Given any wave operator with principle part ∂_t^2 -∂_x^2 +ℬ^xx(ω t,x)∂_x^2, where ℬ^xx:𝕋^ν+1→ℝ is a sufficiently small, quasi-periodic perturbation and ω∈ℝ^ν, we explain how to construct null coordinates that respect the quasi-periodicity of the solutions. As it turns out, in these coordinates, the principal symbol of the wave operator above has constant coefficients. To construct these coordinates, we start by writing the wave operator in geometric form, modulo terms of order 1, meaning as the wave operator arising from an (1+1)-Lorentzian metric and then define null coordinates as solutions to the Eikonal equations, so that the metric is conformally flat in these coordinates. The problem of constructing these coordinates is then eventually reduced to that of straightening a vector field with quasi-periodic coefficients. As an application, we give a novel proof of a recent reducibility result of Berti-Feola-Procesi-Terracina for the quasi-periodically forced linear Klein-Gordon equation, the novelty concerning essentially the analysis of the maximal order terms. In particular, the method we propose here does not rely on any quantitative Egorov-type result.

Given any wave operator with principle part $\partial_{t}^2 -\partial_{x}^2 +\mathcal{B}^{xx}(\omega t,x)\partial_{x}^2$, where $\mathcal{B}^{xx}:\mathbb{T}^{\nu+1} \rightarrow \mathbb{R}$ is a sufficiently small, quasi-periodic perturbation and $\omega \in \mathbb{R}^\nu$, we explain how to construct \emph{null coordinates} that respect the quasi-periodicity of the solutions. As it turns out, in these coordinates, the principal symbol of the wave operator above has constant coefficients. To construct these coordinates, we start by writing the wave operator in geometric form, modulo terms of order $1$, meaning as the wave operator arising from an $(1+1)$-Lorentzian metric and then define null coordinates as solutions to the Eikonal equations, so that the metric is conformally flat in these coordinates. The problem of constructing these coordinates is then eventually reduced to that of straightening a vector field with quasi-periodic coefficients. As an application, we give a novel proof of a recent reducibility result of Berti-Feola-Procesi-Terracina for the quasi-periodically forced linear Klein-Gordon equation, the novelty concerning essentially the analysis of the maximal order terms. In particular, the method we propose here does not rely on any quantitative Egorov-type result.

We consider the initial boundary value problem for the Einstein vacuum equations in the maximal gauge, or more generally, in a gauge where the mean curvature of a timelike foliation is fixed near the boundary. We prove the existence of solutions such that the normal to the boundary is tangent to the time slices, the lapse of the induced time coordinate on the boundary is fixed and the main geometric boundary conditions are given by the 1-parameter family of Riemannian conformal metrics on each two-dimensional section. As in the local existence theory of Christodoulou-Klainerman for the Einstein vacuum equations in the maximal gauge, we use as a reduced system the wave equations satisfied by the components of the second fundamental form of the the time foliation. The main difficulty lies in completing the above set of boundary conditions such that the reduced system is well-posed, but still allows for the recovery of the Einstein equations. We solve this problem by imposing the momentum constraint equations on the boundary, suitably modified by quantities vanishing in the maximal gauge setting. To derive energy estimates for the reduced system at time t, we show that all the terms in the flux integrals on the boundary can be either directly controlled by the boundary conditions or they lead to an integral on the two-dimensional section at time t of the boundary. Exploiting again the maximal gauge condition on the boundary, this contribution to the flux integrals can then be absorbed by a careful trace inequality in the interior energy.

The study of the stability of Minkowksi space has had a tremendous influence on the mathematical analysis of General Relativity and in a wider context, on geometric hyperbolic partial differential equations. This article proposes an overview of the context, ideas and part of the legacy surrounding this subject, focusing particularly on the original proof of Christodoulou–Klainerman.

The vector field method has been a central tool in the analysis of geometric wave equations since its earlier development by Klainerman in the 1980s. We give here an overview of vector field methods in the context of kinetic equations, with applications to classical systems, such as the Vlasov-Poisson system. These notes emanate from the lectures given by the author during the 11th Summer School on Methods and Models of Kinetic Theory, Pesaro, June 2022.

