arXiv subjects

Philippe G. LeFloch

Publications and source records attributed to Philippe G. LeFloch.

At least 55 records · Page 3Linked to original sources

arXiv2019

Asymptotic structure of cosmological fluid flows in one and two space dimensions: a numerical study

We consider an isothermal compressible fluid evolving on a cosmological background which may be either expanding or contracting toward the future. The Euler equations governing such a flow consist of two nonlinear hyperbolic balance laws which we treat in one and in two space dimensions. We design a finite volume scheme which is fourth-order accurate in time and second-order accurate in space. This scheme allows us to compute weak solutions containing shock waves and, by design, is well-balanced in the sense that it preserves exactly a special class of solutions. Using this scheme, we investigate the asymptotic structure of the fluid when the time variable approaches infinity (in the expanding regime) or approaches zero (in the contracting regime). We study these two limits by introducing a suitable rescaling of the density and velocity variables and, in turn, we analyze the effects induced by the geometric terms (of expanding or contracting nature) and the nonlinear interactions between shocks. Extensive numerical experiments in one and in two space dimensions are performed in order to support our observations.

math.AP↗

arXiv2019

Compensated compactness and corrector stress tensor for the Einstein equations in T2 symmetry

We consider the Einstein equations in T2 symmetry, either for vacuum spacetimes or coupled to the Euler equations for a compressible fluid, and we introduce the notion of T2 areal flows on T3 with finite total energy. By uncovering a hidden structure of the Einstein equations, we establish a compensated compactness framework and solve the global evolution problem for vacuum spacetimes as well as for self-gravitating compressible fluids. We study the stability and instability of such flows and prove that, when the initial data are well-prepared, any family of T2 areal flows is sequentially compact in a natural topology. In order to handle general initial data we propose a relaxed notion of T2 areal flows endowed with a corrector stress tensor (as we call it) which is a bounded measure generated by geometric oscillations and concentrations propagating at the speed of light. This generalizes a result for vacuum spacetimes in: Le Floch B. and LeFloch P.G., Arch. Rational Mech. Anal. 233 (2019), 45-86. In addition, we determine the global geometry of the corresponding future Cauchy developments and we prove that the area of the T2 orbits generically approaches infinity in the future-expanding regime. In the future-contracting regime, the volume of the T3 spacelike slices approaches zero and, for generic initial data, the area of the orbits of symmetry approaches zero in Gowdy symmetric matter spacetimes and in T2 vacuum spacetimes.

gr-qc↗

arXiv2019

Augmented hyperbolic models with diffusive-dispersive shocks

Given a first-order nonlinear hyperbolic system of conservation laws endowed with a convex entropy-entropy flux pair, we can consider the class of weak solutions containing shock waves depending upon some small scale parameters. In this Note, we define and derive several classes of entropy-dissipating augmented models, as we call them, which involve (possibly nonlinear) second- and third-order augmentation terms. Such terms typically arise in continuum physics and model the viscosity and capillarity effects in a fluid, for instance. By introducing a new notion of positive entropy production that concerns general functions (rather than solutions) we can easily check the entropy-dissipating property for a broad class of augmented models. The weak solutions associated with the zero diffusion/dispersion limit may contain (nonclassical undercompressive) shocks whose selection is determined from these diffusive and dispersive effects (for instance by using traveling wave solutions), and having a classification of the models, as we propose, is essential for developing a general theory.

math.AP↗

arXiv2019

Mesh-free error integration in arbitrary dimensions: a numerical study of discrepancy functions

