We study the dispersive properties of the wave equation and the massless Dirac equation in three space dimensions, perturbed with electromagnetic potentials. The potentials are assumed to be small but may be rough. For both equations, we prove a dispersive estimate of the form |u(t,x)|< C/t. The constant C can be estimated in terms of a weighted H^s norm of the data, for suitable values of s. As a consequence of our method of proof, we establish the limiting absorption principle for the massless Dirac operator perturbed with a small, rough matrix potential.
We consider two-dimensional Pauli and Dirac operators with a polynomially vanishing magnetic field. The main results of the paper provide resolvent expansions of these operators in the vicinity of their thresholds. It is proved that the nature of these expansions is fully determined by the flux of the magnetic field. The most important novelty of the proof is a comparison between the spatial asymptotics of the zero modes obtained in two different manners. The result of this matching allows to compute explicitly all the singular terms in the associated resolvent expansions. As an application we…
We prove sharp pointwise decay estimates for critical Dirac equations on $\mathbb{R}^n$ with $n\geq 2$. They appear for instance in the study of critical Dirac equations on compact spin manifolds, describing blow-up profiles, and as effective equations in honeycomb structures. For the latter case, we find excited states with an explicit asymptotic behavior. Moreover, we provide some classification results both for ground states and for excited states.
The Cauchy problem is considered for the massive Dirac equation in the non-extreme Kerr-Newman geometry, for smooth initial data with compact support outside the event horizon and bounded angular momentum. We prove that the Dirac wave function decays in L^\infty_loc at least at the rate t^{-5/6}. For generic initial data, this rate of decay is sharp. We derive a formula for the probability p that the Dirac particle escapes to infinity. For various conditions on the initial data, we show that p=0,1 or 0<p<1. The proofs are based on a refined analysis of the Dirac propagator constructed in gr-qc…
We consider the explicit asymptotic profile of massless Dirac fields on a Schwarzschild background. First, we prove for the spin $s=\pm \frac{1}{2}$ components of the Dirac field a uniform bound of a positive definite energy and an integrated local energy decay estimate from a symmetric hyperbolic wave system. Based on these estimates, we further show that these components have globally pointwise decay $fv^{-3/2-s}τ^{-5/2+s}$ as both an upper and a lower bound outside the black hole, with function $f$ finite and explicitly expressed in terms of the initial data and the coordinates. This establ…
We investigate $L^1\to L^\infty$ dispersive estimates for the massless two dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies the natural $t^{-\frac12}$ decay rate, which may be improved to $t^{-\frac12-γ}$ for any $0\leq γ<\frac{3}{2}$ at the cost of spatial weights. We classify the structure of threshold obstructions as being composed of a two dimensional space of p-wave resonances and a finite dimensional space of eigenfunctions at zero energy. We show that, in the presence of a threshold resonance, the Dirac evolution satisfies the natura…
In this paper we consider Dirac operators in $\mathbb R^n$, $n\geq2$, with a potential $V$. Under mild decay and continuity assumptions on $V$ and some spectral assumptions on the operator, we prove a limiting absorption principle for the resolvent, which implies a family of Strichartz estimates for the linear Dirac equation. For large potentials the dynamical estimates are not an immediate corollary of the free case since the resolvent of the free Dirac operator does not decay in operator norm on weighted $L^2$ spaces as the frequency goes to infinity.
We prove local smoothing and weighted Strichartz estimates for the Dirac equation with a Aharonov-Bohm potential. The proof relies on an explicit representation of the solution built in terms of spectral projections.
We prove endpoint estimates with angular regularity for the wave and Dirac equations perturbed with a small potential. The estimates are applied to prove global existence for the cubic Dirac equation perturbed with a small potential, for small initial $H^{1}$ data with additional angular regularity. This implies in particular global existence in the critical energy space $H^{1}$ for small radial data.
In this paper we prove some uniform asymptotic estimates for confluent hypergeometric functions making use of the steepest-descent method. As an application, we obtain Strichartz estimates with loss of angular derivatives for the massless Dirac-Coulomb equation in $3D$.
We consider limiting Carleman weights for Dirac operators and prove corresponding Carleman estimates. In particular, we show that limiting Carleman weights for the Laplacian also serve as limiting weights for Dirac operators. As an application we consider the inverse problem of recovering a Lipschitz continuous magnetic field and electric potential from boundary measurements for the Pauli Dirac operator.
