohr, in
continuation of the paper: M. Kohr and R. Precup. Analysis of Navier-Stokes
models for flows in bidisperse porous media. J. Math. Fluid Mech.
(2023) 25:38.