Non-linear parabolic PDEs with rough coefficients and critical data: existence, uniqueness and regularity of weak solutions
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910178681552896 |
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| author | Auscher, Pascal Bechtel, Sebastian |
| author_facet | Auscher, Pascal Bechtel, Sebastian |
| contents | This article investigates the well-posedness of weak solutions to non-linear parabolic PDEs driven by rough coefficients with rough initial data in critical homogeneous Besov spaces. Well-posedness is understood in the sense of existence and uniqueness of maximal weak solutions in suitable weighted $Z$-spaces in the absence of smallness conditions. We showcase our theory with an application to rough reaction--diffusion equations. Subsequent articles will treat further classes of equations, including equations of Burgers-type and quasi-linear problems, using the same approach. Our toolkit includes a novel theory of hypercontractive singular integral operators (SIOs) on weighted $Z$-spaces and a self-improving property for super-linear reverse Hölder inequalities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_05080 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Non-linear parabolic PDEs with rough coefficients and critical data: existence, uniqueness and regularity of weak solutions Auscher, Pascal Bechtel, Sebastian Analysis of PDEs Classical Analysis and ODEs Functional Analysis This article investigates the well-posedness of weak solutions to non-linear parabolic PDEs driven by rough coefficients with rough initial data in critical homogeneous Besov spaces. Well-posedness is understood in the sense of existence and uniqueness of maximal weak solutions in suitable weighted $Z$-spaces in the absence of smallness conditions. We showcase our theory with an application to rough reaction--diffusion equations. Subsequent articles will treat further classes of equations, including equations of Burgers-type and quasi-linear problems, using the same approach. Our toolkit includes a novel theory of hypercontractive singular integral operators (SIOs) on weighted $Z$-spaces and a self-improving property for super-linear reverse Hölder inequalities. |
| title | Non-linear parabolic PDEs with rough coefficients and critical data: existence, uniqueness and regularity of weak solutions |
| topic | Analysis of PDEs Classical Analysis and ODEs Functional Analysis |
| url | https://arxiv.org/abs/2601.05080 |