We investigate $L^1\to L^\infty$ dispersive estimates for the three dimensional Dirac equation with a potential. We also classify the structure of obstructions at the thresholds of the essential spectrum as being composed of a two dimensional space of resonances and finitely many eigenfunctions. We show that, as in the case of the Schrödinger evolution, the presence of a threshold obstruction generically leads to a loss of the natural $t^{-\frac32}$ decay rate. In this case we show that the solution operator is composed of a finite rank operator that decays at the rate $t^{-\frac12}$ plus a te…
In this paper, we show that massless Dirac waves in the Schwarzschild geometry decay to zero at a rate $t^{-2λ}$, where $λ=1, 2,...$ is the angular momentum. Our technique is to use Chandrasekhar's separation of variables whereby the Dirac equations split into two sets of wave equations. For the first set, we show that the wave decays as $t^{-2λ}$. For the second set, in general, the solutions tend to some explicit profile at the rate $t^{-2λ}$. The decay rate of solutions of Dirac equations is achieved by showing that the coefficient of the explicit profile is exactly zero. The key ingredient…
We are interested in the cubic Dirac equation with mass $m \in [0, 1]$ in two space dimensions, which is also known as the Soler model. We conduct a thorough study on this model with initial data sufficiently small in high regularity Sobolev spaces. First, we show the global existence of the model, which is uniform-in-mass. In addition, we derive a unified pointwise decay result valid for all $m \in [0, 1]$. Last but not least, we prove the cubic Dirac equations scatter linearly with an explicit scattering speed. When the mass $m=0$, we can show an improved pointwise decay result.
We prove that the material parameters in a Dirac system with magnetic and electric potentials are uniquely determined by measurements made on a possibly small subset of the boundary. The proof is based on a combination of Carleman estimates for first and second order systems, and involves a reduction of the boundary measurements to the second order case. For this reduction a certain amount of decoupling is required. To effectively make use of the decoupling, the Carleman estimates are established for coefficients which may become singular in the asymptotic limit.
We study the massive two dimensional Dirac operator with an electric potential. In particular, we show that the $t^{-1}$ decay rate holds in the $L^1\to L^\infty$ setting if the threshold energies are regular. We also show these bounds hold in the presence of s-wave resonances at the threshold. We further show that, if the threshold energies are regular that a faster decay rate of $t^{-1}(\log t)^{-2}$ is attained for large $t$, at the cost of logarithmic spatial weights. The free Dirac equation does not satisfy this bound due to the s-wave resonances at the threshold energies.
In this work we construct self-adjoint extensions of the Dirac operator associated to Hermitian matrix potentials with Coulomb decay and prove that the domain is maximal. The result is obtained by means of a Hardy-Dirac type inequality. In particular, we can work with some electromagnetic potentials such that both, the electric potential and the magnetic one, have Coulomb type singularity.
In this paper we show the existence of infinitely many symmetric solutions for a cubic Dirac equation in two dimensions, which appears as effective model in systems related to honeycomb structures. Such equation is critical for the Sobolev embedding and solutions are found by variational methods. Moreover, we prove also prove smoothness and exponential decay at infinity.
In this paper, we present new estimates for the entropy dissipation of the Landau-Fermi-Dirac equation (with hard or moderately soft potentials) in terms of a weighted relative Fisher information adapted to this equation. Such estimates are used for studying the large time behaviour of the equation, as well as for providing new a priori estimates (in the soft potential case). An important feature of such estimates is that they are uniform with respect to the quantum parameter. Consequently, the same estimations are recovered for the classical limit, that is the Landau equation.
In the present paper we obtain estimates in the modulation spaces for the solutions to the Dirac equation with quadratic and sub-quadratic potentials. We derive a representation for the Dirac operator that permits to solve approximately the perturbed Dirac equation and to obtain the desired estimates for the solution.
We extend to general dimension $n\ge1$ the virial identity proved by Boussaid-D'Ancona-Fanelli for the 3D magnetic Dirac equation. As an application we deduce smoothing and Strichartz estimates for an $n$-dimensional Dirac equation perturbed with a magnetic potential.
We prove local in time Strichartz estimates for the Dirac equation on spherically symmetric manifolds. As an application, we give a result of local well-posedness for some nonlinear models.
We show that global Strichartz estimates for magnetic Dirac operators generally fails, if the potentials do not decay fast enough at infinity. In order to prove this, we construct some explicit examples of homogeneous magnetic potentials with less than Coulomb decay, i. e. with homogeneity-degree more than -1, such that the magnetic field points to a fixed direction, which does not depend on $x\in \mathbb{R}^3$.
