scholarly journals Non-classical Riemann solvers with nucleation

2004 ◽  
Vol 134 (5) ◽  
pp. 961-984 ◽  
Author(s):  
P. G. LeFloch ◽  
M. Shearer
Keyword(s):  
Propagation Speed ◽  
Riemann Solver ◽  
Scalar Conservation Laws ◽  
Wave Interactions ◽  
Kinetic Relation ◽  
Riemann Solvers ◽  
Flux Function ◽  
The Cauchy Problem ◽  
Scalar Conservation ◽  
Film Flows

We introduce a new non-classical Riemann solver for scalar conservation laws with concave–convex flux-function. This solver is based on both a kinetic relation, which determines the propagation speed of (under-compressive) non-classical shock waves, and a nucleation criterion, which makes a choice between a classical Riemann solution and a non-classical one. We establish the existence of (non-classical entropy) solutions of the Cauchy problem and discuss several examples of wave interactions. We also show the existence of a class of solutions, called splitting–merging solutions, which are made of two large shocks and small bounded-variation perturbations. The nucleation solvers, as we call them, are applied to (and actually motivated by) the theory of thin-film flows; they help explain numerical results observed for such flows.

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 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].

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.

scholarly journals Asymptotic Time-Behaviour of Solutions to Scalar Conservation Law with a Convex Flux Function

2019 ◽  
Vol 224 ◽  
pp. 01005
Author(s):  
Natalia Petrosyan
Keyword(s):  
Quasilinear Equation ◽  
Rates Of Convergence ◽  
Mean Value ◽  
Time Behaviour ◽  
Mean Values ◽  
Flux Function ◽  
Scalar Conservation Law ◽  
The Cauchy Problem ◽  
Long Time ◽  
Scalar Conservation

We consider the long-time behaviour of solutions of the Cauchy problem for a quasilinear equation ut + f(u)x = 0 with a strictly convex flux function f(u) and initial function u0(x) having the the one-sided limiting mean values u± that are uniform with respect to translations. The estimates of the rates of convergence to solutions of the Riemann problem depending on the behaviour of the integrals $ \int\limits_a^{a + y} {\left( {{u_0}\left( x \right) - {u^ \pm }} \right)} dx $ as y→±∞ are established. The similar results are obtained for solutions of the mixed problem in the domain x > 0, t > 0 with a constant boundary data u– and initial data having limiting mean value u±.

scholarly journals Radon measure-valued solutions of first order scalar conservation laws

2018 ◽  
Vol 9 (1) ◽  
pp. 65-107
Author(s):  
Michiel Bertsch ◽  
Flavia Smarrazzo ◽  
Andrea Terracina ◽  
Alberto Tesei
Keyword(s):  
Radon Measure ◽  
Singular Part ◽  
Scalar Conservation Laws ◽  
Radon Measures ◽  
Lipschitz Continuous ◽  
First Order ◽  
Nonnegative Solutions ◽  
All Solutions ◽  
The Cauchy Problem ◽  
Scalar Conservation

AbstractWe study nonnegative solutions of the Cauchy problem\left\{\begin{aligned} &\displaystyle\partial_{t}u+\partial_{x}[\varphi(u)]=0&% &\displaystyle\phantom{}\text{in }\mathbb{R}\times(0,T),\\ &\displaystyle u=u_{0}\geq 0&&\displaystyle\phantom{}\text{in }\mathbb{R}% \times\{0\},\end{aligned}\right.where{u_{0}}is a Radon measure and{\varphi\colon[0,\infty)\mapsto\mathbb{R}}is a globally Lipschitz continuous function. We construct suitably defined entropy solutions in the space of Radon measures. Under some additional conditions on φ, we prove their uniqueness if the singular part of{u_{0}}is a finite superposition of Dirac masses. Regarding the behavior of φ at infinity, we give criteria to distinguish two cases: either all solutions are function-valued for positive times (an instantaneous regularizing effect), or the singular parts of certain solutions persist until some positivewaiting time(in the linear case{\varphi(u)=u}this happens for all times). In the latter case, we describe the evolution of the singular parts.

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