Sharp Sobolev Estimates for Concentration of Solutions to an Aggregation–Diffusion Equation
2021
We consider the drift-diffusion equation $$\begin{aligned} u_t-\varepsilon \varDelta u+\nabla \cdot (u\ \nabla K*u)=0 \end{aligned}$$
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 $$\varepsilon $$
studied in our previous paper (Biler et al. in J Differ Equ 271:1092–1108, 2021), obtaining sharp upper and lower bounds for Sobolev norms.
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