Motivated by the study of small amplitudes non-linear waves in the Anti-de-Sitter spacetime and in particular the conjectured existence of periodic in time solutions to the Einstein equations, we construct families of arbitrary small time-periodic solutions to the conformal cubic wave equation and the spherically-symmetric Yang-Mills equations on the Einstein cylinder $\mathbb{R}\times \mathbb{S}^3$. For the conformal cubic wave equation, we consider both spherically-symmetric solutions and complexed-valued aspherical solutions with an ansatz relying on the Hopf fibration of the $3$-sphere. In all three cases, the equations reduce to $1+1$ semi-linear wave equations. Our proof relies on a theorem of Bambusi-Paleari for which the main assumption is the existence of a seed solution, given by a non-degenerate zero of a non-linear operator associated with the resonant system. For the problems that we consider, such seed solutions are simply given by the mode solutions of the linearized equations. Provided that the Fourier coefficients of the systems can be computed, the non-degeneracy conditions then amount to solving infinite dimensional linear systems. Since the eigenfunctions for all three cases studied are given by Jacobi polynomials, we derive the different Fourier and resonant systems using linearization and connection formulas as well as integral transformation of Jacobi polynomials. In the Yang-Mills case, the original version of the theorem of Bambusi-Paleari is not applicable because the non-linearity of smallest degree is nonresonant. The resonant terms are then provided by the next order non-linear terms with an extra correction due to backreaction terms of the smallest degree non-linearity and we prove an analogous theorem in this setting.

We prove the existence of time-periodic solutions to non-linear massive Klein-Gordon equations in Anti-de Sitter as well as their orbital stability over exponentially long times for certain values of the mass corresponding to completely resonant spectrum. We analyse the resonant system in the Fourier space by relying in particular on Zeilberger's algorithm which allows for a systematic way to derive recurrence formulae for the Fourier coefficients. We also show that the derivation and analysis of the Fourier systems easily extends to other semi-linear wave equations such as the co-rotational wave map equation.

We prove the global asymptotic stability of the Minkowski space for the massless Einstein-Vlasov system in wave coordinates. In contrast with previous work on the subject, no compact support assumptions on the initial data of the Vlasov field in space or the momentum variables are required. In fact, the initial decay in $v$ is optimal. The present proof is based on vector field and weighted vector field techniques for Vlasov fields, as developed in previous work of Fajman, Joudioux, and Smulevici, and heavily relies on several structural properties of the massless Vlasov equation, similar to the null and weak null conditions. To deal with the weak decay rate of the metric, we propagate well-chosen hierarchized weighted energy norms which reflect the strong decay properties satisfied by the particle density far from the light cone. A particular analytical difficulty arises at top order, when we do not have access to improved pointwise decay estimates for certain metric components. This difficulty is resolved using a novel hierarchy in the massless Einstein-Vlasov system, which exploits the propagation of different growth rates for the energy norms of different metric components.

We give a short proof of local well-posedness for the initial boundary value problem in general relativity with sole boundary condition the requirement that the boundary is umbilic. This includes as a special case the totally geodesic boundary condition that we had previously addressed in [ 8]. The proof is based on wave coordinates and the key observation that the momentum constraint is always valid for umbilic boundaries. This allows for a greater freedom in the choice of boundary conditions, since imposing the umbilic condition also provides Neumann boundary conditions for three of the four wave coordinates conditions. Moreover, the umbilic condition, being geometric, implies that geometric uniqueness in the sense of Friedrich holds in this specific case.

We prove the well-posedness of the initial boundary value problem for the Einstein equations with sole boundary condition the requirement that the timelike boundary is totally geodesic. This provides the first well-posedness result for this specific geometric boundary condition and the first setting for which geometric uniqueness in the original sense of Friedrich holds for the initial boundary value problem. Our proof relies on the ADM system for the Einstein vacuum equations, formulated with respect to a parallelly propagated orthonormal frame along timelike geodesics. As an independent result, we first establish the well-posedness in this gauge of the Cauchy problem for the Einstein equations, including the propagation of constraints. More precisely, we show that by appropriately modifying the evolution equations, using the constraint equations, we can derive a first order symmetric hyperbolic system for the connection coefficients of the orthonormal frame. The propagation of the constraints then relies on the derivation of a hyperbolic system involving the connection, suitably modified Riemann and Ricci curvature tensors and the torsion of the connection. In particular, the connection is shown to agree with the Levi-Civita connection at the same time as the validity of the constraints. In the case of the initial boundary value problem with totally geodesic boundary, we then verify that the vanishing of the second fundamental form of the boundary leads to homogeneous boundary conditions for our modified ADM system, as well as for the hyperbolic system used in the propagation of the constraints. An additional analytical difficulty arises from a loss of control on the normal derivatives to the boundary of the solution. To resolve this issue, we work with an anisotropic scale of Sobolev spaces and exploit the specific structure of the equations.

