arXiv subjects

Philippe G. LeFloch

Publications and source records attributed to Philippe G. LeFloch.

At least 19 recordsLinked to original sources

arXiv2026

Finite-energy spacetimes with torus symmetry. Cauchy stability of Einstein-Euler and Einstein-Navier-Stokes areal flows

We establish finite-energy Cauchy stability for vacuum and matter spacetimes with $T^2$ symmetry on $T^3$, under general constitutive equations satisfying hyperbolicity and mild asymptotic conditions at vacuum and large mass-energy density. First, using the area of the symmetry orbits as a time function, we introduce the JKL formulation of Einstein areal flows}, a first-order evolution-constraint system coupling a wave-map structure for the geometry to hyperbolic matter balance laws and weighted transport equations for the twists and the momentum tangent to the symmetry orbits. Second, we also define two classes of hyperbolic Einstein-Navier-Stokes models, constructed from (as we call them) a Navier-Stokes potential and a relaxation rate map. The proposed particle-production model has divergence-free matter stress and non-negative particle-number production, while the proposed dissipated-energy model conserves particle number and dissipates fluid mass-energy, while an auxiliary stress restores the total stress-energy conservation required by the contracted Bianchi identity. For regular finite-energy (Einstein, Euler, Navier-Stokes) flows, we establish maximum and invariant-domain principles, geometric and energy monotonicity formulas, spacelike and timelike energy estimates, and total-variation estimates. Third, together, these estimates yield finite-energy Cauchy stability in the future expanding regime and, in the contracting regime, until the spacelike volume collapses. They are independent of the viscosity (or relaxation) mechanisms and cover vacuum, unbounded mass-energy density, and arbitrary finite rapidity. They also extend to weakly regular Einstein-Euler areal flows satisfying a reference mathematical entropy inequality.

math.AP↗

arXiv2026

Optimal localization for the Einstein constraints

We establish the existence of asymptotically Euclidean solutions, or initial data sets, to Einstein's vacuum constraints exhibiting both gravitational shielding and arbitrarily low decay. We resolve a conjecture of Carlotto and Schoen on gluing two solutions across an asymptotically conical domain: in the interpolation region we establish optimal estimates at and beyond harmonic decay, together with ADM invariant estimates. Starting from a seed data set that asymptotically solves the constraints, we project it to an exact solution, whose difference from the seed inherits the natural optimal radial decay rate. The seed-to-solution projection operator composes the linearized constraint operator with its formal adjoint. Its harmonic kernel at infinity generates energy-momentum modulators, which define relative invariants equal to the differences of the usual ADM invariants whenever those exist. We introduce harmonic, radial, and shell stability conditions on the angular gluing function; they ensure the positivity and radial monotonicity of the functionals governing optimal decay. A follow-up paper proves these conditions from weighted Poincaré, Korn, and Hardy inequalities, leading to verifiable conditions on the localization function. As particular cases, our theory applies to solutions without localization and for solutions in the small aperture limit. Our analysis of the linearized scalar curvature operator relies crucially on a new fourth-order curvature functional, and also applies to gluing scalar-flat metrics.

gr-qc↗

arXiv2026

Harmonic, radial, and shell stability of the weighted Einstein constraints on the sphere at infinity

