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- Title
On the regularity of weak solutions to Burgers’ equation with finite entropy production.
- Authors
Lamy, Xavier; Otto, Felix
- Abstract
Bounded weak solutions of Burgers’ equation ∂tu+∂x(u2/2)=0<inline-graphic></inline-graphic> that are not entropy solutions need in general not be BV. Nevertheless it is known that solutions with finite entropy productions have a BV-like structure: a rectifiable jump set of dimension one can be identified, outside which u has vanishing mean oscillation at all points. But it is not known whether all points outside this jump set are Lebesgue points, as they would be for BV solutions. In the present article we show that the set of non-Lebesgue points of u has Hausdorff dimension at most one. In contrast with the aforementioned structure result, we need only one particular entropy production to be a finite Radon measure, namely μ=∂t(u2/2)+∂x(u3/3)<inline-graphic></inline-graphic>. We prove Hölder regularity at points where μ<inline-graphic></inline-graphic> has finite (1+α)<inline-graphic></inline-graphic>-dimensional upper density for some α>0<inline-graphic></inline-graphic>. The proof is inspired by a result of De Lellis, Westdickenberg and the second author : if μ+<inline-graphic></inline-graphic> has vanishing 1-dimensional upper density, then u is an entropy solution. We obtain a quantitative version of this statement: if μ+<inline-graphic></inline-graphic> is small then u is close in L1<inline-graphic></inline-graphic> to an entropy solution.
- Subjects
HAUSDORFF spaces; LINEAR equations; ALGEBRAIC equations; LEBESGUE integral; MEASURE theory; RADON measures
- Publication
Calculus of Variations & Partial Differential Equations, 2018, Vol 57, Issue 4, p1
- ISSN
0944-2669
- Publication type
Article
- DOI
10.1007/s00526-018-1380-4