2024/10/04 by Black, Tobias
#35B65 #35K35 #35K65 #35Q92 #92C17 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2410.03307
We establish the Hölder continuity of bounded nonnegative weak solutions to (Φ-1(w))t=Δw+∇⋅(a(x,t)Φ-1(w))+b(x,t,Φ-1(w)), with convex Φ∈ C0([0,∞))∩ C2((0,∞)) satisfying Φ(0)=0, Φ'>0 on (0,∞) and sΦ''(s)≤ CΦ'(s)\quadfor all s∈[0,s0] for some C>0 and s0∈(0,1]. The functions a and b are only assumed to satisfy integrability conditions of the form aamp;∈ L2q1((0,T);L2q2(Ω;ℝN)),
bamp;∈ M(ΩT×ℝ) such that |b(x,t,ξ)|≤ b(x,t) a.e. for some b∈ Lq1((0,T);Lq2(Ω)) with q1,q2>1 such that (2)/(q1)+(N)/(q2)=2-Nκ\quadfor some κ∈(0,\tfrac2N). Letting w=Φ(u) and assuming the inverse Φ-1:[0,∞)→[0,∞) to be locally Hölder continuous, this entails Hölder regularity for bounded weak solutions of ut=ΔΦ(u)+∇⋅(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.