We consider fourth-order and second-order partial differential operators localized on domains of the sphere in arbitrary dimension. These operators arise as weighted compositions of the linearized Einstein constraint operators and their adjoints, and played a key role in our resolution of the optimal localization problem in general relativity, also referred to as the gravitational shielding problem. To control the asymptotic behavior of solutions to Einstein's constraints in our companion paper (preprint arXiv:2312.17706), we introduced the notions of harmonic, radial, and shell stability. Harmonic stability controls the borderline harmonic modes, radial stability governs the radial evolution of spherical averages, and shell stability controls the coupled radial-angular evolution of solutions. In the present paper, we establish that these stability properties follow from weighted Poincaré, Korn, and Hardy inequalities. Furthermore, in arbitrary dimension, we investigate the behavior of the associated geometric constants, and conclude that the stability conditions hold for a broad class of localization functions; the theory applies to arbitrarily small localization domains, corresponding to gluing cones with arbitrarily small aperture. At the opposite extreme, our conditions also hold on the entire sphere, corresponding to the absence of localization. This completes, for gluing cones of arbitrarily small aperture in every dimension, the program initiated by A. Carlotto and R. Schoen on gravitational shielding and the construction of solutions enjoying super-harmonic decay estimates. In our proofs, we introduce Hamiltonian and momentum functionals, which we call shell functionals, and show that they enjoy monotonicity and semi-coercivity properties; their structure also suggests possible analogies with functionals arising in other curvature-related geometric problems.

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arXiv2026

Physics-informed token transformer methodology for nonlinear balance laws. I. Schwarzschild--Burgers fluid flows

We introduce a Physics-Informed Token Transformer (PITT) methodology for nonlinear hyperbolic balance laws in one space dimension, using piecewise steady-state profiles for the representation of approximate weak solutions. The method combines symbolic equation tokenization, a Fourier neural operator encoder, an explicit Rankine--Hugoniot law for shock motion, and a learned correction term. For clarity, we present it here for the relativistic Schwarzschild--Burgers equation, a scalar model for spherically symmetric fluid flows on a Schwarzschild background. For this model the steady-state invariant and the generalized Riemann solutions are explicit, and they can therefore be built into the neural evolution. In particular, the leading discontinuities are advanced by the analytical jump condition, while the learned part reconstructs smooth regions, rarefaction fans, geometric dependence, and finite-resolution effects. The method is designed to locate wave fronts accurately and to preserve the relevant steady states. We test our PITT method on moving shocks, stationary shocks, rarefaction waves, and compare it with a standard high-order finite-volume approximation. We also analyze the standard Burgers limit (when the Schwarzschild mass tends to zero). The Rankine--Hugoniot prior plays the dominant role in these tests, while equation tokenization gives a systematic additional gain. The method is relevant for problems involving geometric effects and/or complex shock-wave dynamics, and is used here to study the long-time dynamics of perturbations of steady-state solutions. In particular, we exhibit an asymptotic law of propagation for the shock location of perturbed steady-state flows.

math.NA↗

arXiv2026

Stability and instability of torus-symmetric Einstein spacetimes with square-integrable connection

We study the global evolution problem for the Einstein equations under T2 symmetry on T3, allowing vacuum, scalar-field, and compressible-fluid matter models, governed by a general equation of state including isothermal and polytropic fluids. Under this symmetry, we obtain the first non-perturbative, global existence and stability theory with connection coefficients being merely square-integrable, which allows both impulsive gravitational waves and shock waves. In areal gauge, we introduce new fluid and geometric variables and reformulate the Einstein-Euler system as a first-order system of nonlinear balance laws with constraints and an entropy structure. The resulting formulation exhibits hyperbolicity, null forms, entropy currents, div-curl structure, maximum principles, and spacetime estimates. This leads to a notion of tame Einstein-Euler flow for which the essential geometric and fluid variables are square-integrable (finite energy), and the secondary variables are absolutely continuous (or, more generally, of bounded variation). In this non-perturbative and weak regularity setting, the equations remain meaningful even when the Weyl curvature concentrates into Dirac masses along timelike hypersurfaces, and the Ricci curvature remains only integrable. Our main results are a global existence theorem for areal foliations, a nonlinear stability theorem for well-prepared initial data, and a nonlinear instability theorem for geometrically oscillatory data, the latter producing measure corrections to the stress energy tensor. In the future-contracting regime, the areal foliation reaches a geometric singularity where the volume of T3 spatial slices degenerates to zero. The areal function reaches zero generically in the non-vacuum Gowdy-symmetric and vacuum torus-symmetric cases. In the future-expanding regime, the areal foliation is complete.