We are interested in mesh-free formulas based on the Monte-Carlo methodology for the approximation of multi-dimensional integrals, and we investigate their accuracy when the functions belong to a reproducing-kernel space. A kernel typically captures regularity and qualitative properties of functions "beyond" the standard Sobolev regularity class. We are interested in the issue whether quantitative error bounds can be a priori guaranteed in applications (e.g. mathematical finance but also scientific computing and machine learning). Our main contribution is a numerical study of the error discrepancy function based on a comparison between several numerical strategies, when one varies the choice of the kernel, the number of approximation points, and the dimension of the problem. We consider two strategies in order to localize to a bounded set the standard kernels defined in the whole Euclidian space (exponential, multiquadric, Gaussian, truncated), namely, on one hand the class of periodic kernels defined via a discrete Fourier transform on a lattice and, on the other hand, a class of transport-based kernels. First of all, relying on a Poisson formula on a lattice, together with heuristic arguments, we discuss the derivation of theoretical bounds for the discrepancy function of periodic kernels. Second, for each kernel of interest, we perform the numerical experiments that are required in order to generate the optimal distributions of points and the discrepancy error functions. Our numerical results allow us to validate our theoretical observations and provide us with quantitative estimates for the error made with a kernel-based strategy as opposed to a purely random strategy.

math.AP↗

arXiv2019

The Transport-based Mesh-free Method (TMM) and its applications in finance: a review

We review a numerical technique, referred to as the Transport-based Mesh-free Method (TMM), and we discuss its applications to mathematical finance. We recently introduced this method from a numerical standpoint and investigated the accuracy of integration formulas based on the Monte-Carlo methodology: quantitative error bounds were discussed and, in this short note, we outline the main ideas of our approach. The techniques of transportation and reproducing kernels lead us to a very efficient methodology for numerical simulations in many practical applications, and provide some light on the methods used by the artificial intelligence community. For applications in the finance industry, our method allows us to compute many types of risk measures with an accurate and fast algorithm. We propose theoretical arguments as well as extensive numerical tests in order to justify sharp convergence rates, leading to rather optimal computational times. Cases of direct interest in finance support our claims and the importance of the problem of the curse of dimensionality in finance applications is briefly discussed.

math.AP↗

arXiv2019

Asymptotic structure of cosmological Burgers flows in one and two space dimensions: a numerical study

We study the cosmological Burgers model, as we call it, which is a nonlinear hyperbolic balance law (in one and two spatial variables) posed on an expanding or contracting background. We design a finite volume scheme that is fourth-order in time and second-order in space, and allows us to compute weak solutions containing shock waves. Our main contribution is the study of the asymptotic structure of the solutions as the time variable approaches infinity (in the expanding case) or zero (in the contracting case). We discover that a saddle competition is taking place which involves, on one hand, the geometrical effects of expanding or contracting nature and, on the other hand, the nonlinear interactions between shock waves.

math.AP↗

arXiv2019

The finite volume method on a Schwarzschild background

We introduce a class of nonlinear hyperbolic conservation laws on a Schwarzschild black hole background and derive several properties satisfied by (possibly weak) solutions. Next, we formulate a numerical approximation scheme which is based on the finite volume methodology and takes the curved geometry into account. An interesting feature of our model is that no boundary conditions is required at the black hole horizon boundary. We establish that this scheme converges to an entropy weak solution to the initial value problem and, in turn, our analysis also provides us with a theory of existence and stability for a new class of conservation laws.

math.AP↗

arXiv2017

The Euclidian-hyperboidal foliation method and the nonlinear stability of Minkowski spacetime

We introduce a new method for analyzing nonlinear wave-Klein-Gordon systems and establishing global-in-time existence results for the Cauchy problem when the initial data need not have compact support. This method, which we call the Euclidian-Hyperboidal Foliation Method (EHFM), relies on the construction of a spacetime foliation obtained by glueing together asymptotically Euclidian and asymptotically hyperboloidal hypersurfaces. Well-chosen frames of vector fields (null-semi-hyperboloidal, Euclidian-hyperboloidal) allow us to exhibit the structure of the equations under consideration and analyze the decay of solutions in timelike and in spacelike directions. New Sobolev inequalities for Euclidian-hyperboloidal foliations involving the Killing fields of Minkowski spacetime (but not the scaling field), as well as pointwise bounds for wave and Klein-Gordon equations on curved spacetimes are established. Our bootstrap argument involves a hierarchy of (almost sharp) energy and pointwise bounds and distinguishes between low- and high-order derivatives of the solutions. We apply this method to the Einstein equations when the matter model is a massive field and the methods by Christodolou and Klainerman and by Lindblad and Rodnianski do not apply.

