Partial gradient regularity for parabolic systems with degenerate diffusion and Hölder continuous coefficients

2024/07/25 by Fabian Bäuerlein, Bäuerlein, Fabian
Computer Science · Mathematics · #35K55 #35K65 #35K67 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2407.17837

openalex publication_date 2024/07/25 · openalex created_date 2024/09/19 · openalex updated_date 2026/07/28

Abstract

We consider vector valued weak solutions u:Ω_T→ ℝ^N with N∈ ℕ of degenerate or singular parabolic systems of type ∂_t u - div a(z,u,Du) = 0 \qquadin Ω_T= Ω× (0,T), where Ω denotes an open set in ℝⁿ for n≥ 1 and T>0 a finite time. Assuming that the vector field a is not of Uhlenbeck-type structure, satisfies p-growth assumptions and (z,u)↦ a(z,u,ξ) is Hölder continuous for every ξ∈ ℝ^Nn, we show that the gradient Du is partially Hölder continuous, provided the vector field degenerates like that of the p-Laplacian for small gradients.

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