A tree-free approach to regularity structures: The regular case for quasi-linear equations

2022/07/21 by P.G. Linares, Félix Otto, Linares, Pablo +1
Biochemistry, Genetics and Molecular Biology · Computer Science · Engineering · #60H17 #60L30 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Caveolin-1 and cellular processes #FOS: Mathematics #Probability (math.PR) #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2207.10627

openalex publication_date 2022/07/21 · openalex created_date 2022/07/23 · openalex updated_date 2026/07/28

Abstract

We give a motivation and gentle introduction into the regularity structure and model introduced by Otto, Sauer, Smith and Weber, which fall into the framework of Hairer, but have a greedier index set than the one given by trees. We do this here for a simple quasi-linear parabolic equation and assume that the driving noise is so regular that no renormalization is needed. We introduce the abstract model space T and its grading, the pre-model \mathbfΠ, the centered model Π_x, the structure group G, and the re-centering transformations Γ_xy. Using integration and reconstruction, we establish the desired estimates on Π_x and Γ_xy, which here are deterministic since we deal with the regular case.

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