scholarly journals On the spreading of characteristics for non-convex conservation laws

2001 ◽  
Vol 131 (4) ◽  
pp. 909-925 ◽  
Author(s):  
Helge Kristian Jenssen ◽  
Carlo Sinestrari
Keyword(s):  
Conservation Laws ◽  
Error Term ◽  
Scalar Conservation Laws ◽  
One Dimensional ◽  
Flux Function ◽  
Characteristic Speed ◽  
Convex Case ◽  
Lipschitz Estimate ◽  
Scalar Conservation ◽  
Hölder Estimate

We study the spreading of characteristics for a class of one-dimensional scalar conservation laws for which the flux function has one point of inflection. It is well known that in the convex case the characteristic speed satisfies a one-sided Lipschitz estimate. Using Dafermos' theory of generalized characteristics, we show that the characteristic speed in the non-convex case satisfies an Hölder estimate. In addition, we give a one-sided Lipschitz estimate with an error term given by the decrease of the total variation of the solution.

One-Dimensional Scalar Conservation Laws with Regulated Data

2020 ◽  
Vol 52 (3) ◽  
pp. 3114-3130
Author(s):  
Helge Kristian Jenssen ◽  
Johanna Ridder
Keyword(s):  
Conservation Laws ◽  
Scalar Conservation Laws ◽  
One Dimensional ◽  
Scalar Conservation

scholarly journals Fractional BV spaces and applications to scalar conservation laws

2014 ◽  
Vol 11 (04) ◽  
pp. 655-677 ◽  
Author(s):  
C. Bourdarias ◽  
M. Gisclon ◽  
S. Junca
Keyword(s):  
Conservation Laws ◽  
Stability Result ◽  
Entropy Solutions ◽  
Smoothing Effect ◽  
Scalar Conservation Laws ◽  
One Dimensional ◽  
Regular Functions ◽  
Bv Functions ◽  
Scalar Conservation ◽  
First Time

We obtain new fine properties of entropy solutions to scalar nonlinear conservation laws. For this purpose, we study the "fractional BV spaces" denoted by BVs(ℝ) (for 0 < s ≤ 1), which were introduced by Love and Young in 1937 and closely related to the critical Sobolev space Ws,1/s(ℝ). We investigate these spaces in connection with one-dimensional scalar conservation laws. The BVs spaces allow one to work with less regular functions than BV functions and appear to be more natural in this context. We obtain a stability result for entropy solutions with BVs initial data. Furthermore, for the first time, we get the maximal Ws,p smoothing effect conjectured by Lions, Perthame and Tadmor for all nonlinear (possibly degenerate) convex fluxes.

scholarly journals EXISTENCE RESULT FOR THE COUPLING PROBLEM OF TWO SCALAR CONSERVATION LAWS WITH RIEMANN INITIAL DATA

2010 ◽  
Vol 20 (10) ◽  
pp. 1859-1898 ◽  
Author(s):  
BENJAMIN BOUTIN ◽  
CHRISTOPHE CHALONS ◽  
PIERRE-ARNAUD RAVIART
Keyword(s):  
Conservation Laws ◽  
Weak Sense ◽  
Scalar Conservation Laws ◽  
Existence Of A Solution ◽  
Coupling Condition ◽  
Coupling Problem ◽  
Flux Function ◽  
Scalar Conservation Law ◽  
Fixed Interface ◽  
Scalar Conservation

This paper is devoted to the coupling problem of two scalar conservation laws through a fixed interface located for instance at x = 0. Each scalar conservation law is associated with its own (smooth) flux function and is posed on a half-space, namely x < 0 or x > 0. At interface x = 0 we impose a coupling condition whose objective is to enforce in a weak sense the continuity of a prescribed variable, which may differ from the conservative unknown (and the flux functions as well). We prove the existence of a solution to the coupled Riemann problem using a constructive approach. The latter allows in particular to highlight interesting features like non-uniqueness of both continuous and discontinuous (at interface x = 0) solutions. The behavior of some numerical scheme is also investigated.

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.

HIGH-FIELD LIMIT FROM A KINETIC EQUATION TO MULTIDIMENSIONAL SCALAR CONSERVATION LAWS

2007 ◽  
Vol 04 (01) ◽  
pp. 123-145 ◽  
Author(s):  
F. BERTHELIN ◽  
N. J. MAUSER ◽  
F. POUPAUD
Keyword(s):  
Kinetic Equation ◽  
Conservation Laws ◽  
High Field ◽  
Distribution Functions ◽  
Scalar Conservation Laws ◽  
Field Limit ◽  
Flux Function ◽  
Transport Operator ◽  
Scalar Conservation

We study the relaxation of kinetic BGK models involving a high-field term within the transport operator. They lead us to multidimensional scalar conservation laws with a flux-function which is perturbed with respect to classical relaxation. The proof of the relaxation limit makes modified Maxwellians to appear. We consider "pseudo distribution functions" which can take negative values and we introduce appropriate admissible states. This is a first step towards adapting this analysis to quantum kinetic BGK models.

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