In this work we prove global well-posedness for the massive Maxwell-Dirac system in the Lorenz gauge in $\mathbb{R}^{1+3}$, for small, sufficiently smooth and decaying initial data, as well as modified scattering for the solutions. Heuristically we exploit the close connection between the massive Maxwell-Dirac and the wave-Klein-Gordon equations, while developing a novel approach which applies directly at the level of the Dirac equations. The modified scattering result follows from a precise description of the asymptotic behavior of the solutions inside the light cone, which we derive via the method of testing with wave packets of Ifrim-Tataru.
In this paper, we prove global well-posedness of the massless Maxwell-Dirac equation in Coulomb gauge on $\mathbb{R}^{1+d}$ $(d \geq 4)$ for data with small scale-critical Sobolev norm, as well as modified scattering of the solutions. Main components of our proof are A) uncovering null structure of Maxwell-Dirac in the Coulomb gauge, and B) proving solvability of the underlying covariant Dirac equation. A key step for achieving both is to exploit (and justify) a deep analogy between Maxwell-Dirac and Maxwell-Klein-Gordon (for which an analogous result was proved earlier by Krieger-Sterbenz-Tat…
We consider the Cauchy problem of massless Dirac-Maxwell equations on an asymptotically flat background and give a global existence and uniqueness theorem for initial values small in an appropriate weighted Sobolev space. The result can be extended via analogous methods to Dirac-Higgs-Yang-Mills theories.
The aim of this Note is to prove by a perturbation method the existence of solutions of the coupled Einstein-Dirac-Maxwell equations for a static, spherically symmetric system of two fermions in a singlet spinor state and with the electromagnetic coupling constant $(\frac{e}{m})^2<1$. We show that the nondegenerate solution of Choquard's equation generates a branch of solutions of the Einstein-Dirac-Maxwell equations.
Maxwell's equation, Dirac's equation and the equation of gravito-electromagnetism are shown to be particular instances of the extended Maxwell system. The equations are discussed in the framework of the theory of evolutionary equations. Their formal relationship are systematically analyzed. Applications to coupled systems such as the Maxwell-Dirac system are also discussed.
The Maxwell-Dirac equations with nonzero charge mass in one space dimension are studied under the Lorentz gauge condition. The global existence and uniqueness of solution in $C([0,+\infty);L^2(R^1))\times C_b(R^1 \times [0,\infty))$ for initial value problem of Maxwell-Dirac equations are proved.
We prove the existence of a $L^2$-normalized solitary wave solution for the Maxwell-Dirac equations in (3+1)-Minkowski space. In addition, for the Coulomb-Dirac model, describing fermions with attractive Coulomb interactions in the mean-field limit, we prove the existence of the (positive) energy minimizer.
We study the coupled system of Maxwell and Dirac equations from a semiclassical point of view. A rigorous nonlinear WKB-analysis, locally in time, for solutions of (critical) order $O(\sqrtε)$ is performed, where the small semiclassical parameter $ε$ denotes the microscopic/macroscopic scale ratio.
The Maxwell-Dirac equations in one space dimension are proved to be well posed in the charge class, that is, with $L^2$ data for the spinor. We also prove that this result is sharp, in the sense that well-posedness fails for spinor data in $H^s$ with $s<0$, as well as in $L^p$ with $1 \le p < 2$. More precisely, we give an explicit example of such data for which no local solution can exist. Our proof of well-posedness applies to a class of systems which includes also the Dirac-Klein-Gordon system, but it does not require any null structure in the system.
We consider an invariant formulation of the system of Maxwell's equations for an anisotropic medium on a compact orientable Riemannian 3-manifold $(M,g)$ with nonempty boundary. The system can be completed to a Dirac type first order system on the manifold. We show that the Betti numbers of the manifold can be recovered from the dynamical response operator for the Dirac system given on a part of the boundary. In the case of the original physical Maxwell system, assuming that the entire boundary is known, all Betti numbers of the manifold can also be determined from the dynamical response opera…
We uncover the full null structure of the Maxwell-Dirac system in Lorenz gauge. This structure, which cannot be seen in the individual component equations, but only when considering the system as a whole, is expressed in terms of tri- and quadrilinear integral forms with cancellations measured by the angles between spatial frequencies. In the 3D case, we prove frequency-localized $L^2$ space-time estimates for these integral forms at the scale invariant regularity up to a logarithmic loss, hence we obtain almost optimal local well-posedness of the system by iteration.