gr-qc↗

arXiv2026

Scattering laws for interfaces in self-gravitating matter flows

We consider the evolution of self-gravitating matter fields that may undergo phase transitions, and we connect ideas from phase transition dynamics with concepts from bouncing cosmology. Our framework introduces scattering maps prescribed on two classes of hypersurfaces: a gravitational singularity hypersurface and a fluid-discontinuity hypersurface. By analyzing the causal structures induced by the light cone and the acoustic cone, we formulate a local evolution problem for the Einstein-Euler system in the presence of such interfaces. We explain how suitable scattering relations must supplement the field equations in order to ensure uniqueness and thus yield a complete macroscopic description of the evolution. This viewpoint builds on a theory developed in collaboration with G. Veneziano for quiescent (velocity-dominated) singularities in solutions of the Einstein equations coupled to a scalar field, where the passage across the singular hypersurface is encoded by a singularity scattering map. The guiding question is to identify junction prescriptions that are compatible with the Einstein and Euler equations, in particular with the propagation of constraints. The outcome is a rigid set of universal relations, together with a family of model-dependent parameters. Under physically motivated requirements (general covariance, causality, constraint compatibility, and ultra-locality), we aim to classify admissible scattering relations arising from microscopic physics and characterizing, at the macroscopic level, the dynamics of a fluid coupled to Einstein gravity.

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arXiv2026

A first-order formulation of f(R) gravity in spherical symmetry

We develop an augmented characteristic, first-order formulation of the field equations in f(R) gravity governing the global evolution of a (possibly) massive scalar field phi under spherical symmetry. This formulation is designed to isolate the genuine dynamical degrees of freedom while preserving the geometric structure of the theory. By treating the spacetime scalar curvature as an independent unknown, we obtain a closed first-order nonlocal system for the pair (phi,R). This augmentation eliminates the higher-derivative character of the original equations at the level of the principal part. Our formulation allows us to pose the characteristic initial value problem and to establish several structural properties of solutions. More precisely, we work in generalized Bondi-Sachs coordinates and prescribe initial data on an asymptotically flat, future light cone with vertex at the center of symmetry, and we identify the minimal regularity conditions required at the center. These regularity conditions are shown to be precisely those ensuring equivalence between the reduced system and the full f(R) equations. Extending Christodoulou's method for the Einstein-scalar-field system, we recast the f(R) field equations as an integro-differential system of two coupled, first-order, nonlocal, nonlinear hyperbolic equations, whose principal unknowns are the scalar field and the spacetime scalar curvature.

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arXiv2025

The global nonlinear stability of Minkowski spacetime with self-gravitating massive Dirac fields

We consider the Einstein-Dirac system for a massive field, which describes the evolution of self-gravitating massive spinor fields, and we investigate the global evolution problem, when the initial data set is sufficiently close to data describing a spacelike, asymptotically Euclidean slice of the Minkowski spacetime. We establish the gauge-invariant nonlinear stability of such fields, namely the existence of a globally hyperbolic development, which remains asymptotic to Minkowski spacetime in future timelike, null, and spacelike directions. Previous results on this problem have been limited to the Einstein-Dirac system in the massless case. Our analysis follows the asymptotically hyperboloidal-Euclidean framework introduced by LeFloch and Y. Ma for the massive Klein-Gordon-Einstein system. The structure specific to spinor fields and the Dirac equation necessitates significantly new elements in the proof. In contrast with prior approaches, our treatment of spinor fields and the Dirac equation is gauge-invariant, relying on the formalism of Lorentz Clifford algebras, principal fiber bundles, etc. Our analysis is carried out with the metric expressed in light-bending wave coordinates, as we call them. This leads us to the study of a global existence problem for a system of wave equations with constraints and a Klein-Gordon-type equations. We derive L2 estimates for the Dirac equation and its coupling with the Einstein equations, along with $pointwise estimates. New Sobolev inequalities are proven for spinor fields in a gauge-invariant manner in the hyperboloidal-Euclidean foliation. The nonlinear coupling between the massive Dirac equation and the Einstein equations is investigated, and we establish a hierarchy of estimates, which distinguish between translations, rotations, and boosts.

