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21
Nonclassical Riemann solvers with nucleation
 Proc. Royal Soc
"... Abstract. We introduce a new nonclassical Riemann solver for scalar conservation laws with concaveconvex fluxfunction. This solver is based on both a kinetic relation, which determines the propagation speed of (undercompressive) nonclassical shock waves, and a nucleation criterion, which makes a c ..."
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Cited by 13 (2 self)
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Abstract. We introduce a new nonclassical Riemann solver for scalar conservation laws with concaveconvex fluxfunction. This solver is based on both a kinetic relation, which determines the propagation speed of (undercompressive) nonclassical shock waves, and a nucleation criterion, which makes a choice between a classical Riemann solution and a nonclassical one. We establish the existence of (nonclassical entropy) solutions of the Cauchy problem and discuss several examples of wave interactions. We also show the existence of a class of solutions, called splittingmerging solutions, which are made of two large shocks and small BV (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. 1.
A convergent and conservative schemes for nonclassical solutions based on kinetic relations
 I. Interfaces and Free Bound
, 2008
"... AbstractWe propose a new numerical approach to compute nonclassical solutions to hyperbolic conservation laws. The class of finite difference schemes presented here is fully conservative and keep nonclassical shock waves as sharp interfaces, contrary to standard finite difference schemes. The main ..."
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Cited by 8 (4 self)
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AbstractWe propose a new numerical approach to compute nonclassical solutions to hyperbolic conservation laws. The class of finite difference schemes presented here is fully conservative and keep nonclassical shock waves as sharp interfaces, contrary to standard finite difference schemes. The main challenge is to achieve, at the discretization level, a consistency property with respect to a prescribed kinetic relation. The latter is required for the selection of physically meaningful nonclassical shocks. Our method is based on a reconstruction technique performed in each computational cell that may contain a nonclassical shock. To validate this approach, we establish several consistency and stability properties, and we perform careful numerical experiments. The convergence of the algorithm toward the physically meaningful solutions selected by a kinetic relation is demonstrated numerically for several test cases, including concaveconvex as well as convexconcave fluxfunctions. Résume ́ Nous proposons un nouvel algorithme pour approcher les solutions non classiques de lois de conservation hyperboliques. Le schéma aux différences finies présente ́ ici est conservatif et transporte de manière exacte les chocs non classiques, a ̀ la différences des algorithmes standard. La principale difficulte ́ est de garantir, au niveau discret, la consistance avec une re
PHASE TRANSITIONS AND SHARPINTERFACE LIMITS FOR THE 1DELASTICITY SYSTEM WITH NONLOCAL ENERGY
"... Abstract. The onedimensional system of elasticity with a nonmonotone or convexconcave stressstrain relation provides a model to describe the longitudinal dynamics of solidsolid phase transitions in a bar. If dissipative effects are neglected it takes the form of a system of firstorder nonlinea ..."
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Cited by 6 (0 self)
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Abstract. The onedimensional system of elasticity with a nonmonotone or convexconcave stressstrain relation provides a model to describe the longitudinal dynamics of solidsolid phase transitions in a bar. If dissipative effects are neglected it takes the form of a system of firstorder nonlinear conservation laws and dynamical phase boundaries appear as shock wave solutions. In the physically most relevant cases these shocks are of the nonclassical undercompressive type and therefore entropy solutions of the associated Cauchy problem are not uniquely determined. Important dissipative effects that lead to unique regular solutions are viscosity and capillarity where the latter effect is usually modelled by at least thirdorder spatial derivatives. Differently from these models we consider a novel type of nonlocal regularization that models both effects but avoids highorder derivatives. We suggest a particular scaling for the dissipative terms and conjecture that with this scaling the regular solutions single out unique physically relevant weak solutions of the firstorder conservation law in the limit of vanishing dissipation parameter. We verify the conjecture first by proving that the nonlocal system admits special solutions of travelingwave type that correspond to dynamical phase boundaries. Moreover it is proven that regular solutions of a general Cauchy problem converge to weak solutions of the system of firstorder conservation laws. The proof is achieved by the method of compensated compactness. Key words. Nonlocal energy, viscositycapillarity, sharp interface limit, undercompressive shock waves. 1. Introduction. We
DIMINISHING FUNCTIONALS FOR NONCLASSICAL ENTROPY SOLUTIONS SELECTED BY KINETIC RELATIONS
, 812
"... Abstract. We consider nonclassical entropy solutions to scalar conservation laws with concaveconvex flux functions, whose set of left and righthand admissible states ul, ur across undercompressive shocks is selected by a kinetic function ur = ϕ ♭ (ul). We introduce a new definition for the (genera ..."
