Maxwell Klein Gordon Equations Research Articles

Asymptotics and Scattering for Massive Maxwell–Klein–Gordon Equations

Asymptotics and Scattering for Massive Maxwell–Klein–Gordon Equations

Probabilistic Small Data Global Well-Posedness of the Energy-Critical Maxwell–Klein–Gordon Equation

Probabilistic Small Data Global Well-Posedness of the Energy-Critical Maxwell–Klein–Gordon Equation

Convergence of a discretization of the Maxwell–Klein–Gordon equation based on finite element methods and lattice gauge theory

AbstractThe Maxwell–Klein–Gordon equations are a set of coupled nonlinear time‐dependent wave equations, used to model the interaction of an electromagnetic field with a particle. The solutions, expressed with a magnetic vector potential, are invariant under gauge transformations. This characteristic implies a constraint on the solution fields that might be broken at the discrete level. In this article, we propose and study a constraint preserving numerical scheme for this set of equations in dimension 2. At the semidiscrete level, we combine conforming Finite Element discretizations with the so‐called Lattice Gauge Theory to design a compatible gauge invariant discretization, leading to preservation of a discrete constraint. Relying on energy techniques and compactness arguments, we establish the convergence of this semidiscrete scheme, without a priori knowledge of the solution. Finally, at the fully discrete level, we propose a compatible explicit time‐integration strategy of leapfrog type. We implement the resulting fully discrete scheme and provide assessment on academic scenarios.

Nonrelativistic limit of solitary waves for nonlinear Maxwell–Klein–Gordon equations

We study the nonrelativistic limit of solitary waves from Nonlinear Maxwell–Klein–Gordon equations (NMKG) to Nonlinear Schrödinger–Poisson equations (NSP). It is known that the existence or multiplicity of positive solutions depends on the choices of parameters the equations contain. In this paper, we prove that for a given positive solitary wave of NSP, which is found in Ruiz’s work (J Funct Anal 237(2):655–674, 2006), there corresponds a family of positive solitary waves of NMKG under the nonrelativistic limit. Notably, our results contain a new result of existence of positive solutions to (NMKG) with lower order nonlinearity.

L∞ bounds for Maxwell-gauged equations in [formula omitted] and their applications

By making use of local energy conservation, we estimate the L∞ norm of the first derivative of the solution to the Maxwell-gauged O(3) sigma model in R1+1. As an application of the bound, we obtain the growth of the H2 norm of the solution. Quadratic polynomial growth is obtained by improving exponential growth. A similar idea can be applied to the Maxwell–Klein–Gordon equations in R1+1.

Nonlinear wave equations from a non-local complex backward–forward derivative operator

ABSTRACT In this study, we argue that it is possible to construct non-local nonlinear wave equations with random propagation as extensions to some well-known wave equations found in field theory with cubic interactions like Maxwell–Klein–Gordon equations, Dirac–Klein–Gordon equations, wave maps, solitons and Yang–Mills equations. Our construction is based on the extended complex backward–forward derivative operator which represents the basic differentiable operator tool to deal with non-local Lagrangians.

Global Well-Posedness for the Massive Maxwell–Klein–Gordon Equation with Small Critical Sobolev Data

In this paper we prove global well-posedness and modified scattering for the massive Maxwell–Klein–Gordon equation in the Coulomb gauge on $$\mathbb {R}^{1+d}$$ $$(d \ge 4)$$ for data with small critical Sobolev norm. This extends to the general case $$ m^2 > 0 $$ the results of Krieger–Sterbenz–Tataru ( $$d=4,5 $$ ) and Rodnianski–Tao ( $$ d \ge 6 $$ ), who considered the case $$ m=0$$ . We proceed by generalizing the global parametrix construction for the covariant wave operator and the functional framework from the massless case to the Klein–Gordon setting. The equation exhibits a trilinear cancelation structure identified by Machedon–Sterbenz. To treat it one needs sharp $$L^{2}$$ null form bounds, which we prove by estimating renormalized solutions in null frames spaces similar to the ones considered by Bejenaru–Herr. To overcome logarithmic divergences we rely on an embedding property of $$ \Box ^{-1} $$ in conjunction with endpoint Strichartz estimates in Lorentz spaces.

Concentration Compactness for Critical Radial Wave Maps

We consider radially symmetric, energy critical wave maps from $$(1+2)$$ -dimensional Minkowski space into the unit sphere $$\mathbb {S}^m$$ , $$m \ge 1$$ , and prove global regularity and scattering for classical smooth data of finite energy. In addition, we establish a priori bounds on a suitable scattering norm of the radial wave maps and exhibit concentration compactness properties of sequences of radial wave maps with uniformly bounded energies. This extends and complements the beautiful classical work of Christodoulou and Tahvildar-Zadeh (Duke Math J 71(1):31–69, 1993; Pure Appl Math 46(7):1041–1091, 1993) and Struwe (Math Z 242(3):407–414, 2002; Calc Var Partial Differ Equ 16(4):431–437, 2003) as well as of Nahas (Calc Var Partial Differ Equ 46(1–2):427–437, 2013) on radial wave maps in the case of the unit sphere as the target. The proof is based upon the concentration compactness/rigidity method of Kenig and Merle (Invent Math 166(3):645–675, 2006; Acta Math 201(2):147–212, 2008) and a “twisted” Bahouri–Gerard type profile decomposition (Am J Math 121(1):131–175, 1999), following the implementation of this strategy by the second author and Schlag (Concentration compactness for critical wave maps. EMS monographs in mathematics, European Mathematical Society (EMS), Zurich, 2012) for energy critical wave maps into the hyperbolic plane as well as by the last two authors (Ann PDE 1(1):1–208, 2015) for the energy critical Maxwell–Klein–Gordon equation.

