Extend the mass-concentration method to multidimensional weak-Lebesgue drifts

Extend the mass concentration function method based on pointwise comparison in the mass variable to the multidimensional nonlinear Fokker–Planck equation with drift fields satisfying b\in L^\infty((0,\infty);L^{p,\infty}(\mathbb{R}^d)) for finite p>d, in order to handle the critical case k=1/d-1/p.

Background

The paper studies the nonlinear Fokker–Planck equation $$\partial_tu+\operatorname{div}(b(t,x)u^{1+k})=\Delta u$$ in dimensions $d\ge 2$, under the assumption that the prescribed drift belongs to $L^\infty((0,\infty);L^{p,\infty}(\mathbb{R}^d))$ with $p>d$. The critical exponent is $k_c=1/d-1/p$.

The authors explain that a previous approach based on the mass concentration function $m(t,s)=\int_0^s u^*(t,r)\,dr$ cannot presently be extended to the finite-p drift assumption considered here. That method requires uniformly controlling the drift contribution over all sets of a fixed measure, a requirement that appears difficult to reconcile with an integral condition on the drift. The paper resolves the critical case by using a different comparison argument, so this extension remains unresolved.