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Philippe G. LeFloch

Publications and source records attributed to Philippe G. LeFloch.

At least 109 records · Page 6Linked to original sources
arXiv2011

A framework for late-time/stiff relaxation asymptotics

We consider solutions to nonlinear hyperbolic systems of balance laws with stiff relaxation and formally derive a parabolic-type effective system describing the late-time asymptotics of these solutions. We show that many examples from continuous physics fall into our framework, including the Euler equations with (possibly nonlinear) friction. We then turn our attention to the discretization of these stiff problems and introduce a new finite volume scheme which preserves the late-time asymptotic regime. Importantly, our scheme requires only the classical CFL (Courant, Friedrichs, Lewy) condition associated with the hyperbolic system under consideration, rather than the more restrictive, parabolic-type stability condition.

math.AP↗
arXiv2011

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 (i.e., 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.

gr-qc↗
arXiv2011

The characteristic initial value problem for plane symmetric spacetimes with weak regularity

We investigate the existence and the global causal structure of plane symmetric spacetimes with weak regularity when the matter consists of an irrotational perfect fluid with pressure equal to its mass-energy density. Our theory encompasses the class of weakly regular spacetimes whose metric coefficients have square-integrable first-order derivatives and whose curvature must be understood in the sense of distributions. We formulate the characteristic initial value problem with data posed on two null hypersurfaces intersecting along a two-plane. Relying on Newman-Penrose's formalism and expressing our weak regularity conditions in terms of the Newman-Penrose scalars, we arrive at a fully geometrical formulation in which, along each initial hypersurface, two scalar fields describing the incoming radiation must be prescribed. To analyze the future boundary of such a spacetime and identify its global causal structure, we introduce a gauge that reduces the Einstein equations to a coupled system of wave equations and ordinary differential equations for well-chosen unknowns. We prove that, within the weak regularity class under consideration and for generic initial data, a true spacetime singularity forms in finite proper time. Our formulation is robust enough so that propagating discontinuities in the curvature or in the matter variables do not prevent us from constructing a spacetime whose curvature generically blows-up on the future boundary. Earlier work on the problem studied here was restricted to sufficiently regular and vacuum spacetimes.

gr-qc↗
arXiv2011

Weakly regular T2 symmetric spacetimes. The global geometry of future developments

Under weak regularity assumptions, only, we develop a fully geometric theory of vacuum Einstein spacetimes with T2 symmetry, establish the global well-posedness of the initial value problem for Einstein's field equations, and investigate the global causal structure of the constructed spacetimes. Our weak regularity assumptions are the minimal ones allowing to give a meaning to the Einstein equations under the assumed symmetry and to solve the initial value problem. First of all, we introduce a frame adapted to the symmetry in which each Christoffel symbol can be checked to belong to some Lp space. We identify certain cancellation properties taking place in the expression of the Riemann and Ricci curvatures, and this leads us to a reformulation of the initial value problem for the Einstein field equations when the initial data set has weak regularity. Second, we investigate the future development of a weakly regular initial data set. We check that the area R of the orbits of symmetry must grow to infinity in the future timelike directions, and we establish the existence of a global foliation by the level sets of R. Our weak regularity assumptions only require that R is Lipschitz continuous while the metric coefficients describing the initial geometry of the orbits of symmetry are in the Sobolev space H1 and the remaining coefficients have even weaker regularity. We develop here the compactness arguments required to cover the natural level of regularity associated with the energy of the system of partial differential equations determined from Einstein's field equations.

gr-qc↗
arXiv2011

A geometric approach to error estimates for conservation laws posed on a spacetime

We consider a hyperbolic conservation law posed on an (N+1)-dimensional spacetime, whose flux is a field of differential forms of degree N. Generalizing the classical Kuznetsov's method, we derive an L1 error estimate which applies to a large class of approximate solutions. In particular, we apply our main theorem and deal with two entropy solutions associated with distinct flux fields, as well as with an entropy solution and an approximate solution. Our framework encompasses, for instance, equations posed on a globally hyperbolic Lorentzian manifold.

math.AP↗
arXiv2011

Why many theories of shock waves are necessary. Kinetic relations for nonconservative systems

