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

2020/09/25 by Piotr Biler, Biler, Piotr, Alexandre Boritchev +5
Computer Science · Mathematics · Medicine · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2009.12173

openalex publication_date 2020/09/25 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

We consider the drift-diffusion equation u_t-εΔu + ∇ ⋅(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.

Citations

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