Radu Precup
Babeş-Bolyai University, Cluj-Napoca, Romania

We discuss the localization of velocity for a problem of the type \begin{equation*} \left\{ \begin{array}{l} -\text{div\ }\left( A\left( x\right) \nabla u\right) +\eta _{0}\left( x\right) u+\kappa _{0}\left( x\right) \left( u\cdot \nabla \right) u+\nabla p=\Phi \left( x,u\right) \ \ \ \text{in }\Omega \\ \text{div\ }u=0\ \ \ \text{in\ }\Omega \\ u=0\ \ \text{on }\partial \Omega , \end{array} \right. \end{equation*} where $\Phi $ is a reaction term dependent on velocity. First we obtain the localization of the enstrophy, namely $\ r\leq \left\vert u\right\vert _{H_{0}^{1}\left( \Omega \right) }\leq R,\ $and then, the localization of the kinetic energy, that is $\ r\leq \left\vert u\right\vert _{L^{2}\left( \Omega \right) }\leq R.$ The bounds $r$ and $R$ are estimated in terms of the reaction force $\Phi $ 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.