gr-qc↗

arXiv2025

Reproducing kernel methods for machine learning, PDEs, and statistics

This monograph develops a unified, application-driven framework for kernel methods grounded in reproducing kernel Hilbert spaces (RKHS) and optimal transport (OT). Part I lays the theoretical and numerical foundations on positive-definite kernels; discrete and continuous RKHS; kernel engineering and scaling maps; error assessment via kernel discrepancy/maximum mean discrepancy (MMD); and a systematic operator view of kernels. In this viewpoint, projection, gradient, divergence, and Laplace-Beltrami operators are built directly from kernels, enabling discrete analogues of differential operators and variational tools that connect learning with PDE-style modeling. Part II turns to practice across four domains. In machine learning, we treat supervised and unsupervised tasks, then develop RKHS-based generative modeling, contrasting density and projection approaches and enhancing them with OT and scalable, combinatorial assignments. We introduce clustering strategies that reduce computational burden and support large-scale regression and transport. In physics-informed modeling, we present mesh-free kernel discretizations for elliptic and time-dependent PDEs, discuss automatic differentiation, and propose high-order discrete approximations. In reinforcement learning, we formulate kernel Q-learning and non-parametric HJB methods, and show how kernel operators yield sample-efficient baselines on continuous-state, discrete-action tasks. In mathematical finance, we build nonparametric time-series models and market generators, study benchmarking and extrapolation for pricing, and apply the framework to stress testing and portfolio methods.

math.NA↗

arXiv2024

A class of kernel-based scalable algorithms for data science

We present several generative and predictive algorithms based on the RKHS (reproducing kernel Hilbert spaces) methodology, which, most importantly, are scale up efficiently with large datasets or high-dimensional data. It is well recognized that the RKHS methodology leads one to efficient and robust algorithms for numerous tasks in data science, statistics, and scientific computation. However, the implementations existing the literature are often difficult to scale up for encompassing large datasets. In this paper, we introduce a simple and robust, divide-and-conquer methodology. It applies to large scale datasets and relies on several kernel-based algorithms, which distinguish between various extrapolation, interpolation, and optimal transport steps. We argue how to select the suitable algorithm in specific applications thanks to a feedback of performance criteria. Our primary focus is on applications and problems arising in industrial contexts, such as generating meshes for efficient numerical simulations, designing generators for conditional distributions, constructing transition probability matrices for statistical or stochastic applications, and addressing various tasks relevant to the Artificial Intelligence community. The proposed algorithms are highly relevant to supervised and unsupervised learning, generative methods, as well as reinforcement learning.

math.NA↗

arXiv2024

Optimal shielding for Einstein gravity

To construct asymptotically-Euclidean Einstein's initial data sets, we introduce the localized seed-to-solution method, which projects from approximate to exact solutions of the Einstein constraints. The method enables us to glue together initial data sets in multiple asymptotically-conical regions, and in particular construct data sets that exhibit the gravity shielding phenomenon, specifically that are localized in a cone and exactly Euclidean outside of it. We achieve optimal shielding in the sense that the metric and extrinsic curvature { are controlled at a super-harmonic rate, regardless of how slowly they decay (even} beyond the standard ADM formalism), and the gluing domain can be a collection of arbitrarily narrow nested cones. We also uncover several notions of independent interest: silhouette functions, localized ADM modulator, and relative energy-momentum vector. An axisymmetric example is provided numerically.

gr-qc↗

arXiv2024

The Euclidean-hyperboloidal foliation method. Application to f(R) modified gravity

