| Title: |
Non-linear parabolic PDEs with rough data and coefficients: existence, uniqueness and regularity of weak solutions in critical spaces |
| Authors: |
Bechtel, Sebastian; Auscher, Pascal |
| Contributors: |
Université Paris-Saclay; European Project: 101034255,H2020-MSCA-COFUND-2020,H2020-MSCA-COFUND-2020,MathInGreaterParis(2021) |
| Source: |
https://hal.science/hal-05449658 ; 2026. |
| Publisher Information: |
CCSD |
| Publication Year: |
2026 |
| Subject Terms: |
critical spaces; rough coefficients; weak solutions; reaction--diffusion equations; singular integral operators; self-improving properties; function spaces; [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] |
| Description: |
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 $\mathrm{Z}$-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 $\mathrm{Z}$-spaces and a self-improving property for super-linear reverse Hölder inequalities. |
| Document Type: |
report |
| Language: |
English |
| Relation: |
info:eu-repo/grantAgreement//101034255/EU/MathInGreaterParis Fellowship Programme/MathInGreaterParis |
| Availability: |
https://hal.science/hal-05449658; https://hal.science/hal-05449658v1/document; https://hal.science/hal-05449658v1/file/First%20version.pdf |
| Rights: |
https://creativecommons.org/licenses/by/4.0/ ; info:eu-repo/semantics/OpenAccess |
| Accession Number: |
edsbas.2EE5700B |
| Database: |
BASE |