Gradient regularity for a class of doubly nonlinear parabolic partial differential equations
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909453402505216 |
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| author | Strunk, Michael |
| author_facet | Strunk, Michael |
| contents | In this paper, we study the local gradient regularity of non-negative weak solutions to doubly nonlinear parabolic partial differential equations of the type \begin{align*}
\partial_t u^q - \mbox{div}\, A(x,t,Du)=0
\qquad\mbox{in $Ω_T$}, \end{align*} with $q>0$, $Ω_T=Ω\times(0,T)\subset\mathbb{R}^{n+1}$ a space-time cylinder, and $A=A(x,t,ξ)$ a vector field satisfying standard $p$-growth conditions. Our main result establishes the local Hölder continuity of the spatial gradient of non-negative weak solutions in the super-critical fast diffusion regime $$0<p-1<q<\frac{n(p-1)}{(n-p)_+}.$$ This result is achieved by utilizing a time-insensitive Harnack inequality and Schauder estimates that are developed for equations of parabolic $p$-Laplacian type. Additionally, we establish a local $L^{\infty}$-bound for the spatial gradient. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_05631 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gradient regularity for a class of doubly nonlinear parabolic partial differential equations Strunk, Michael Analysis of PDEs In this paper, we study the local gradient regularity of non-negative weak solutions to doubly nonlinear parabolic partial differential equations of the type \begin{align*} \partial_t u^q - \mbox{div}\, A(x,t,Du)=0 \qquad\mbox{in $Ω_T$}, \end{align*} with $q>0$, $Ω_T=Ω\times(0,T)\subset\mathbb{R}^{n+1}$ a space-time cylinder, and $A=A(x,t,ξ)$ a vector field satisfying standard $p$-growth conditions. Our main result establishes the local Hölder continuity of the spatial gradient of non-negative weak solutions in the super-critical fast diffusion regime $$0<p-1<q<\frac{n(p-1)}{(n-p)_+}.$$ This result is achieved by utilizing a time-insensitive Harnack inequality and Schauder estimates that are developed for equations of parabolic $p$-Laplacian type. Additionally, we establish a local $L^{\infty}$-bound for the spatial gradient. |
| title | Gradient regularity for a class of doubly nonlinear parabolic partial differential equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.05631 |