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) #Bounded function #Diffusion #Diffusion equation #Econometrics #Economics #Economy #FOS: Mathematics #Mathematical Biology Tumor Growth #Mathematical analysis #Mathematical and Theoretical Epidemiology and Ecology Models #Mathematics #Nabla symbol #Nonlinear Partial Differential Equations #Nonlinear system #Omega #Physics #Quantum mechanics #Sobolev space #Space (punctuation) #Statistical physics #Statistics #Thermal diffusivity #Thermodynamics #Upper and lower bounds #math.AP

paper · pdf · open access · doi:10.48550/arxiv.2009.12173

published in arXiv (Cornell University) (Cornell University)

arxiv created 2020/09/25 · openalex publication_date 2020/09/25 · arxiv updated 2020/09/28 · openalex created_date 2022/10/01 · openalex updated_date 2026/08/06

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

• A user’s guide to PDE models for chemotaxis

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