math.AP↗

arXiv2017

Self-gravitating fluid flows with Gowdy symmetry near cosmological singularities

We consider self-gravitating fluids in cosmological spacetimes with Gowdy symmetry on the torus $T^3$ and, in this class, we solve the singular initial value problem for the Einstein-Euler system of general relativity, when an initial data set is prescribed on the hypersurface of singularity. We specify initial conditions for the geometric and matter variables and identify the asymptotic behavior of these variables near the cosmological singularity. Our analysis of this class of nonlinear and singular partial differential equations exhibits a condition on the sound speed, which leads us to the notion of sub-critical, critical, and super-critical regimes. Solutions to the Einstein-Euler systems when the fluid is governed by a linear equation of state are constructed in the first two regimes, while additional difficulties arise in the latter one. All previous studies on inhomogeneous spacetimes concerned vacuum cosmological spacetimes only.

gr-qc↗

arXiv2017

The global nonlinear stability of Minkowski space for self-gravitating massive fields

The theory presented in this monograph establishes the first mathematically rigorous result on the global nonlinear stability of self-gravitating matter under small perturbations of an asymptotically flat, spacelike hypersurface of Minkowski spacetime. It allows one to exclude the existence of dynamically unstable, self-gravitating massive fields and, therefore, solves a long-standing open problem in General Relativity. By a significant extension of the Hyperboloidal Foliation Method they introduced in 2014, the authors establish global-in-time existence for the Einstein equations expressed as a coupled wave-Klein-Gordon system of partial differential equations. The metric and matter fields are sought for in Sobolev-type functional spaces, suitably defined from the translations and the boosts of Minkowski spacetime.

gr-qc↗

arXiv2017

Ergodicity of spherically symmetric fluid flows outside of a Schwarzschild black hole with random boundary forcing

We consider the Burgers equation posed on the outer communication region of a Schwarzschild black hole spacetime. Assuming spherical symmetry for the fluid flow under consideration, we study the propagation and interaction of shock waves under the effect of random forcing. First of all, considering the initial and boundary value problem with boundary data prescribed in the vicinity of the horizon, we establish a generalization of the Hopf--Lax--Oleinik formula, which takes the curved geometry into account and allows us to establish the existence of bounded variation solutions. To this end, we analyze the global behavior of the characteristic curves in the Schwarzschild geometry, including their behavior near the black hole horizon. In a second part, we investigate the long-term statistical properties of solutions when a random forcing is imposed near the black hole horizon and study the ergodicity of the fluid flow under consideration. We prove the existence of a random global attractor and, for the Burgers equation outside of a Schwarzschild black hole, we are able to validate the so-called `one-force-one-solution' principle. Furthermore, all of our results are also established for a pressureless Euler model which consists of two balance laws and includes a transport equation satisfied by the integrated fluid density.

math.AP↗

arXiv2016

The global nonlinear stability of Minkowski spacetime for self-gravitating massive fields. A brief overview

This is a short review of a series of papers which, in collaboration with Yue Ma, establish several novel existence results for systems of coupled wave-Klein-Gordon equation. Our method, the Hyperbolic Hyperboloidal Method, has allowed us to address the global evolution problem for the Einstein equations of general relativity and investigate the global geometry of matter spacetimes that are initially close to Minkowski spacetime.

math.AP↗

arXiv2016

Weakly regular fluid flows with bounded variation on the domain of outer communication of a Schwarzschild black hole spacetime. A numerical study

We study the dynamical behavior of compressible fluids evolving on the outer domain of communication of a Schwarzschild background. To this end, we design several numerical methods which take the Schwarzschild geometry into account and we treat, both, the relativistic Burgers equation and the relativistic Euler system under the assumption that the flow is spherically symmetric. All the schemes we construct are proven to be well-balanced and therefore to preserve the family of steady state solutions for both models. They enable us to study the nonlinear stability of fluid equilibria, and in particular to investigate the behavior of the fluid near the blackhole horizon. We state and numerically demonstrate several conjectures about the late-time behavior of perturbations of steady solutions.

