Philippe G. LeFloch

Convergent and conservative schemes for nonclassical solutions based on kinetic relations. I

We 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 concave-convex as well as convex-concave flux-functions.

A Global Foliation of Einstein–Euler Spacetimes with Gowdy-Symmetry on T3

We investigate the initial value problem for the Einstein–Euler equations of general relativity under the assumption of Gowdy symmetry on T3, and we construct matter spacetimes with low regularity. These spacetimes admit both impulsive gravitational waves in the metric (for instance, Dirac mass curvature singularities propagating at light speed) and shock waves in the fluid (that is, discontinuities propagating at about the sound speed). Given an initial data set, we establish the existence of a future development, and we provide a global foliation in terms of a globally and geometrically defined time-function, closely related to the area of the orbits of the symmetry group. The main difficulty lies in the low regularity assumed on the initial data set which requires a distributional formulation of the Einstein–Euler equations.

The global nonlinear stability of Minkowski space for self-gravitating massive fields. The Wave-Klein-Gordon Model

The Hyperboloidal Foliation Method (introduced by the authors in 2014) is extended here and applied to the Einstein equations of general relativity. Specifically, we establish the nonlinear stability of Minkowski spacetime for self-gravitating massive scalar fields, while existing methods only apply to massless scalar fields. First of all, by analyzing the structure of the Einstein equations in wave coordinates, we exhibit a nonlinear wave-Klein–Gordon model defined on a curved background, which is the focus of the present paper. For this model, we prove here the existence of global-in-time solutions to the Cauchy problem, when the initial data have sufficiently small Sobolev norms. A major difficulty comes from the fact that the class of conformal Killing fields of Minkowski space is significantly reduced in the presence of a massive scalar field, since the scaling vector field is not conformal Killing for the Klein–Gordon operator. Our method relies on the foliation (of the interior of the light cone) of Minkowski spacetime by hyperboloidal hypersurfaces and uses Lorentz-invariant energy norms. We introduce a frame of vector fields adapted to the hyperboloidal foliation and we establish several key properties: Sobolev and Hardy-type inequalities on hyperboloids, as well as sup-norm estimates, which correspond to the sharp time decay for the wave and the Klein–Gordon equations. These estimates allow us to control interaction terms associated with the curved geometry and the massive field by distinguishing between two levels of regularity and energy growth and by a successive use of our key estimates in order to close a bootstrap argument.

Quasilinear hyperbolic Fuchsian systems and AVTD behavior in T 2 -symmetric vacuum spacetimes

We set up the singular initial value problem for quasilinear hyperbolic Fuchsian systems of first order and establish an existence and uniqueness theory for this problem with smooth data and smooth coefficients (and with even lower regularity). We apply this theory in order to show the existence of smooth (generally not analytic) T2-symmetric solutions to the vacuum Einstein equations, which exhibit asymptotically velocity term dominated behavior in the neighborhood of their singularities and are polarized or half-polarized.

Symmetries and Global Solvability of the Isothermal Gas Dynamics Equations

Abstract.We study the Cauchy problem associated with the system of two conservation laws arising in isothermal gas dynamics, in which the pressure and the density are related by the γ-law equation p(ρ)∼ργ with γ=1. Our results complete those obtained earlier for γ>1. We prove the global existence and compactness of entropy solutions generated by the vanishing viscosity method. The proof relies on compensated compactness arguments and symmetry group analysis. Interestingly, we make use here of the fact that the isothermal gas dynamics system is invariant modulo a linear scaling of the density. This property enables us to reduce our problem to that with a small initial density.One symmetry group associated with the linear hyperbolic equations describing all entropies of the Euler equations gives rise to a fundamental solution with initial data imposed on the line ρ=1. This is in contrast to the common approach (when γ>1) which prescribes initial data on the vacuum line ρ=0. The entropies we construct here are weak entropies, i.e., they vanish when the density vanishes.Another feature of our proof lies in the reduction theorem, which makes use of the family of weak entropies to show that a Young measure must reduce to a Dirac mass. This step is based on new convergence results for regularized products of measures and functions of bounded variation.

