Related papers: A commuting-vector-field approach to some dispersi…
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,…
In this note, we prove pointwise decay in time of solutions to the 3D energy-critical nonlinear Schr\"odinger equations assuming data in $L^1\cap H^3$. The main ingredients are the boundness of the Schr\"odinger propagators in Hardy space…
The first article in a two-part series (the second article being [arXiv:2205.13197]) assumes a weak local energy decay estimate holds and proves that solutions to the linear wave equation with variable coefficients in $\mathbb R^{1+3}$,…
We study the global decay properties of solutions to the linear wave equation in 1+3 dimensions on time-dependent, weakly asymptotically flat spacetimes. Assuming non-trapping of null geodesics and a local energy decay estimate, we prove…
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…
We prove pointwise in time decay estimates via an abstract conjugate operator method. This is then applied to a large class of dispersive equations.
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…
We present a new vector field approach to almost-sharp decay for the wave equation on spherically symmetric, stationary and asymptotically flat spacetimes. Specifically, we derive a new hierarchy of higher-order weighted energy estimates by…
We study the linear wave equation on a class of spatially homogeneous and isotropic Friedmann-Lema\^itre-Robertson-Walker (FLRW) spacetimes in the decelerated regime with spatial topology $\mathbb{R}^3$. Employing twisted $t$-weighted…
We study the asymptotic properties of the small data solutions of the Vlasov-Maxwell system in dimension three. No neutral hypothesis nor compact support assumptions are made on the data. In particular, the initial decay in the velocity…
In this article, we present a vector field method for the study of solutions to massless relativistic transport equations. Compared to the methodologie developped by Fajman-Joudioux-Smulevici, we remove the Lorentz boosts of the commutation…
This paper proves the existence of a bounded energy and integrated energy decay for solutions of the massless Vlasov equation in the exterior of a very slowly rotating Kerr spacetime. This combines methods previously developed to prove…
We consider the Vlasov equation on slowly expanding isotropic homogeneous tori, described by the Friedmann--Lema\^itre--Robertson--Walker cosmological spacetimes. For expansion rate $t^q$, with $0< q<\frac{1}{2}$ (excluding certain…
We prove a dispersive estimate for the one-dimensional Schroedinger equation, mapping between weighted $L^p$ spaces with stronger time-decay ($t^{-3/2}$ versus $t^{-1/2}$) than is possible on unweighted spaces. To satisfy this bound, the…
We prove pointwise-in-time dispersive estimates for solutions to the generalized Korteweg--de Vries (gKdV) equation. In particular, for solutions to the mass-critical model, we assume only that initial data lie in $\dot{H}^{\frac{1}{4}}…
We consider the Cauchy problem for the two-dimensional Novikov-Veselov equation integrable via the inverse scattering problem for the Schr\"odinger operator with fixed negative energy. The associated linear equation is characterized by a…
We use a novel physical space method to prove relatively non-degenerate integrated energy estimates for the wave equation on subextremal Schwarzschild-de Sitter spacetimes with parameters $(M,\Lambda)$. These are integrated decay statements…
Studied here is the large-time behavior of solutions of the Korteweg-de Vries equation posed on the right half-line under the effect of a localized damping. Assuming as in \cite{linares-pazoto} that the damping is active on a set…
In this paper, we are concerned with the global well-posedness and time-asymptotic decay of the Vlasov-Fokker-Planck equation with local alignment forces. The equation can be formally derived from an agent-based model for self-organized…
We prove new time decay estimates for the linearized $\beta$-plane equation near the Couette flow on the plane that combine inviscid damping and the dispersion of Rossby waves. Specifically, we show that the profiles of the velocity field…