Refining H\"older regularity theory in degenerate drift-diffusion equations

Abstract

We establish the H\"older continuity of bounded nonnegative weak solutions to \begin{align*} \big(\Phi^{-1}(w)\big)_t=\Delta w+\nabla\cdot\big(a(x,t)\Phi^{-1}(w)\big)+b\big(x,t,\Phi^{-1}(w)\big), \end{align*} with convex $\Phi\in C^0([0,\infty))\cap C^2((0,\infty))$ satisfying $\Phi(0)=0$, $\Phi'>0$ on $(0,\infty)$ and $$s\Phi''(s)\leq C\Phi'(s)\quad\text{for all }s\in[0,s_0]$$ 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}(\Omega;\mathbb{R}^N)\big),\\ b&\in M\big(\Omega_T\times\mathbb{R}\big)\ \text{such that }\big|b(x,t,\xi)\big|\leq \hat{b}(x,t)\ \text{a.e. for some }\hat{b}\in L^{q_1}\big((0,T);L^{q_2}(\Omega)\big) \end{align*} with $q_1,q_2>1$ such that $$\frac{2}{q_1}+\frac{N}{q_2}=2-N\kappa\quad\text{for some }\kappa\in(0,\tfrac{2}{N}).$$ Letting $w=\Phi(u)$ and assuming the inverse $\Phi^{-1}:[0,\infty)\to[0,\infty)$ to be locally H\"older continuous, this entails H\"older regularity for bounded weak solutions of $$u_t=\Delta\Phi(u)+\nabla\cdot\big(a(x,t)u\big)+b(x,t,u)$$ 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 H\"older regularity, we also extend the regularity result to the associated initial-boundary value problem for boundary conditions of flux-type.