Transport-collapse scheme for heterogeneous scalar conservation laws

2018 ◽  
Vol 15 (01) ◽  
pp. 119-132
Author(s):  
Darko Mitrović ◽  
Andrej Novak
Keyword(s):  
Cauchy Problem ◽  
Kinetic Equation ◽  
Conservation Laws ◽  
Admissible Solution ◽  
Scalar Conservation Laws ◽  
Numerical Examples ◽  
The Cauchy Problem ◽  
Scalar Conservation

We extend Brenier’s transport collapse scheme on the Cauchy problem for heterogeneous scalar conservation laws i.e. for the conservation laws with spacetime-dependent coefficients. The method is based on averaging out the solution to the corresponding kinetic equation, and it necessarily converges toward the entropy admissible solution. We also provide numerical examples.

scholarly journals A Bhatnagar–Gross–Krook approximation to scalar conservation laws with discontinuous flux

2010 ◽  
Vol 140 (5) ◽  
pp. 953-972 ◽  
Author(s):  
F. Berthelin ◽  
J. Vovelle
Keyword(s):  
Conservation Laws ◽  
Entropy Solution ◽  
Limit Problem ◽  
Scalar Conservation Laws ◽  
First Order ◽  
The Cauchy Problem ◽  
Discontinuous Flux ◽  
Scalar Conservation ◽  
Bgk Approximation ◽  
Well Posed

AbstractWe study the Bhatnagar–Gross–Krook (BGK) approximation to first-order scalar conservation laws with a flux which is discontinuous in the space variable. We show that the Cauchy problem for the BGK approximation is well posed and that, as the relaxation parameter tends to 0, it converges to the (entropy) solution of the limit problem.

scholarly journals Well-posedness theory for stochastically forced conservation laws on Riemannian manifolds

2019 ◽  
Vol 16 (03) ◽  
pp. 519-593
Author(s):  
L. Galimberti ◽  
K. H. Karlsen
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Generalized Solutions ◽  
Well Posedness ◽  
Scalar Conservation Laws ◽  
Stochastic Forcing ◽  
Rigidity Result ◽  
Hyperbolic Conservation ◽  
The Cauchy Problem ◽  
Scalar Conservation

We investigate a class of scalar conservation laws on manifolds driven by multiplicative Gaussian (Itô) noise. The Cauchy problem defined on a Riemanian manifold is shown to be well-posed. We prove existence of generalized kinetic solutions using the vanishing viscosity method. A rigidity result àla Perthame is derived, which implies that generalized solutions are kinetic solutions and that kinetic solutions are uniquely determined by their initial data ([Formula: see text] contraction principle). Deprived of noise, the equations we consider coincide with those analyzed by Ben-Artzi and LeFloch [Well-posedness theory for geometry-compatible hyperbolic conservation laws on manifolds, Ann. Inst. H. Poincaré Anal. Non Linéaire 24(6) (2007) 989–1008], who worked with Kružkov–DiPerna solutions. In the Euclidian case, the stochastic equations agree with those examined by Debussche and Vovelle [Scalar conservation laws with stochastic forcing, J. Funct. Anal. 259(4) (2010) 1014–1042].

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.

Non-classical global solutions for a class of scalar conservation laws

2004 ◽  
Vol 134 (2) ◽  
pp. 225-240
Author(s):  
Paolo Baiti
Keyword(s):  
Conservation Laws ◽  
Global Solutions ◽  
Entropy Inequality ◽  
Travelling Wave ◽  
Scalar Conservation Laws ◽  
Kinetic Relation ◽  
Bounded Variation Functions ◽  
The Cauchy Problem ◽  
Scalar Conservation ◽  
The One

We consider the Cauchy problem for a class of scalar conservation laws with flux having a single inflection point. We prove existence of global weak solutions satisfying a single entropy inequality together with a kinetic relation, in a class of bounded variation functions. The kinetic relation is obtained by the travelling-wave criterion for a regularization consisting of balanced diffusive and dispersive terms. The result is applied to the one-dimensional Buckley-Leverett equation.

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.

scholarly journals On the Cauchy problem for scalar conservation laws in the class of Besicovitch almost periodic functions: Global well-posedness and decay property

2016 ◽  
Vol 13 (03) ◽  
pp. 633-659 ◽  
Author(s):  
Evgeny Yu. Panov
Keyword(s):  
Cauchy Problem ◽  
Conservation Laws ◽  
Almost Periodic Functions ◽  
Entropy Solutions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Scalar Conservation Law ◽  
The Cauchy Problem ◽  
Scalar Conservation ◽  
Necessary And Sufficient

We study the Cauchy problem for a multidimensional scalar conservation law with merely continuous flux vector in the class of Besicovitch almost periodic functions. The existence and uniqueness of entropy solutions are established. We also uncover the necessary and sufficient condition for the decay of almost periodic entropy solutions as the time variable [Formula: see text]. Our results are then interpreted in the framework of conservation laws on the Bohr compact.

scholarly journals On uniqueness of solutions to conservation laws verifying a single entropy condition

2019 ◽  
Vol 16 (01) ◽  
pp. 157-191 ◽  
Author(s):  
Sam G. Krupa ◽  
Alexis F. Vasseur
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Systems ◽  
Quasilinear Equation ◽  
Scalar Conservation Laws ◽  
Uniqueness Of Solutions ◽  
The Cauchy Problem ◽  
Systems Of Conservation Laws ◽  
Uniqueness Of The Solution ◽  
Scalar Conservation ◽  
Jacobi Equations

For hyperbolic systems of conservation laws, uniqueness of solutions is still largely open. We aim to expand the theory of uniqueness for systems of conservation laws. One difficulty is that many systems have only one entropy. This contrasts with scalar conservation laws, where many entropies exist. It took until 1994 to show that one entropy is enough to ensure uniqueness of solutions for the scalar conservation laws (see [E. Yu. Panov, Uniqueness of the solution of the Cauchy problem for a first order quasilinear equation with one admissible strictly convex entropy, Mat. Z. 55(5) (1994) 116–129 (in Russian), Math. Notes 55(5) (1994) 517–525]. This single entropy result was proven again by De Lellis, Otto and Westdickenberg about 10 years later [Minimal entropy conditions for Burgers equation, Quart. Appl. Math. 62(4) (2004) 687–700]. These two proofs both rely on the special connection between Hamilton–Jacobi equations and scalar conservation laws in one space dimension. However, this special connection does not extend to systems. In this paper, we prove the single entropy result for scalar conservation laws without using Hamilton–Jacobi. Our proof lays out new techniques that are promising for showing uniqueness of solutions in the systems case.

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