For a class of nonconservative hyperbolic systems of partial differential equations endowed with a strictly convex mathematical entropy, we formulate the initial value problem by supplementing the equations with a kinetic relation prescribing the rate of entropy dissipation across shock waves. Our condition can be regarded as a generalization to nonconservative systems of a similar concept introduced by Abeyaratne, Knowles, and Truskinovsky for subsonic phase transitions and by LeFloch for nonclassical undercompressive shocks to nonlinear hyperbolic systems. The proposed kinetic relation for nonconservative systems turns out to be equivalent, for the class of systems under consideration at least, to Dal Maso, LeFloch, and Murat's definition based on a prescribed family of Lipschitz continuous paths. In agreement with previous theories, the kinetic relation should be derived from a phase plane analysis of traveling solutions associated with an augmented version of the nonconservative system. We illustrate with several examples that nonconservative systems arising in the applications fit in our framework, and for a typical model of turbulent fluid dynamics, we provide a detailed analysis of the existence and properties of traveling waves which yields the corresponding kinetic function.

math.AP↗
arXiv2011

Second-order hyperbolic Fuchsian systems and applications

We introduce a new class of singular partial differential equations, referred to as the second-order hyperbolic Fuchsian systems, and we investigate the associated initial value problem when data are imposed on the singularity. First, we establish a general existence theory of solutions with asymptotic behavior prescribed on the singularity, which relies on a new approximation scheme, suitable also for numerical purposes. Second, this theory is applied to the (vacuum) Einstein equations for Gowdy spacetimes, and allows us to recover, by more direct arguments, well-posedness results established earlier by Rendall and collaborators. Another main contribution in this paper is the proposed approximation scheme, which we refer to as the Fuchsian numerical algorithm and is shown to provide highly accurate numerical approximations to the singular initial value problem. For the class of Gowdy spacetimes, the numerical experiments presented here show the interest and efficiency of the proposed method and demonstrate the existence of a class of Gowdy spacetimes containing a smooth, incomplete, and non-compact Cauchy horizon.

gr-qc↗
arXiv2011

Einstein spacetimes with weak regularity

We review recent work on the Einstein equations of general relativity when the curvature is defined in a weak sense. Weakly regular spacetimes are constructed, in which impulsive gravitational waves, as well as shock waves, propagate.

gr-qc↗
arXiv2011

Second-order hyperbolic Fuchsian systems. General theory

We introduce a class of singular partial differential equations, the second-order hyperbolic Fuchsian systems, and we investigate the associated initial value problem when data are imposed on the singularity. First of all, we analyze a class of equations in which hyperbolicity is not assumed and we construct asymptotic solutions of arbitrary order. Second, for the proposed class of second-order hyperbolic Fuchsian systems, we establish the existence of solutions with prescribed asymptotic behavior on the singularity. Our proof is based on a new scheme which is also suitable to design numerical approximations. Furthermore, as shown in a follow-up paper, the second-order Fuchsian framework is appropriate to handle Einstein's field equations for Gowdy symmetric spacetimes and allows us to recover (and slightly generalize) earlier results by Rendall and collaborators, while providing a direct approach leading to accurate numerical solutions. The proposed framework is also robust enough to encompass matter models arising in general relativity.

gr-qc↗
arXiv2011

Second-order hyperbolic Fuchsian systems. Gowdy spacetimes and the Fuchsian numerical algorithm

This is the second part of a series devoted to the singular initial value problem for second-order hyperbolic Fuchsian systems. In the first part, we defined and investigated this general class of systems, and we established a well-posedness theory in weighted Sobolev spaces. This theory is applied here to the vacuum Einstein equations for Gowdy spacetimes admitting, by definition, two Killing fields satisfying certain geometric conditions. We recover, by more direct and simpler arguments, the well-posedness results established earlier by Rendall and collaborators. In addition, in this paper we introduce a natural approximation scheme, which we refer to as the Fuchsian numerical algorithm and is directly motivated by our general theory. This algorithm provides highly accurate, numerical approximations of the solution to the singular initial value problem. In particular, for the class of Gowdy spacetimes under consideration, various numerical experiments are presented which show the interest and efficiency of the proposed method. Finally, as an application, we numerically construct Gowdy spacetimes containing a smooth, incomplete, non-compact Cauchy horizon.

gr-qc↗
arXiv2010

Existence of naked singularities in Brans-Dicke theory of gravitation. An analytical and numerical study

