On the structure of weak solutions to scalar conservation laws with finite entropy production. (English) Zbl 1486.35294
The author studies properties of weak solutions with finite entropy production to multidimensional scalar conservation laws \(\partial_t u + \mathrm{div}_x F(u)=0\), \(u=u(t,x)\), \(x\in\mathbb{R}^d\), \(t\in (0,T)\). Based on the kinetic formulation, it is established under suitable nonlinearity assumption on the flux vector \(F\) that the set of non Lebesgue points of \(u\) has Hausdorff dimension at most \(d\). The author also introduces a notion of Lagrangian representation, which allows to give a new interpretation of the entropy dissipation measure.
Reviewer: Evgeniy Panov (Veliky Novgorod)
MSC:
| 35L65 | Hyperbolic conservation laws |
| 35D30 | Weak solutions to PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35L03 | Initial value problems for first-order hyperbolic equations |
Keywords:
scalar conservation laws; weak solutions; finite entropy production; kinetic formulation; Lagrangian representationReferences:
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