Stability and Instability of the Sub-extremal Reissner\u2013Nordstr\xf6m Black Hole Interior for the Einstein\u2013Maxwell\u2013Klein\u2013Gordon Equations in Spherical Symmetry

We show non-linear stability and instability results in spherical symmetry for the interior of a charged black hole—approaching a sub-extremal Reissner–Nordström background fast enough—in presence of a massive and charged scalar field, motivated by the strong cosmic censorship conjecture in that setting:Stability We prove that spherically symmetric characteristic initial data to the Einstein–Maxwell–Klein–Gordon equations approaching a Reissner–Nordström background with a sufficiently decaying polynomial decay rate on the event horizon gives rise to a space–time possessing a Cauchy horizon in a neighbourhood of time-like infinity. Moreover, if the decay is even stronger, we prove that the space–time metric admits a continuous extension to the Cauchy horizon. This generalizes the celebrated stability result of Dafermos for Einstein–Maxwell-real-scalar-field in spherical symmetry.Instability We prove that for the class of space–times considered in the stability part, whose scalar field in addition obeys a polynomial averaged-L2 (consistent) lower bound on the event horizon, the scalar field obeys an integrated lower bound transversally to the Cauchy horizon. As a consequence we prove that the non-degenerate energy is infinite on any null surface crossing the Cauchy horizon and the curvature of a geodesic vector field blows up at the Cauchy horizon near time-like infinity. This generalizes an instability result due to Luk and Oh for Einstein–Maxwell-real-scalar-field in spherical symmetry.This instability of the black hole interior can also be viewed as a step towards the resolution of the C2 strong cosmic censorship conjecture for one-ended asymptotically flat initial data.

On the global behavior of solutions of the Maxwell–Klein–Gordon equations

It is known that the Maxwell–Klein–Gordon equations in R3+1 admit global solutions with finite energy data. In this paper, we present a new approach to study the asymptotic behavior of these global solutions. We show the quantitative energy flux decay of the solutions with data merely bounded in some weighted energy space. We also establish an integrated local energy decay and a hierarchy of r-weighted energy decay. The results in particular hold in the presence of large total charge. This is the first result to give a complete and precise description of the global behavior of large nonlinear charged scalar fields.

Decay of solutions of Maxwell–Klein–Gordon equations with arbitrary Maxwell field

In the author’s previous work, it has been shown that solutions of Maxwell–Klein–Gordon equations in ℝ3+1 possess some form of global strong decay properties with data bounded in some weighted energy space. In this paper, we prove pointwise decay estimates for the solutions for the case when the initial data are merely small on the scalar field but can be arbitrarily large on the Maxwell field. This extends the previous result of Lindblad and Sterbenz, in which smallness was assumed both for the scalar field and the Maxwell field.

Efficient time integration of the Maxwell–Klein–Gordon equation in the non-relativistic limit regime

The Maxwell–Klein–Gordon equation describes the interaction of a charged particle with an electromagnetic field. Solving this equation in the non-relativistic limit regime, i.e. the speed of light c formally tending to infinity, is numerically very delicate as the solution becomes highly-oscillatory in time. In order to resolve the oscillations, standard numerical time integration schemes require severe time step restrictions depending on the large parameter c2.The idea to overcome this numerical challenge is to filter out the high frequencies explicitly by asymptotically expanding the exact solution with respect to the small parameter c−2. This allows us to reduce the highly-oscillatory problem to its corresponding non-oscillatory Schrödinger–Poisson limit system. On the basis of this expansion we are then able to construct efficient numerical time integration schemes, which do NOT suffer from any c-dependent time step restriction.

Local Well-Posedness of the (4 + 1)-Dimensional Maxwell–Klein–Gordon Equation at Energy Regularity

This paper is the first part of a trilogy [22, 23] dedicated to a proof of global well-posedness and scattering of the \((4+1)\)-dimensional mass-less Maxwell–Klein–Gordon equation (MKG) for any finite energy initial data. The main result of the present paper is a large energy local well-posedness theorem for MKG in the global Coulomb gauge, where the lifespan is bounded from below by the energy concentration scale of the data. Hence the proof of global well-posedness is reduced to establishing non-concentration of energy. To deal with non-local features of MKG we develop initial data excision and gluing techniques at critical regularity, which might be of independent interest.

