2017/04/25 by Xavier Lamy, Félix Otto, Lamy, Xavier +1
Engineering · Mathematics · #35L60 #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1704.07633
openalex publication_date 2017/04/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Bounded weak solutions of Burgers' equation \∂tu+\∂x(u2/2)=0\nthat are not entropy solutions need in general not be BV. Nevertheless it is\nknown that solutions with finite entropy productions have a BV-like\nstructure: a rectifiable jump set of dimension one can be identified, outside\nwhich u has vanishing mean oscillation at all points. But it is not known\nwhether all points outside this jump set are Lebesgue points, as they would be\nfor BV solutions. In the present article we show that the set of non-Lebesgue\npoints of u has Hausdorff dimension at most one. In contrast with the\naforementioned structure result, we need only one particular entropy production\nto be a finite Radon measure, namely \μ=\∂t\n(u2/2)+\∂x(u3/3). We prove H "older regularity at points where \μ\nhas finite (1+\α)-dimensional upper density for some \α>0. The\nproof is inspired by a result of De Lellis, Westdickenberg and the second\nauthor : if \μ+ has vanishing 1-dimensional upper density, then u is an\nentropy solution. We obtain a quantitative version of this statement: if\n\μ+ is small then u is close in L1 to an entropy solution.\n