We consider the low regularity well-posedness problem for the Maxwell-Dirac system in n+1 dimensions in the cases $n=3$ and $n=2$ : \begin{align*} \partial^μ F_{μν} & = - \langle ψ,α_ν ψ\rangle \\ -i α^μ \partial_μ ψ& = A_μ α^μ ψ\, , \end{align*} where $ F_{μν} = \partial^μ A_ν - \partial^ν A_μ$ , and $α^μ$ are the Dirac matrices. We assume the temporal gauge $A_0=0$ and make use of the fact that some of the nonlinearities fulfill a null condition. Because we work in the temporal gauge we also apply a method, which was used by Tao for the Yang-Mills system in this gauge.
We deal with the ``nonrelativistic limit'', i.e. the limit c to infinity, where c is the speed of light, of the nonlinear PDE system obtained by coupling the Dirac equation for a 4-spinor to the Maxwell equations for the self-consistent field created by the ``moving charge'' of the spinor. This limit, sometimes also called ``Post-Newtonian'' limit, yields a Schroedinger-Poisson system, where the spin and the magnetic field no longer appear. However, our splitting of the 4-spinor into two 2-spinors preserves the symmetry of "electrons'' and "positrons''; the latter obeying a Schroedinger equati…
We study nonlinear bound states, or solitary waves, in the Dirac-Maxwell system proving the existence of solutions in which the Dirac wave function is of the form $ϕ(x,ω)e^{-iωt}$, $ω\in(-m,ω_*)$, with some $ω_*>-m$, such that $ϕ_ω\in H^1(\mathbb{R}^3,\mathbb{C}^4)$, $\Vertϕ_ω\Vert^2_{L^2}=O(m-|ω|)$, and $\Vertϕ_ω\Vert_{L^\infty}=O(m-|ω|)$. The method of proof is an implicit function theorem argument based on an identification of the nonrelativistic limit as the ground state of the Choquard equation.
In recent work, Grünrock and Pecher proved that the Dirac-Klein-Gordon system in 2d is globally well-posed in the charge class (data in $L^2$ for the spinor and in a suitable Sobolev space for the scalar field). Here we obtain the analogous result for the full Maxwell-Dirac system in 2d. Making use of the null structure of the system, found in earlier joint work with Damiano Foschi, we first prove local well-posedness in the charge class. To extend the solutions globally we build on an idea due to Colliander, Holmer and Tzirakis. For this we rely on the fact that MD is charge subcritical in tw…
The Maxwell-Dirac system describes the interaction of an electron with its self-induced electromagnetic field. In space dimension $d=3$ the system is charge-critical, that is, $L^2$-critical for the spinor with respect to scaling, and local well-posedness is known almost down to the critical regularity. In the charge-subcritical dimensions $d=1,2$, global well-posedness is known in the charge class. Here we prove that these results are sharp (or almost sharp, if $d=3$), by demonstrating ill-posedness below the charge regularity. In fact, for $d \le 3$ we exhibit a spinor datum belonging to $H^…
In this paper we consider a modified fractional Maxwell model based on the application of Hadamard-type fractional derivatives. The model is physically motivated by the fact that we can take into account at the same time memory effects and the time-dependence of the viscosity coefficient. We obtain an ultra-slow relaxation response whose explicit analytic form is given by the Mittag-Leffler function with a logarithmic argument. We show graphically the main properties of this relaxation response, also with the asymptotic behaviour.
