A note on strong solutions to the variational kinetic equation for scalar conservation laws

2014 ◽  
Vol 11 (03) ◽  
pp. 621-632
Author(s):  
Misha Perepelitsa
Keyword(s):  
Conservation Laws ◽  
Differential Inclusion ◽  
Maximal Monotone ◽  
Strong Solutions ◽  
Indicator Function ◽  
Closed Convex Cone ◽  
Scalar Conservation Laws ◽  
Kinetic Formulation ◽  
Degeneracy Condition ◽  
Scalar Conservation

We consider a variational kinetic formulation for weak, entropy solutions of scalar conservation laws due to Brenier. The solutions in this formulation are represented by a kinetic density function Y that solves a differential inclusion ∂tY ∈ -A(Y) = -∂vf ⋅ ∇xY -∂ IK(Y), where IK is the indicator function of a closed, convex cone K. Under a certain "non-degeneracy" condition we determine a maximal monotone extension of A and use it to prove the existence of strong and weak solutions of the differential inclusion for a general, possibly degenerate, flux ∂vf(v). Furthermore, we discuss several properties of strong solutions.

scholarly journals Regularity estimates for scalar conservation laws in one space dimension

2018 ◽  
Vol 15 (04) ◽  
pp. 623-691 ◽  
Author(s):  
Elio Marconi
Keyword(s):  
Conservation Laws ◽  
Space Dimension ◽  
Scalar Conservation Laws ◽  
Kinetic Formulation ◽  
Flux Function ◽  
Regularity Estimates ◽  
Degeneracy Condition ◽  
Inflection Points ◽  
Scalar Conservation ◽  
One Space Dimension

We deal with the regularizing effect that, in scalar conservation laws in one space dimension, the nonlinearity of the flux function [Formula: see text] has on the entropy solution. More precisely, if the set [Formula: see text] is dense, the regularity of the solution can be expressed in terms of [Formula: see text] spaces, where [Formula: see text] depends on the nonlinearity of [Formula: see text]. If moreover the set [Formula: see text] is finite, under the additional polynomial degeneracy condition at the inflection points, we prove that [Formula: see text] for every [Formula: see text] and that this can be improved to [Formula: see text] regularity except an at most countable set of singular times. Finally, we present some examples that show the sharpness of these results and counterexamples to related questions, namely regularity in the kinetic formulation and a property of the fractional BV spaces.

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