This paper is a part of a series devoted to the Euclidean-hyperboloidal foliation method introduced by the authors for investigating the global existence problem associated with nonlinear systems of coupled wave-Klein-Gordon equations with small data. This method was developed especially for investigating the initial value problem for the Einstein-massive field system in wave gauge. Here, we study the (fourth-order) field equations of f(R) modified gravity and investigate the global dynamical behavior of the gravitational field in the near-Minkowski regime. We establish the existence of a globally hyperbolic Cauchy development approaching Minkowski spacetime (in spacelike, null, and timelike directions), when the initial data set is sufficiently close to an asymptotically Euclidean and spacelike hypersurface in Minkowski spacetime. We cast the (fourth-order) f(R)-field equations in the form of a second-order wave-Klein-Gordon system, which has an analogous structure to the Einstein-massive field system but, in addition, involves a (possibly small) effective mass parameter. We establish the nonlinear stability of the Minkowski spacetime in the context of f(R) gravity, when the integrand f(R) in the action functional can be taken to be arbitrarily close to the integrand R of the standard Hilbert-Einstein action.

gr-qc↗

arXiv2024

Extrapolation and generative algorithms for three applications in finance

For three applications of central interest in finance, we demonstrate the relevance of numerical algorithms based on reproducing kernel Hilbert space (RKHS) techniques. Three use cases are investigated. First, we show that extrapolating from few pricer examples leads to sufficiently accurate and computationally efficient results so that our algorithm can serve as a pricing framework. The second use case concerns reverse stress testing, which is formulated as an inversion function problem and is treated here via an optimal transport technique in combination with the notions of kernel-based encoders, decoders, and generators. Third, we show that standard techniques for time series analysis can be enhanced by using the proposed generative algorithms. Namely, we use our algorithm in order to extend the validity of any given quantitative model. Our approach allows for conditional analysis as well as for escaping the `Gaussian world'. This latter property is illustrated here with a portfolio investment strategy.

math.NA↗

arXiv2024

Nonlinear stability of self-gravitating massive fields

We consider the global evolution problem for Einstein's field equations in the near-Minkowski regime and study the long-time dynamics of a massive scalar field evolving under its own gravitational field. We establish the existence of a globally hyperbolic Cauchy development associated with any initial data set that is sufficiently close to a data set in Minkowski spacetime. In addition to applying to massive fields, our theory allows us to cover metrics with slow decay in space. The strategy of proof, proposed here and referred to as the Euclidean-Hyperboloidal Foliation Method, applies, more generally, to nonlinear systems of coupled wave and Klein-Gordon equations. It is based on a spacetime foliation defined by merging together asymptotically Euclidean hypersurfaces (covering spacelike infinity) and asymptotically hyperboloidal hypersurfaces (covering timelike infinity). A transition domain (reaching null infinity) limited by two asymptotic light cones is introduced in order to realize this merging. On the one hand, we exhibit a boost-rotation hierarchy property (as we call it) which is associated with Minkowski's Killing fields and is enjoyed by commutators of curved wave operators and, on the other hand, we exhibit a metric hierarchy property (as we call it) enjoyed by components of Einstein's field equations in frames associated with our Euclidean-hyperboloidal foliation. The core of the argument is, on the one hand, the derivation of novel integral and pointwise estimates which lead us to almost sharp decay properties (at timelike, null, and spacelike infinity) and, on the other hand, the control of the (quasi-linear and semi-linear) coupling between the geometric and matter parts of the Einstein equations.

gr-qc↗

arXiv2023

The seed-to-solution method for the Einstein constraints and the asymptotic localization problem