We study the semi-classical ground states of the nonlinear Maxwell-Dirac system: \[ \left\{ \begin{aligned} &\al\cdot\big(i\hbar\nabla+ q(x)\fa(x)\big) w-a\bt w -ωw - q(x)ϕ(x) w = P(x)g(\jdz{w}) w\\ &-Δϕ=q(x)\jdz{w}^2\\ &-Δ{A_k}=q(x)(α_k w)\cdot \bar w\ \ \ \ k=1,2,3 \end{aligned}\right. \] for $x\in\R^3$, where $\fa$ is the magnetic field, $ϕ$ is the electron field and $q$ describes the changing pointwise charge distribution. We develop a variational method to establish the existence of least energy solutions for $\hbar$ small. We also describe the concentration behavior of the solutions as $…
In this monograph we prove that the nonlinear Lie algebra representation given by the manifestly covariant Maxwell-Dirac (M-D) equations is integrable to a global nonlinear representation $U$ of the Poincaré group ${\cal P}_0$ on a differentiable manifold ${\cal U}_\infty$ of small initial conditions for the M-D equations. This solves, in particular, the Cauchy problem for the M-D equations, namely existence of global solutions for initial data in ${\cal U}_\infty$ at $t=0$. The existence of modified wave operators $Ω_+$ and $Ω_-$ and asymptotic completeness is proved. The asymptotic represent…
We study the theory of scattering for the Maxwell-Schr"odinger system in space dimension 3,in the Coulomb gauge.In the special case of vanishing asymptotic magnetic field,we prove the existence of modified wave operators for that system with no size restriction on the Schr"odinger data and we determine the asymptotic behaviour in time of solutions in the range of the wave operators.The method consists in partially solving the Maxwell equations for the potentials,substituting the result into the Schr"odinger equation,which then becomes both nonlinear and nonlocal in time,and treating the latter…
In this paper, we consider the Maxwell-Dirac system in 3 dimension under zero magnetic field. We prove the global well-posedness and modified scattering for small solutions in the weighted Sobolev class. Imposing the Lorenz gauge condition, (and taking the Dirac projection operator), it becomes a system of Dirac equations with Hartree type nonlinearity with a long range potential as $|x|^{-1} $. We perform the weighted energy estimates. In this procedure, we have to deal with various resonance functions that stem from the Dirac projections. We use the spacetime resonance argument of Germain-Ma…
We study the theory of scattering for the Maxwell-Schr"odinger system in space dimension 3, in the Coulomb gauge. We prove the existence of modified wave operators for that system with no size restriction on the Schr"odinger and Maxwell asymptotic data and we determine the asymptotic behaviour in time of solutions in the range of the wave operators. The method consists in partially solving the Maxwell equations for the potentials, substituting the result into the Schr"odinger equation, which then becomes both nonlinear and nonlocal in time. The Schr"odinger function is then parametrized in ter…
The 1D Cauchy problem for the Dirac-Klein-Gordon system is shown to be locally well-posed for low regularity Dirac data in $\hat{H^{s,p}}$ and wave data in $\hat{H^{r,p}} \times \hat{H^{r-1,p}}$ for $1<p\le 2$ under certain assumptions on the parameters r and s, where $\|f\|_{\hat{H^{s,p}}} := \| < ξ>^s \hat{f}\|_{L^{p'}}$, generalizing the results for $p=2$ by Selberg and Tesfahun. Especially we are able to improve the results from the scaling point of view with respect to the Dirac part.
In recent decades qualitative inverse scattering methods with eigenvalues as target signatures received much attention. To understand those methods a knowledge on the properties of the related eigenvalue problems is essential. However, even the existence of eigenvalues for such (nonselfadjoint) problems is a challenging question and existing results for absorbing media are usually established under unrealistic assumptions or a smoothing of the eigenvalue problem. We present a technique to prove the existence of infinitely many eigenvalues for such problems under realistic assumptions. In parti…
In this paper we study a generalized class of Maxwell-Boltzmann equations which in addition to the usual collision term contains a linear deformation term described by a matrix A. This class of equations arises, for instance, from the analysis of homoenergetic solutions for the Boltzmann equation considered by many authors since 1950s. Our main goal is to study a large time asymptotics of solutions under assumption of smallness of the matrix A. The main result of the paper is formulated in Theorem 2.1. Informally stated, this Theorem says that, for sufficiently small norm of A, any non-negativ…
