Refining Hölder regularity theory in degenerate drift-diffusion equations

Abstract

We establish the Hölder continuity of bounded nonnegative weak solutions to \begin{align*} \big(Φ^{-1}(w)\big)_t=Δw+\nabla\cdot\big(a(x,t)Φ^{-1}(w)\big)+b\big(x,t,Φ^{-1}(w)\big), \end{align*} with convex $\hat{I}¦\in C^0([0,\infty ))\cap C^2((0,\infty ))$ satisfying $\hat{I}¦(0)=0$, $\hat{I}{¦}^{\prime }>0$ on $(0,\infty )$ and $$s\hat{I}{¦}^{\prime \prime }(s)\le C\hat{I}{¦}^{\prime }(s)\text{for all }s\in [0,s_0]$$ for some $C>0$ and $s_0\in (0,1]$. The functions $a$ and $b$ are only assumed to satisfy integrability conditions of the form \begin{align*} a&\in L^{2q_1}\big((0,T);L^{2q_2}(Ω;\mathbb{R}^N)\big),\\ b&\in M\big(Ω_T\times\mathbb{R}\big)\ \text{such that }\big|b(x,t,ξ)\big|\leq \hat{b}(x,t)\ \text{a.e. for some }\hat{b}\in L^{q_1}\big((0,T);L^{q_2}(Ω)\big) \end{align*} with $q_1,q_2>1$ such that $$\frac{2}{q_1}+\frac{N}{q_2}=2-N\hat{I}º\text{for some }\hat{I}º\in (0,\frac{2}{N}).$$ Letting $w=\hat{I}¦(u)$ and assuming the inverse $\hat{I}{¦}^{-1}:[0,\infty )\to [0,\infty )$ to be locally Hölder continuous, this entails Hölder regularity for bounded weak solutions of $$u_t=\hat{I}”\hat{I}¦(u)+\nabla \cdot (a(x,t)u)+b(x,t,u)$$ and, accordingly, covers a wide array of taxis type structures. In particular, many chemotaxis frameworks with nonlinear diffusion, which cannot be covered by the standard literature, fall into this category. After rigorously treating local Hölder regularity, we also extend the regularity result to the associated initial-boundary value problem for boundary conditions of flux-type.

49 pages, v2: added more details for the boundary regularity