Static gravitational equations of general relativity and “the fifth force”

Einstein’s static field equations are investigated in various coordinate charts. After comparing Newtonian gravitational theory (in a curvilinear coordinate chart) with various charts of Einstein’s static gravitational equations, the most appropriate choice of the coordinate chart for Einstein’s static field equations is made. As a consequence, Einstein’s equations imply the non-linear potential equation $$\begin{aligned} e^{2\omega ({\mathbf {x}})} \,{{\mathbf {\cdot }}}\, (\mathop {\nabla }\limits ^{\circ })^2 \omega ({\mathbf {x}}) = 4\pi G \,{{\mathbf {\cdot }}}\, \big [T^\alpha _\alpha ({\mathbf {x}}) - T^4_4 ({\mathbf {x}})\big ], \end{aligned}$$ instead of the usual Poisson’s equation of the Newtonian theory. Investigating the non-linear potential equation above in the spherically symmetric cases, the corresponding potentials \(\omega (r)\) yield scenarios comparable to “the fifth force”. Next, static gravitational and electric fields generated by an incoherent charged dust are investigated. The corresponding non-linear potential equation is derived. Finally, the static Einstein–Maxwell–Klein–Gordon equations are explored and again, the corresponding non-linear potential equation is obtained. This potential resembles the static Higgs boson field.

Global well-posedness for the Maxwell–Klein–Gordon equation in 4+1 dimensions: Small energy

We prove that the critical Maxwell-Klein Gordon equation on R4+1 is globally well-posed for smooth initial data which are small in the energy. This reduces the problem of global regularity for large, smooth initial data to precluding concentration of energy.

On the Bondi mass of Maxwell–Klein–Gordon spacetimes

In this paper we calculate the Bondi mass of asymptotically flat spacetimes with interacting electromagnetic and scalar fields. The system of coupled Einstein–Maxwell–Klein–Gordon equations is investigated and corresponding field equations are written in the spinor form and in the Newman–Penrose formalism. Asymptotically flat solution of the resulting system is found near null infinity. Finally we use the asymptotic twistor equation to find the Bondi mass of the spacetime and derive the Bondi mass-loss formula. We compare the results with our previous work (Bičák et al. in Class Quantum Gravity 27(17):175011, 2010) and show that, unlike the conformal scalar field, the (Maxwell–)Klein–Gordon field has negatively semi-definite mass-loss formula.

Spinning gauged boson stars in anti-de Sitter spacetime

We study axially symmetric solutions of the Einstein–Maxwell–Klein–Gordon equations describing spinning gauged boson stars in a (3+1)-dimensional asymptotically AdS spacetime. These smooth horizonless solutions possess an electric charge and a magnetic dipole moment, their angular momentum being proportional to the electric charge. A special class of solutions with a self-interacting scalar field, corresponding to static axially symmetric solitons with a nonzero magnetic dipole moment, is also investigated.

Vortex Dynamics for the Nonlinear Maxwell–Klein–Gordon Equation

We derive the vortex dynamics for the nonlinear Maxwell–Klein–Gordon equation with the Ginzburg–Landau type potential. In particular, we consider the case when the external electric fields are of order \({O( | \log \epsilon |^{\frac{1}{2}})}\). We study the convergence of the space–time Jacobian \({\partial_t \psi \cdot i \nabla \psi}\) as an interaction term between the vortices and electric fields. An explicit form of the limiting vector measure is shown.

Discretizing the Maxwell-Klein-Gordon equation by the lattice gauge theory formalism

In this article we study the discretization of the Maxwell–Klein–Gordon equation from a variational point of view. We first solve the problem with an action corresponding to the Yee scheme for the Maxwell part, which is automatically gauge invariant, and a gauge invariant action for the Klein–Gordon part given by the lattice gauge theory. The action is showed to be consistent with the continuous formulation, and the equations to be solved are derived from a discrete stationary action principle. Due to the gauge invariance, the local electric charge can be shown to be conserved. As this is an essential feature of the continuous model, this conservation can be viewed as the main achievement of this scheme. Thereafter we compare the above-described scheme with a scheme that uses a standard finite-difference approximation of the derivatives and where the coupling between the scalar field and the gauge field is done in the simplest way. This scheme will possess a global gauge symmetry that ensures the conservation of global charge as in the hybrid case, but the scheme has no local symmetry and no locally conserved charge. Finally, we present some numerical results in the temporal gauge, shedding light on the theoretical discussion.

Constraint Preserving Schemes for Some Gauge Invariant Wave Equations

We study discretizations of the Maxwell–Klein–Gordon equation as an example of a constrained geometric nonlinear evolution partial differential equation. For the temporal gauge we propose a fully discrete scheme which preserves the nonlinear constraint thanks to a special application of Lagrange multipliers. We show that the method generalizes to Hamiltonian wave equations whose kinetic and potential energy are both invariant under a group of transformations, even though the Galerkin spaces are not invariant. We then extend the method to the Lorenz gauge. Numerical results illustrate the discussion.