Global regularity results for the fractional heat equation and application to a class of non-linear KPZ problems
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866913884808413184 |
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| author | Abdellaoui, Boumediene Atmani, Somia Biroud, Kheireddine Laamri, El-Haj |
| author_facet | Abdellaoui, Boumediene Atmani, Somia Biroud, Kheireddine Laamri, El-Haj |
| contents | In the first part of this paper, we prove the global regularity, in an adequate parabolic Bessel-Potential space and then in the corresponding parabolic fractional Sobolev space, of the unique solution to following fractional heat equation $ w_t+(-Δ)^sw= h\;;\; w(x,t)=0 \text{ in } \; (\mathbb{R}^N\setminusΩ)\times(0,T)\;;\; w(x,0)=w_0(x) \; \text{in}\; Ω$, where $Ω$ is an open bounded subset of $\mathbb{R}^N$. The proof is based on a new pointwise estimate on the fractional gradient of the corresponding kernel. Moreover, we establish the compactness of $(w_0,h)\mapsto w$. As a majeur application, in the second part , we establish existence and regularity of solutions to a class of Kardar--Parisi--Zhang equations with fractional diffusion and a nonlocal gradient term. Additionally, several auxiliary results of independent interest are obtained. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_06875 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global regularity results for the fractional heat equation and application to a class of non-linear KPZ problems Abdellaoui, Boumediene Atmani, Somia Biroud, Kheireddine Laamri, El-Haj Analysis of PDEs 35B05, 35K15, 35B40, 35K55, 35K65 In the first part of this paper, we prove the global regularity, in an adequate parabolic Bessel-Potential space and then in the corresponding parabolic fractional Sobolev space, of the unique solution to following fractional heat equation $ w_t+(-Δ)^sw= h\;;\; w(x,t)=0 \text{ in } \; (\mathbb{R}^N\setminusΩ)\times(0,T)\;;\; w(x,0)=w_0(x) \; \text{in}\; Ω$, where $Ω$ is an open bounded subset of $\mathbb{R}^N$. The proof is based on a new pointwise estimate on the fractional gradient of the corresponding kernel. Moreover, we establish the compactness of $(w_0,h)\mapsto w$. As a majeur application, in the second part , we establish existence and regularity of solutions to a class of Kardar--Parisi--Zhang equations with fractional diffusion and a nonlocal gradient term. Additionally, several auxiliary results of independent interest are obtained. |
| title | Global regularity results for the fractional heat equation and application to a class of non-linear KPZ problems |
| topic | Analysis of PDEs 35B05, 35K15, 35B40, 35K55, 35K65 |
| url | https://arxiv.org/abs/2506.06875 |