We prove the global stability of the Minkowski space viewed as the trivial solution of the Einstein-Vlasov system. To estimate the Vlasov field, we use the vector field and modified vector field techniques developed in [FJS15; FJS17]. In particular, the initial support in the velocity variable does not need to be compact. To control the effect of the large velocities, we identify and exploit several structural properties of the Vlasov equation to prove that the worst non-linear terms in the Vlasov equation either enjoy a form of the null condition or can be controlled using the wave coordinate gauge. The basic propagation estimates for the Vlasov field are then obtained using only weak interior decay for the metric components. Since some of the error terms are not time-integrable, several hierarchies in the commuted equations are exploited to close the top order estimates. For the Einstein equations, we use wave coordinates and the main new difficulty arises from the commutation of the energy-momentum tensor, which needs to be rewritten using the modified vector fields.

We study the global dynamics of the wave equation, Maxwell's equation and the linearized Bianchi equations on a fixed anti-de Sitter (AdS) background. Provided dissipative boundary conditions are imposed on the dynamical fields we prove uniform boundedness of the natural energy as well as both degenerate (near the AdS boundary) and non-degenerate integrated decay estimates. Remarkably, the non-degenerate estimates lose a derivative. We relate this loss to a trapping phenomenon near the AdS boundary, which itself originates from the properties of (approximately) gliding rays near the boundary. Using the Gaussian beam approximation we prove that non-degenerate energy decay without loss of derivatives does not hold. As a consequence of the non-degenerate integrated decay estimates, we also obtain pointwise-in-time decay estimates for the energy. Our paper provides the key estimates for a proof of the non-linear stability of the anti-de Sitter spacetime under dissipative boundary conditions. Finally, we contrast our results with the case of reflecting boundary conditions.

This paper proves almost-sharp asymptotics for small data solutions of the Vlasov-Nordstr\"om system in dimension three. This system consists of a wave equation coupled to a transport equation and describes an ensemble of relativistic, self-gravitating particles. We derive sharp decay estimates using a variant of the vector-field method introduced in previous work. More precisely, we construct modified vector fields, depending on the solutions, to propagate $L^1$-bounds for the distribution function and its derivatives. The modified vector fields are designed to have improved commutation properties with the transport operator and yet to still provide sufficient control on the solutions to allow for a sharp Klainerman-Sobolev type inequality. Our method does not require any compact support assumption in the velocity variable nor do we need strong interior decay for the solution to the wave equation.

We adapt the vector field method of Klainerman to the study of relativistic transport equations. First, we prove robust decay estimates for velocity averages of solutions to the relativistic massive and massless transport equations, without any compact support requirements (in $x$ or $v$) for the distribution functions. In the second part of this article, we apply our method to the study of the massive and massless Vlasov-Nordstr\"om systems. In the massive case, we prove global existence and (almost) optimal decay estimates for solutions in dimensions $n \geq 4$ under some smallness assumptions. In the massless case, the system decouples and we prove optimal decay estimates for the solutions in dimensions $n \geq 4$ for arbitrarily large data, and in dimension $3$ under some smallness assumptions, exploiting a certain form of the null condition satisfied by the equations. The $3$-dimensional massive case requires an extension of our method and will be treated in future work.

The aim of this talk is to present an extension of the vector field method of Klainerman, which is typically applied in the context of non-linear wave equations, to the case of kinetic equations of Vlasov type. We first describe how our method yields sharp decay estimates for velocity averages for the linear classical and relativistic transport equations and then explain how it can be applied to various models of mathematical physics, such as the Vlasov-Poisson, Vlasov-Nordström and Vlasov-Einstein systems.

