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

TL;DR

This paper proves H"older continuity of solutions to a class of degenerate drift-diffusion equations with nonlinear diffusion, covering chemotaxis models, under minimal regularity assumptions on coefficients.

Contribution

It establishes the first H"older regularity results for solutions to degenerate drift-diffusion equations with nonlinear diffusion and minimal coefficient regularity.

Findings

Proved local H"older continuity of solutions.

Extended regularity results to initial-boundary value problems.

Applicable to a wide range of chemotaxis models.

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.