Sharp Sobolev estimates for concentration of solutions to an aggregation-diffusion equation

Piotr Biler (1) , Alexandre Boritchev (2) , Grzegorz Karch (1) , Philippe Laurençot (3)

1 Uniwersytet Wroclawski = University of Wroclaw
2 MMCS - Modélisation mathématique, calcul scientifique
3 IMT - Institut de Mathématiques de Toulouse UMR5219

We consider the drift-diffusion equation u t − ε∆u + ∇ · (u ∇K * u) = 0 in the whole space with global-in-time solutions bounded in all Sobolev spaces; for simplicity, we restrict ourselves to the model case K(x) = −|x|. We quantify the mass concentration phenomenon, a genuinely nonlinear effect, for radially symmetric solutions of this equation for small diffusivity ε studied in our previous paper [3], obtaining optimal sharp upper and lower bounds for Sobolev norms.

Mots clés: nonlocal drift-diffusion equation; small diffusivity; concentration of solutions

Domaines: Equations aux dérivées partielles [math.AP]

Piotr Biler, Alexandre Boritchev, Grzegorz Karch, Philippe Laurençot. Sharp Sobolev estimates for concentration of solutions to an aggregation-diffusion equation. Journal of Dynamics and Differential Equations, 2022, 34, pp.3131--3141. ⟨10.1007/s10884-021-09998-w⟩. ⟨hal-02948312⟩