scholarly journals Null Lagrangian Measures in Subspaces, Compensated Compactness and Conservation Laws

2019 ◽  
Vol 234 (2) ◽  
pp. 857-910 ◽  
Author(s):  
Andrew Lorent ◽  
Guanying Peng
Keyword(s):  
Conservation Laws ◽  
Compensated Compactness ◽  
Null Lagrangian

scholarly journals Pairings between bounded divergence-measure vector fields and BV functions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Graziano Crasta ◽  
Virginia De Cicco ◽  
Annalisa Malusa
Keyword(s):  
Vector Field ◽  
Conservation Laws ◽  
Bounded Variation ◽  
Vector Fields ◽  
Hyperbolic Conservation Laws ◽  
Divergence Measure ◽  
Compensated Compactness ◽  
Hyperbolic Conservation ◽  
Bv Functions ◽  
Bounded Functions

AbstractWe introduce a family of pairings between a bounded divergence-measure vector field and a function u of bounded variation, depending on the choice of the pointwise representative of u. We prove that these pairings inherit from the standard one, introduced in [G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl. (4) 135 1983, 293–318], [G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal. 147 1999, 2, 89–118], all the main properties and features (e.g. coarea, Leibniz, and Gauss–Green formulas). We also characterize the pairings making the corresponding functionals semicontinuous with respect to the strict convergence in \mathrm{BV}. We remark that the standard pairing in general does not share this property.

Convergence of the viscosity solutions for weakly strictly hyperbolic conservation laws

1997 ◽  
Vol 127 (5) ◽  
pp. 1103-1110
Author(s):  
Zhu Changjiang
Keyword(s):  
Approximate Solution ◽  
Conservation Laws ◽  
Viscosity Solutions ◽  
Hyperbolic Conservation Laws ◽  
Compensated Compactness ◽  
Hyperbolic Conservation ◽  
Solution Sequence ◽  
Uniformly Bounded

SynopsisThis paper is an extension of papers [14–16]. Using the theory of compensated compactness, we establish the convergence of the uniformly bounded approximate solution sequence for a class of ‘weakly strictly hyperbolic’ conservation laws.

scholarly journals ON THE DIRICHLET-NEUMANN BOUNDARY PROBLEM FOR SCALAR CONSERVATION LAWS

2016 ◽  
Vol 21 (5) ◽  
pp. 685-698
Author(s):  
Marin Mišur ◽  
Darko Mitrovic ◽  
Andrej Novak
Keyword(s):  
Approximate Solution ◽  
Conservation Laws ◽  
Boundary Problem ◽  
Neumann Boundary ◽  
Compensated Compactness ◽  
Scalar Conservation Laws ◽  
Basic Tool ◽  
Numerical Examples ◽  
Elliptic Approximation ◽  
Scalar Conservation

We consider a Dirichlet-Neumann boundary problem in a bounded domain for scalar conservation laws. We construct an approximate solution to the problem via an elliptic approximation for which, under appropriate assumptions, we prove that the corresponding limit satisfies the considered equation in the interior of the domain. The basic tool is the compensated compactness method. We also provide numerical examples.

COMPENSATED COMPACTNESS FOR 2D CONSERVATION LAWS

2005 ◽  
Vol 02 (03) ◽  
pp. 697-712 ◽  
Author(s):  
EITAN TADMOR ◽  
MICHEL RASCLE ◽  
PATRIZIA BAGNERINI
Keyword(s):  
Conservation Laws ◽  
Entropy Production ◽  
Large Family ◽  
Approximate Solutions ◽  
Translation Invariance ◽  
Minimal Amount ◽  
Compensated Compactness ◽  
Convergence Results ◽  
Spatial Regularity ◽  
Compactness Arguments

We introduce a new framework for studying two-dimensional conservation laws by compensated compactness arguments. Our main result deals with 2D conservation laws which are nonlinear in the sense that their velocity fields are a.e. not co-linear. We prove that if uε is a family of uniformly bounded approximate solutions of such equations with H-1-compact entropy production and with (a minimal amount of) uniform time regularity, then (a subsequence of) uε convergences strongly to a weak solution. We note that no translation invariance in space — and in particular, no spatial regularity of u(·, t) is required. Our new approach avoids the use of a large family of entropies; by a judicious choice of entropies, we show that only two entropy production bounds will suffice. We demonstrate these convergence results in the context of vanishing viscosity, kinetic BGK and finite volume approximations. Finally, the intimate connection between our 2D compensated compactness arguments and the notion of multi-dimensional nonlinearity based on kinetic formulation is clarified.

scholarly journals Convergence of second-order, entropy stable methods for multi-dimensional conservation laws

2020 ◽  
Vol 54 (4) ◽  
pp. 1415-1428
Author(s):  
Neelabja Chatterjee ◽  
Ulrik Skre Fjordholm
Keyword(s):  
Weak Solution ◽  
Numerical Methods ◽  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Compensated Compactness ◽  
Entropy Stability ◽  
Entropy Condition ◽  
Hyperbolic Conservation ◽  
Convergence Results ◽  
Entropy Stable

High-order accurate, entropy stable numerical methods for hyperbolic conservation laws have attracted much interest over the last decade, but only a few rigorous convergence results are available, particularly in multiple space dimensions. In this paper we show how the entropy stability of one such method, which is semi-discrete in time, yields a (weak) bound on oscillations. Under the assumption of L∞-boundedness of the approximations we use compensated compactness to prove convergence to a weak solution satisfying at least one entropy condition.

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