Sylvie Benzoni-Gavage

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-populations model for traffic flow

-population generalization of the Lighthill–Whitham and Richards traffic flow model. This model is analytically interesting because of several non-standard features. For instance, it leads to non-classical shocks and enjoys an unexpected stability in spite of the presence of umbilic points. Furthermore, while satisfying all the minimal ‘common sense’ requirements, it also allows for a description of phenomena often neglected by other models, such as overtaking.

Alternate Evans Functions and Viscous Shock Waves

The Evans function is known as a helpful tool for locating the spectrum of some variational differential operators. This is of special interest regarding the stability analysis of traveling waves, ...

Structure of Korteweg models and stability of diffuse interfaces

The models considered are supposed to govern the motion of compressible fluids such as liquid-vapor mixtures endowed with a variable internal capillarity. Several formulations and simplifica- tions are discussed, from the full multi-dimensional equations for non-isothermal motions in Eulerian coordinates to the one-dimensional equations for isothermal motions in Lagrangian coordinates. Hamil- tonian structures are pointed out in each case, and in the one-dimensional isothermal case, they are used to study the stability of two kinds of non-linear waves: the solitary, or homoclinic waves, and the het- eroclinic waves, which correspond to propagating phase boundaries of non-zero thickness, also called diffuse interfaces. It is known from an earlier work by Benzoni-Gavage (Physica D, 2001) that the lat- ter are (weakly) spectrally stable. Here, diffuse interfaces are shown to be orbitally stable. The proof relies on their interpretation as critical points of the Hamiltonian under constraints, whose justification requires some care because of the different endstates at infinity. Another difficulty comes from higher order derivatives that are not controlled by the Hamiltonian. In the case of a variable capillarity, our stability result unfortunately does not imply global existence. As regards the solitary waves, which come into families parametrized by the wave speed, they are not stable from the variational point of view. However, using a method due to Grillakis, Shatah and Strauss (Journal of Functional Analysis, 1987), it is possible to show that some solitary waves, depending on their speed, are orbitally stable. Namely, the convexity of a function of the wave speed called moment of instability determines the stability of soli- tary waves. This approach, already used by Bona and Sachs (Communications in Mathematical Physics, 1988) for the Boussinesq equation, is here adapted to solitary waves in Korteweg models, which are first classified according to their endstate and internal structure. The corresponding moments of instability are computed by quadrature. They exhibit both convexity and concavity regions. AMS classification. 76T10; 35B35; 35Q51; 37C29.

Generic types and transitions in hyperbolic initial–boundary-value problems

The stability of linear initial–boundary-value problems for hyperbolic systems (with constant coefficients) is linked to the zeros of the so-called Lopatinskii determinant. Depending on the location of these zeros, problems may be either unstable, strongly stable or weakly stable. The first two classes are known to be ‘open’, in the sense that the instability or the strong stability persists under a small change of coefficients in the differential operator and/or in the boundary condition. Here we show that a third open class exists, which we call ‘weakly stable of real type’. Many examples of physical or mathematical interest depend on one or more parameters, and the determination of the stability class as a function of these parameters usually needs an involved computation. We simplify it by characterizing the transitions from one open class to another one. These boundaries are easier to determine since they must solve some overdetermined algebraic system. Applications to the wave equation, linear elasticity, shock waves and phase boundaries in fluid mechanics are given.

Semi-discrete shock profiles for hyperbolic systems of conservation laws

Abstract The existence of semi-discrete shock profiles for a general hyperbolic system of conservation laws is proved. Such profiles are regarded as heteroclinic orbits of a retarded functional differential equation (RFDE). The proof relies on the Hale center manifold theorem and holds for shocks of small strength.

Nonlinear stability of semidiscrete shock waves

The orbital stability of possibly large semidiscrete shock waves is considered. These waves are traveling wave solutions of discrete in space and continuous in time systems of conservation laws, which constitute a class of lattice dynamical systems (LDSs). The underlying lattice

Stability of Semi-Discrete Shock Profiles by Means of an Evans Function in Infinite Dimensions

\Delta x \mathbb{Z}

Note on a paper by Robinet, Gressier, Casalis & Moschetta

is by nature not invariant by change of frame. Thus semidiscrete shock waves cannot really be transformed into stationary waves, unlike other kinds of approximate shock waves (e.g., viscous or relaxation shocks). This implies that the linearization of the LDS about a given semidiscrete shock wave yields a nonautonomous linear LDS, which cannot be tackled by means of Laplace transform in time. However, viewing the LDS as a finite-difference PDE and performing afterall the change of frame, the profile becomes a stationary solution of the transformed equation. Then, linearizing about the profile, we get an evolution finite-difference PDE in which the spatial operator L, a delayed and advanced differential operat...