Weakly Regular Einstein–Euler Spacetimes with Gowdy Symmetry: The Global Areal Foliation

We consider weakly regular Gowdy-symmetric spacetimes on T3 satisfying the Einstein–Euler equations of general relativity, and we solve the initial value problem when the initial data set has bounded variation, only so that the corresponding spacetime may contain impulsive gravitational waves as well as shock waves. By analyzing both future expanding and future contracting spacetimes, we establish the existence of a global foliation by spacelike hypersurfaces so that the time function coincides with the area of the surfaces of symmetry and asymptotically approaches infinity in the expanding case and zero in the contracting case. More precisely, the latter property in the contracting case holds provided the mass density does not exceed a certain threshold, which is a natural assumption since certain exceptional data with sufficiently large mass density are known to give rise to a Cauchy horizon, on which the area function attains a positive value. An earlier result by LeFloch and Rendall assumed a different class of weak regularity and did not determine the range of the area function in the contracting case. Our method of proof is based on a version of the random choice scheme adapted to the Einstein equations for the symmetry and regularity class under consideration. We also analyze the Einstein constraint equations under weak regularity.

SINGULAR LIMITS IN PHASE DYNAMICS WITH PHYSICAL VISCOSITY AND CAPILLARITY

Following pioneering work by Fan and Slemrod who studied the e ect of artificial viscosity terms, we consider the sys- tem of conservation laws arising in liquid-vapor phase dynamics with physical viscosity and capillarity e ects taken into account. Following Dafermos we consider self-similar solutions to the Rie- mann problem and establish uniform total variation bounds, al- lowing us to deduce new existence results. Our analysis cover both the hyperbolic and the hyperbolic-elliptic regimes and apply to arbitrarily large Riemann data. The proofs rely on a new technique of reduction to two coupled scalar equations associated with the two wave fans of the system. Strong L 1 convergence to a weak solution of bounded variation is established in the hyperbolic regime, while in the hyperbolic- elliptic regime a stationary singularity near the axis separating the two wave fans, or more generally an almost-stationary oscil- lating wave pattern (of thickness depending upon the capillarity- viscosity ratio) are observed which prevent the solution to have globally bounded variation.

Nonlinear Hyperbolic Systems: Nondegenerate Flux, Inner Speed Variation, and Graph Solutions

We study the Cauchy problem for general nonlinear strictly hyperbolic systems of partial differential equations in one space variable. First, we re-visit the construction of the solution to the Riemann problem and introduce the notion of a nondegenerate (ND) system. This is the optimal condition guaranteeing, as we show it, that the Riemann problem can be solved with finitely many waves only; we establish that the ND condition is generic in the sense of Baire (for the Whitney topology), so that any system can be approached by a ND system. Second, we introduce the concept of inner speed variation and we derive new interaction estimates on wave speeds. Third, we design a wave front tracking scheme and establish its strong convergence to the entropy solution of the Cauchy problem; this provides a new existence proof as well as an approximation algorithm. As an application, we investigate the time regularity of the graph solutions (X,U) introduced by LeFloch, and propose a geometric version of our scheme; in turn, the spatial component X of a graph solution can be chosen to be continuous in both time and space, while its component U is continuous in space and has bounded variation in time.

Hyperbolic Conservation Laws on Spacetimes

We present a generalization of Kruzkov’s theory to manifolds. Nonlinear hyperbolic conservation laws are posed on a differential (n + 1)-manifold, called a spacetime, and the flux field is defined as a field of n-forms depending on a parameter. The entropy inequalities take a particularly simple form as the exterior derivative of a family of n-form fields. Under a global hyperbolicity condition on the spacetime, which allows arbitrary topology for the spacelike hypersurfaces of the foliation, we establish the existence and uniqueness of an entropy solution to the initial value problem, and we derive a geometric version of the standard L 1 semi-group property. We also discuss an alternative framework in which the flux field consists of a parametrized family of vector fields.

Coupling techniques for nonlinear hyperbolic equations. I. Self-similar diffusion for thin interfaces

We investigate various analytical and numerical techniques for the coupling of nonlinear hyperbolic systems and, in particular, we introduce here an augmented formulation which allows for the modeling of the dynamics of interfaces between fluid flows. The main technical difficulty to be overcome lies in the possible resonance effect when wave speeds coincide and global hyperbolicity is lost. As a consequence, non-uniqueness of weak solutions is observed for the initial value problem which need to be supplemented with further admissibility conditions. This first paper is devoted to investigating these issues in the setting of self-similar vanishing viscosity approximations to the Riemann problem for general hyperbolic systems. Following earlier works by Joseph, LeFloch, and Tzavaras, we establish an existence theorem for the Riemann problem under fairly general structural assumptions on the nonlinear hyperbolic system and its regularization. Our main contribution consists of nonlinear wave interaction estimates for solutions which apply to resonant wave patterns.