Maximal $L_p$-regularity for $x$-dependent fractional heat equations with Dirichlet conditions
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| Format: | Preprint |
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2023
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| author | Abels, Helmut Grubb, Gerd |
| author_facet | Abels, Helmut Grubb, Gerd |
| contents | We prove optimal regularity results in $L_p$-based function spaces in space and time for a large class of linear parabolic equations with a nonlocal elliptic operator in bounded domains with limited smoothness. Here the nonlocal operator is given by a strongly elliptic and even pseudodifferential operator of order $2a$ ($0<a<1$) with nonsmooth $x$-dependent coefficients. This includes the prominent case of the fractional Laplacian $(-Δ)^a$, as well as elliptic operators $(-\nabla \cdot A(x)\nabla+b(x))^a$. The proofs are based on general results on maximal $L_p$-regularity and its relation to $\mathcal{R}$-boundedness of the resolvent of the associated (elliptic) operator. Finally, we apply these results to show existence of strong solutions locally in time for a class of nonlinear nonlocal parabolic equations, which include a fractional nonlinear diffusion equation and a fractional porous medium equation after a transformation.
The nonlinear results are new for operators on domains with boundary; the linear results are so when $P$ is $x$-dependent nonsymmetric. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_01864 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Maximal $L_p$-regularity for $x$-dependent fractional heat equations with Dirichlet conditions Abels, Helmut Grubb, Gerd Analysis of PDEs Functional Analysis Primary: 35S15, 35R11, Secondary: 35K61, 35S16, 47G30, 60G52 We prove optimal regularity results in $L_p$-based function spaces in space and time for a large class of linear parabolic equations with a nonlocal elliptic operator in bounded domains with limited smoothness. Here the nonlocal operator is given by a strongly elliptic and even pseudodifferential operator of order $2a$ ($0<a<1$) with nonsmooth $x$-dependent coefficients. This includes the prominent case of the fractional Laplacian $(-Δ)^a$, as well as elliptic operators $(-\nabla \cdot A(x)\nabla+b(x))^a$. The proofs are based on general results on maximal $L_p$-regularity and its relation to $\mathcal{R}$-boundedness of the resolvent of the associated (elliptic) operator. Finally, we apply these results to show existence of strong solutions locally in time for a class of nonlinear nonlocal parabolic equations, which include a fractional nonlinear diffusion equation and a fractional porous medium equation after a transformation. The nonlinear results are new for operators on domains with boundary; the linear results are so when $P$ is $x$-dependent nonsymmetric. |
| title | Maximal $L_p$-regularity for $x$-dependent fractional heat equations with Dirichlet conditions |
| topic | Analysis of PDEs Functional Analysis Primary: 35S15, 35R11, Secondary: 35K61, 35S16, 47G30, 60G52 |
| url | https://arxiv.org/abs/2312.01864 |