We investigate the future asymptotic behavior of Gowdy spacetimes on T3, when the metric satisfies weak regularity conditions, so that the metric coefficients (in suitable coordinates) are only in the Sobolev space H1 or have even weaker regularity. The authors recently introduced this class of spacetimes in the broader context of T2-symmetric spacetimes and established the existence of a global foliation by spacelike hypersurfaces when the time function is chosen to be the area of the surfaces of symmetry. In the present paper, we identify the global causal geometry of these spacetimes and, in particular, establish that weakly regular Gowdy spacetimes are future timelike geodesically complete. This result extends a theorem by Ringström for metrics with sufficiently high regularity. We emphasize that our proof of the energy decay is based on an energy functional inspired by the Gowdy-to-Ernst transformation. In order to establish the geodesic completeness property, we prove a higher regularity property concerning the metric coefficients along timelike curves and we provide a novel analysis of the geodesic equation for Gowdy spacetimes, which does not require high-order regularity estimates. Even when sufficient regularity is assumed, our proof provides an alternative and shorter proof of the energy decay and of the geodesic completeness property for Gowdy spacetimes.

We investigate the late-time asymptotics of future-expanding, polarized vacuum Einstein spacetimes with T-2-symmetry on T-3, which, by definition, admit two spacelike Killing fields. Our main result is the existence of a stable asymptotic regime within this class; that is, we provide here a full description of the late-time asymptotics of the solutions to the Einstein equations when the initial data set is close to the asymptotic regime. Our proof is based on several energy functionals with lower-order corrections (as is standard for such problems) and the derivation of a simplified model that we exhibit here. Roughly speaking, the Einstein equations in the symmetry class under consideration consist of a system of wave equations coupled to constraint equations plus a system of ordinary differential equations. The unknowns involved in the system of ordinary equations are blowing up in the future timelike directions. One of our main contributions is the derivation of novel effective equations for suitably renormalized unknowns. Interestingly, this renormalization is not performed with respect to a fixed background, but does involve the energy of the coupled system of wave equations. In addition, we construct an open set of initial data that are arbitrarily close to the expected asymptotic behavior. We emphasize that, in comparison, the class of Gowdy spacetimes exhibits a very different dynamical behavior to the one we uncover in the present work for general polarized T-2-symmetric spacetimes. Furthermore, all the conclusions of this paper are valid within the framework of weakly T-2-symmetric spacetimes previously introduced by the authors.

The aim of this article is to demonstrate how the vector field method of Klainerman can be adapted to the study of transport equations. After an illustration of the method for the free transport operator, we apply the vector field method to the Vlasov-Poisson system in dimension 3 or greater. The main results are optimal decay estimates and the propagation of global bounds for commuted fields associated with the conservation laws of the free transport operators, under some smallness assumption. Similar decay estimates had been obtained previously by Hwang, Rendall and Velázquez using the method of characteristics, but the results presented here are the first to contain the global bounds for commuted fields and the optimal spatial decay estimates. In dimension 4 or greater, it suffices to use the standard vector fields commuting with the free transport operator while in dimension 3, the rate of decay is such that these vector fields would generate a logarithmic loss. Instead, we construct modified vector fields where the modification depends on the solution itself. The methods of this paper, being based on commutation vector fields and conservation laws, are applicable in principle to a wide range of systems, including the Einstein-Vlasov and the Vlasov-Nordström system.

We provide a geometric well-posedness theory for the Einstein equations within the class of weakly regular vacuum spacetimes with T-2-symmetry, as defined in the present paper, and we investigate their global causal structure. Our assumptions allow us to give a meaning to the Einstein equations under weak regularity as well as to solve the initial value problem under the assumed symmetry. First, introducing a frame adapted to the symmetry and identifying certain cancellation properties taking place in the standard expressions of the connection and the curvature, we formulate the initial value problem for the Einstein field equations under the proposed weak regularity assumptions. Second, considering the Cauchy development of any weakly regular initial data set and denoting by R the area of the orbits of symmetry, we establish the existence of a global foliation by the level sets of R such that R grows to infinity in the future direction. Our weak regularity assumptions only require that R is Lipschitz continuous while the metric coefficients describing the initial geometry of the symmetry orbits are in the Sobolev space H-1 and the remaining coefficients have even weaker regularity.