Within the framework of the scalar-tensor models of gravitation and by relying on analytical and numerical techniques, we establish the existence of a class of spherically symmetric spacetimes containing a naked singularity. Our result relies on and extends a work by Christodoulou on the existence of naked singularities for the Einstein-scalar field equations. We establish that a key parameter in Christodoulou's construction couples to the Brans-Dicke field and becomes a dynamical variable, which enlarges and modifies the phase space of solutions. We recover analytically many properties first identified by Christodoulou, in particular the loss of regularity (especially at the center), and then investigate numerically the properties of these spacetimes.

gr-qc↗
arXiv2010

Global geometry of T2 symmetric spacetimes with weak regularity

We define the class of weakly regular spacetimes with T2 symmetry, and investigate their global geometry structure. We formulate the initial value problem for the Einstein vacuum equations with weak regularity, and establish the existence of a global foliation by the level sets of the area R of the orbits of symmetry, so that each leaf can be regarded as an initial hypersurface. Except for the flat Kasner spacetimes which are known explicitly, R takes all positive values. Our weak regularity assumptions only require that the gradient of R is continuous while the metric coefficients belong to the Sobolev space H1 (or have even less regularity).

gr-qc↗
arXiv2010

Uniqueness of Kottler spacetime and Besse conjecture

We establish a black hole uniqueness theorem for Schwarzschild-de Sitter spacetime, also called Kottler spacetime, which satisfies Einstein's field equations of general relativity with positive cosmological constant. Our result concerns the class of static vacuum spacetimes with compact spacelike slices and regular maximal level set of the lapse function. We provide a characterization of the interior domain of communication of the Kottler spacetime, which surrounds an inner horizon and is surrounded by a cosmological horizon. The proposed proof combines arguments from the theory of partial differential equations and differential geometry, and is centered on a detailed study of a possibly singular foliation. We also apply our technique in the Riemannian setting, and establish the validity of the so-called Besse conjecture.

gr-qc↗
arXiv2010

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 L1 semi-group property. We also discuss an alternative framework in which the flux field consists of a parametrized family of vector fields.

math.AP↗
arXiv2010

Kinetic relations for undercompressive shock waves. Physical, mathematical, and numerical issues

Kinetic relations are required in order to characterize nonclassical undercompressive shock waves and formulate a well-posed initial value problem for nonlinear hyperbolic systems of conservation laws. Such nonclassical waves arise in weak solutions of a large variety of physical models: phase transitions, thin liquid films, magnetohydrodynamics, Camassa-Holm model, martensite-austenite materials, semi-conductors, combustion theory, etc. This review presents the research done in the last fifteen years which led the development of the theory of kinetic relations for undercompressive shocks and has now covered many physical, mathematical, and numerical issues. The main difficulty overcome here in our analysis of nonclassical entropy solutions comes from their lack of monotonicity with respect to initial data. Undercompressive shocks of hyperbolic conservation laws turn out to exhibit features that are very similar to shocks of nonconservative hyperbolic systems, who were investigated earlier by the author.

math.AP↗
arXiv2008

Local foliations and optimal regularity of Einstein spacetimes

We investigate the local regularity of pointed spacetimes, that is, time-oriented Lorentzian manifolds in which a point and a future-oriented, unit timelike vector (an observer) are selected. Our main result covers the class of Einstein vacuum spacetimes. Under curvature and injectivity bounds only, we establish the existence of a local coordinate chart defined in a ball with definite size in which the metric coefficients have optimal regularity. The proof is based on quantitative estimates, on one hand, for a constant mean curvature (CMC) foliation by spacelike hypersurfaces defined locally near the observer and, on the other hand, for the metric in local coordinates that are spatially harmonic in each CMC slice. The results and techniques in this paper should be useful in the context of general relativity for investigating the long-time behavior of solutions to the Einstein equations.

gr-qc↗
arXiv2008

Sharp L1 stability estimates for hyperbolic conservation laws

In this paper, we introduce a generalization of Liu-Yang's weighted norm to linear and to nonlinear hyperbolic equations. Extending a result by Hu and LeFloch for piecewise constant solutions, we establish sharp L1 continuous dependence estimates for general solutions of bounded variation. Two different strategies are successfully investigated. On one hand, we justify passing to the limit in an L1 estimate valid for piecewise constant wave-front tracking approximations. On the other hand, we use the technique of generalized characteristics and, following closely an approach by Dafermos, we derive the sharp L1 estimate directly from the equation.

math.AP↗