math.AP↗

arXiv2016

A class of solutions to the Einstein equations with AVTD behavior in generalized wave gauges

We establish the existence of smooth vacuum Gowdy solutions, which are asymptotically velocity term dominated (AVTD) and have T3-spatial topology, in an infinite dimensional family of generalized wave gauges. These results show that the AVTD property, which is known to hold for solutions in areal coordinates, is stable to perturbations with respect to the gauge source functions. Our proof is based on an analysis of the singular initial value problem for the Einstein vacuum equations in the generalized wave gauge formalism, and provides a framework which we anticipate to be useful for more general spacetimes.

gr-qc↗

arXiv2016

Nonlocal conservation laws. I. A new class of monotonicity-preserving models

We introduce a new class of nonlocal nonlinear conservation laws in one space dimension that allow for nonlocal interactions over a finite horizon. The proposed model, which we refer to as the nonlocal pair interaction model, inherits at the continuum level the unwinding feature of finite difference schemes for local hyperbolic conservation laws, so that the maximum principle and certain monotonicity properties hold and, consequently, the entropy inequalities are naturally satisfied. We establish a global-in-time well-posedness theory for these models which covers a broad class of initial data. Moreover, in the limit when the horizon parameter approaches zero, we are able to prove that our nonlocal model reduces to the conventional class of local hyperbolic conservation laws. Furthermore, we propose a numerical discretization method adapted to our nonlocal model, which relies on a monotone numerical flux and a uniform mesh, and we establish that these numerical solutions converge to a solution, providing as by-product both the existence theory for the nonlocal model and the convergence property relating the nonlocal regime and the asymptotic local regime.

math.AP↗

arXiv2016

Boundedness of the total energy of relativistic membranes evolving in a curved spacetime

We establish a global existence theory for the equation governing the evolution of a relativistic membrane in a (possibly curved) Lorentzian manifold, when the spacetime metric is a perturbation of the Minkowski metric. Relying on the Hyperboloidal Foliation Method introduced by LeFloch and Ma in 2014, we revisit a theorem established earlier by Lindblad (who treated membranes in the flat Minkowski spacetime) and we provide a simpler proof of existence, which is also valid in a curved spacetime and, most importantly, leads to the important property that the total energy of the membrane is globally bounded in time.

math.AP↗

arXiv2016

An algorithm (CoDeFi) for overcoming the curse of dimensionality in mathematical finance

We present an algorithm (CoDeFi) which overcomes the curse of dimensionality (CoD) in scientific computations and, especially, in mathematical finance (Fi). Our method applies a broad class of partial differential equations such as Kolmogorov-type equations and, for instance, the Black and Scholes equation. As a main feature, our algorithm allows one to solve partial differential equations in large dimensions and provides a general framework for stochastic problems. In insurance or finance applications, the number of dimensions corresponds to the number of risk sources and it is crucial to have a numerical method that remains robust and reliable in large dimensions.

math.NA↗

arXiv2016

The global nonlinear stability of Minkowski space for the Einstein equations in presence of a massive field

We provide a significant extension of the Hyperboloidal Foliation Method introduced by the authors in 2014 in order to establish global existence results for systems of quasilinear wave equations posed on a curved space, when wave equations and Klein-Gordon equations are coupled. This method is based on a (3+1) foliation (of the interior of a future light cone in Minkowski spacetime) by spacelike and asymptotically hyperboloidal hypersurfaces. In the new formulation of the method, we succeed to cover wave-Klein-Gordon systems containing "strong interaction" terms at the level of the metric, and then generalize our method in order to establish a new existence theory for the Einstein equations of general relativity. Following pioneering work by Lindblad and Rodnianski on the Einstein equations in wave coordinates, we establish the nonlinear stability of Minkowski spacetime for self-gravitating massive scalar fields.

math.AP↗