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Cited by 2 (2 self)
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Abstract. We consider nonclassical entropy solutions to scalar conservation laws with concaveconvex flux functions, whose set of left and righthand admissible states ul, ur across undercompressive shocks is selected by a kinetic function ur = ϕ ♭ (ul). We introduce a new definition for the (generalized) strength of classical and nonclassical shocks, allowing us to propose a generalized notion of total variation functional. Relying only upon the natural assumption that the composite function ϕ ♭ ◦ ϕ ♭ is uniformly contracting, we prove that the generalized total variation of fronttracking approximations is nonincreasing in time, and we conclude with the existence of nonclassical solutions to the initialvalue problem. We also propose two definitions of generalized interaction potentials which are adapted to handle nonclassical entropy solutions and we investigate their monotonicity properties. In particular, we exhibit an interaction functional which is globally nonincreasing along a splittingmerging interaction pattern. Key words. hyperbolic conservation law; entropy solution; nonconvex fluxfunction; nonclassical shock; kinetic relation; total variation diminishing; interaction potential.
Moderate dispersion in scalar conservation laws
 Communications in Mathematical Sciences
, 2007
"... We consider the weakly dissipative and weakly dispersive BurgersHopfKortewegdeVries equation with the diffusion coefficient ε and the dispersion rate δ in the range δ/ε → 0. We study the travelling wave connecting u(−∞) = 1 to u(+∞) = 0 and show that it converges strongly to the entropic shock ..."
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Cited by 1 (0 self)
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We consider the weakly dissipative and weakly dispersive BurgersHopfKortewegdeVries equation with the diffusion coefficient ε and the dispersion rate δ in the range δ/ε → 0. We study the travelling wave connecting u(−∞) = 1 to u(+∞) = 0 and show that it converges strongly to the entropic shock profile as ε, δ → 0. Keywords Travelling waves, moderate dispersion, Korteweg de Vries equation, entropy solutions, dissipativedispersive scalar conservation laws.
The Riemann problem for the multipressure Euler system
"... Abstract. We prove the existence and uniqueness of the Riemann solutions to the Euler equations closed byN independent constitutive pressure laws. This model stands as a natural asymptotic system for the multipressure Navier–Stokes equations in the regime of infinite Reynolds number. Due to the inh ..."
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Abstract. We prove the existence and uniqueness of the Riemann solutions to the Euler equations closed byN independent constitutive pressure laws. This model stands as a natural asymptotic system for the multipressure Navier–Stokes equations in the regime of infinite Reynolds number. Due to the inherent lack of conservation form in the viscous regularization, the limit system exhibits measurevalued source terms concentrated on shock discontinuities. These nonpositive bounded measures, called kinetic relations, are known to provide a suitable tool to encode the smallscale sensitivity in the singular limit. Considering N independent polytropic pressure laws, we show that these kinetic relations can be derived by solving a simple algebraic problem which governs the endpoints of the underlying viscous shock profiles, for any given but prescribed ratio of viscosity coefficient in the viscous perturbation. The analysis based on traveling wave solutions allows us to introduce the asymptotic Euler system in the setting of piecewise Lipschitz continuous functions and to study the Riemann problem.
après avis des rapporteurs et devant le jury
, 2009
"... pour l’obtention du titre de Docteur de l’Université Pierre et Marie Curie Paris 6 Sujet de la thèse: ..."
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pour l’obtention du titre de Docteur de l’Université Pierre et Marie Curie Paris 6 Sujet de la thèse: