scholarly journals Structure of entropy solutions to general scalar conservation laws in one space dimension

2015 ◽  
Vol 428 (1) ◽  
pp. 356-386 ◽  
Author(s):  
Stefano Bianchini ◽  
Lei Yu
Keyword(s):  
Conservation Laws ◽  
Space Dimension ◽  
Entropy Solutions ◽  
Scalar Conservation Laws ◽  
General Scalar ◽  
Scalar Conservation ◽  
One Space Dimension

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.

STRUCTURE OF ENTROPY SOLUTIONS TO SCALAR CONSERVATION LAWS WITH STRICTLY CONVEX FLUX

2012 ◽  
Vol 09 (04) ◽  
pp. 571-611 ◽  
Author(s):  
ADIMURTHI ◽  
SHYAM SUNDAR GHOSHAL ◽  
G. D. VEERAPPA GOWDA
Keyword(s):  
Shock Wave ◽  
Initial Data ◽  
Conservation Laws ◽  
Compact Support ◽  
Space Dimension ◽  
Structure Theorem ◽  
Regions Of Interest ◽  
Entropy Solutions ◽  
Scalar Conservation Laws ◽  
Scalar Conservation

We consider scalar conservation laws in one space dimension with convex flux and we establish a new structure theorem for entropy solutions by identifying certain shock regions of interest, each of them representing a single shock wave at infinity. Using this theorem, we construct a smooth initial data with compact support for which the solution exhibits infinitely many shock waves asymptotically in time. Our proof relies on Lax–Oleinik explicit formula and the notion of generalized characteristics introduced by Dafermos.

scholarly journals AN EXTENSION OF OLEINIK's INEQUALITY FOR GENERAL 1D SCALAR CONSERVATION LAWS

2008 ◽  
Vol 05 (01) ◽  
pp. 113-165 ◽  
Author(s):  
OLIVIER GLASS
Keyword(s):  
Conservation Laws ◽  
Type Inequality ◽  
Entropy Solutions ◽  
One Dimension ◽  
Scalar Conservation Laws ◽  
Dimension Formula ◽  
Almost Convex ◽  
General Scalar ◽  
Scalar Conservation

We establish an Oleinik-type inequality concerning BV entropy solutions of general scalar conservation laws in one dimension: [Formula: see text] This inequality reads, for x ≤ y and t > 0: [Formula: see text] This contains Oleinik's inequality for convex fluxes; in particular, almost convex and almost concave fluxes yield solutions that almost satisfy Oleinik's estimate. We also show that this inequality is not satisfied in general when one replaces the factor [Formula: see text] with ‖(f″)-‖∞ or with ‖(f″)+‖∞.

scholarly journals Lagrangian Representations for Linear and Nonlinear Transport

2017 ◽  
Vol 63 (3) ◽  
pp. 418-436
Author(s):  
Stefano Bianchini ◽  
Paolo Bonicatto ◽  
Elio Marconi
Keyword(s):  
Conservation Laws ◽  
Continuity Equation ◽  
Space Dimension ◽  
Nonlinear Transport ◽  
Scalar Conservation Laws ◽  
First Order ◽  
Entropy Dissipation ◽  
Linear And Nonlinear ◽  
Scalar Conservation ◽  
The One

In this note we present a unifying approach for two classes of first order partial differential equations: we introduce the notion of Lagrangian representation in the settings of continuity equation and scalar conservation laws. This yields, on the one hand, the uniqueness of weak solutions to transport equation driven by a two dimensional BV nearly incompressible vector field. On the other hand, it is proved that the entropy dissipation measure for scalar conservation laws in one space dimension is concentrated on countably many Lipschitz curves.

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