We establish the existence of a class of asymptotically Euclidean solutions to Einstein's constraint equations, whose asymptotic behavior at infinity is arbitrarily prescribed. The proposed seed-to-solution method relies on iterations based on the linearized Einstein operator and its dual. It generates a Riemannian manifold (with finitely many asymptotically Euclidean ends) from any seed data set consisting of (1): a Riemannian metric and a symmetric two-tensor and (2): a (density) field and a (momentum) vector field representing the matter content. We distinguish between tame and strongly tame seed data sets, depending whether the data provides a rough or an accurate asymptotic Ansatz at infinity. We encompass classes of metrics and matter fields with low decay (with infinite ADM mass) or strong decay (with Schwarzschild behavior). Our analysis is motivated by Carlotto and Schoen's pioneering work on the localization problem for Einstein's vacuum equations. Dealing with metrics with very low decay and establishing estimates beyond harmonic decay require significantly new arguments. We analyze the nonlinear coupling between the Hamiltonian and momentum constraints. By establishing elliptic estimates for the linearized Einstein operator, we uncover the notion of mass-momentum correctors which is related to the ADM mass of the manifold. We derive precise estimates for the difference between the seed data and the actual solution, a result that should be of interest for future numerical investigation. Furthermore, we introduce here and study the asymptotic localization problem in which we replace Carlotto-Schoen's exact localization requirement by an asymptotic condition at a super-harmonic rate. With a suitably constructed, parametrized family of seed data, we solve this problem by exhibiting mass-momentum correctors with harmonic decay.

math.AP↗

arXiv2023

A class of mesh-free algorithms for some problems arising in finance and machine learning

We introduce a numerical methodology, referred to as the transport-based mesh-free method, which allows us to deal with continuous, discrete, or statistical models in the same unified framework, and leads us to a broad class of numerical algorithms recently implemented in a Python library (namely, CodPy). Specifically, we propose a mesh-free discretization technique based on the theory of reproducing kernels and the theory of transport mappings, in a way that is reminiscent of Lagrangian methods in computational fluid dynamics. We introduce kernel-based discretizations of a variety of differential and discrete operators (gradient, divergence, Laplacian, Leray projection, extrapolation, interpolation, polar factorization). The proposed algorithms are nonlinear in nature and enjoy quantitative error estimates based on the notion of discrepancy error, which allows one to evaluate the relevance and accuracy of, both, the given data and the numerical solutions. Our strategy is relevant when a large number of degrees of freedom are present as is the case in mathematical finance and machine learning. We consider the Fokker-Planck-Kolmogorov system (relevant for problems arising in finance and material dynamics) and a class of neural networks based on support vector machines.

math.NA↗

arXiv2022

Global evolution in spherical symmetry for self-gravitating massive fields

We are interested in the global dynamics of a massive scalar field evolving under its own gravitational field and, in this paper, we study spherically symmetric solutions to Einstein's field equations coupled with a Klein-Gordon equation with quadratic potential. For the initial value problem we establish a global existence theory when initial data are prescribed on a future light cone with vertex at the center of symmetry. A suitably generalized solution in Bondi coordinates is sought which has low regularity and possibly large but finite Bondi mass. A similar result was established first by Christodoulou for massless fields. In order to deal with massive fields, we must overcome several challenges and significantly modify Christodoulou's original method. First of all, we formulate the Einstein-Klein-Gordon system in spherical symmetry as a non-local and nonlinear hyperbolic equation and, by carefully investigating the global dynamical behavior of the massive field, we establish various estimates concerning the Einstein operator, the Hawking mass, and the Bondi mass, including positivity and monotonicity properties. Importantly, in addition to a regularization at the center of symmetry we find it necessary to also introduce a regularization at null infinity. We also establish new energy and decay estimates for, both, regularized and generalized solutions.

gr-qc↗

arXiv2022

Boundedness of the conformal hyperboloidal energy for a wave-Klein-Gordon model

We consider the global evolution problem for a model which couples together a nonlinear wave equation and a nonlinear Klein-Gordon equation, and was independently introduced by LeFloch and Y. Ma and by Q. Wang. By revisiting the Hyperboloidal Foliation Method, we establish that a weighted energy of the solutions remains (almost) bounded for all times. The new ingredient in the proof is a hierarchy of fractional Morawetz energy estimates (for the wave component of the system) which is defined from two conformal transformations. The optimal case for these energy estimates corresponds to using the scaling vector field as a multiplier for the wave component.

math.AP↗