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3. Real and Complex Analysis, Potential Theory

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.