In this paper we prove Strichartz estimates for the Dirac equation on asymptotically flat manifolds. The proof combines the weak dispersive estimates proved by the first two authors with the Strichartz and smoothing estimates for the wave and Klein-Gordon flows, exploiting some recent results in the same geometrical setting.
In this paper we study global-in-time, weighted Strichartz estimates for the Dirac equation on warped product spaces in dimension $n\geq3$. In particular, we prove estimates for the dynamics restricted to eigenspaces of the Dirac operator on the compact spin manifolds defining the ambient manifold under some explicit sufficient condition on the metric, and estimates with loss of angular derivatives for general initial data in the setting of spherically symmetric and asymptotically flat manifolds.
In this {\bf draft version} we prove inhomogeneous Strichartz estimates with spherical symmetry in the abstract setting via duality arguments. Then we derive some new explicit estimates in the context of the wave equation. This allows us to prove global existence of spherically symmetric solutions to the Dirac-Klein-Gordon (DKG) system in two space dimensions.
We prove generalized Strichartz estimates for wave and massless Dirac equations in Aharonov-Bohm magnetic fields. Following a well established strategy to deal with scaling critical perturbations of dispersive PDEs, we make use of Hankel transform and rely on some precise estimates on Bessel functions. As a complementary result, we prove a local smoothing estimate for the Klein-Gordon equation in the same magnetic field.
We determine the largest non-trivial rate of exponential decay at infinity for solutions to the Dirac equation \begin{equation*} \mathcal{D}_n ψ+ \mathbb{V} ψ= 0 \quad \text{ in }\mathbb{R}^n, \end{equation*} being $\mathcal{D}_n$ the massless Dirac operator in dimension $n\geq 2$ and $\mathbb{V}$ a (possibly non-Hermitian) matrix-valued perturbation such that $|\mathbb{V}(x)| \sim |x|^{-ε}$ at infinity, for $-\infty < ε< 1$. Moreover, we show that our results are sharp for $n =2,3$, providing explicit examples of solutions that have the prescripted decay, in presence of a potential with the r…
We prove new bilinear estimates for the X^{s, b}_\pm(R^2) spaces which are optimal up to endpoints. These estimates are often used in the theory of nonlinear Dirac equations on R^{1+1}. The proof of the bilinear estimates follows from a dyadic decomposition in the spirit of Tao [21] and D'Ancona, Foschi, and Selberg [11]. As an application, by using the I-method of Colliander, Keel, Staffilani, Takaoka, and Tao, we extend the work of Tesfahun [23] on global existence below the charge class for the Dirac-Klein- Gordon equation on R^{1+1}.
In this paper we study Strichartz estimates for the half wave, the half Klein-Gordon and the Dirac Equations on compact manifolds without boundary, proving in particular for each of these flows local in time estimates both for the wave and Schroedinger admissible couples (in this latter case with an additional loss of regularity). The strategy for the proof is based on a refined version of the WKB approximation.
In this paper we obtain some new Strichartz estimates for the Dirac flow in the context of Wiener amalgam spaces which control the local regularity of a function and its decay at infinity separately unlike $L^p$ spaces. While it is well understood recently for some flows such as the Schrödinger and wave flows that work in the non-relativistic regime, nothing is known about the Dirac flow which governs a physical system in the case of relativistic fields.
We are interested in the cubic Dirac equation in two space dimensions. We establish the small data global existence and sharp pointwise decay results for general cubic nonlinearities without additional structure. We also prove the scattering of the Dirac equation for certain classes of nonlinearities. In all the above results we do not require the initial data to have compact support.
A Fokker-Planck type equation for interacting particles with exclusion principle is analysed. The nonlinear drift gives rise to mathematical difficulties in controlling moments of the distribution function. Assuming enough initial moments are finite, we can show the global existence of weak solutions for this problem. The natural associated entropy of the equation is the main tool to derive uniform in time a priori estimates for the kinetic energy and entropy. As a consequence, long-time asymptotics in $L^1$ are characterized by the Fermi-Dirac equilibrium with the same initial mass. This resu…
In this document we discuss the long time behaviour for the homogeneous Landau-Fermi-Dirac equation in the hard potential case. Uniform in time estimates for statistical moments and Sobolev regularity are presented and used to prove exponential relaxation of non degenerate distributions to the Fermi-Dirac statistics. All these results are valid for rather general initial datum. An important feature of the estimates is the independence with respect to the quantum parameter. Consequently, in the classical limit the same estimates are recovered for the Landau equation.
