Distributional solutions of Burgers' type equations for intrinsic graphs in Carnot groups of step 2

2020/08/02 by Gioacchino Antonelli, Antonelli, Gioacchino, Daniela Di Donato +3 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #Dermatological and Skeletal Disorders #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG) #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.2008.00519

openalex publication_date 2020/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that in arbitrary Carnot groups \mathbb G of step 2, with a splitting \mathbb G=\mathbb W⋅\mathbb L with \mathbb L one-dimensional, the graph of a continuous function φ\colon U⊆ \mathbb W→ \mathbb L is C¹_H-regular precisely when φ satisfies, in the distributional sense, a Burgers' type system D^φφ=ω, with a continuous ω. We stress that this equivalence does not hold already in the easiest step-3 Carnot group, namely the Engel group. As a tool for the proof we show that a continuous distributional solution φ to a Burgers' type system D^φφ=ω, with ω continuous, is actually a broad solution to D^φφ=ω. As a by-product of independent interest we obtain that all the continuous distributional solutions to D^φφ=ω, with ω continuous, enjoy 1/2-little Hölder regularity along vertical directions.

Cited by

• On the regularity of continuous solutions to multidimensional scalar conservation laws with bounded source

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