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

Abstract

In the author's work [30], it has been shown that solutions of Maxwell-Klein-Gordon equations in R 3+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-Sterbenz [16], in which smallness was assumed both for the scalar field and the Maxwell field.We make several remarks:Remark 1. The second order derivatives of the initial data is the minimum regularity we need to derive the above pointwise decay of the solution. Similar decay estimates hold for the higher order derivatives of the solution if higher order weighted Sobolev norms of the initial data are known.Remark 2. The restriction on γ 0 , that is 0 < γ 0 < 1, is merely for the sake of brevity. If γ 0 ≥ 1, then the decay property of the solutions propagates in the exterior region (t + 2 ≤ r). In other words, we have the same decay estimates as in the theorem for τ ≤ 0. However in the interior region where τ > 0, the maximal decay rate is τ −2 + (corresponding to γ 0 = 1), that is, the decay rate in the interior region for γ 0 ≥ 1 in general can not be better than that of γ 0 = 1.Remark 3. Since we assume the scalar field is small, the charge is also small by definition. Combined with the techniques in [30], our approach can be adapted to the case with large charge. There are two ways of generalizations. The first one is to relax the assumption on the scalar field as in [28] so that the charge can be large. Secondly we can consider the following unphysical equations:for some constant λ.Compared to the previous result of Lindblad-Sterbenz [16], we have made the following improvements: first of all, we obtain pointwise decay estimates for solutions of (MKG) for a class of large initial data. We only require smallness on the scalar field. In particular our initial data for (MKG) can be arbitrarily large. Combining the method in [28], we can even make the data on the scalar field large in the energy space. Secondly we have lower regularity on the initial data. In [16], it was assumed that the derivative of the initial data decays one order better, that is, ∇ I (E df , H), D I (Dφ 0 , φ 1 ) belong to the weighted Sobolev space with weights (1 + r) 1+γ0+2|I| , while in this paper we only assume that the angular derivatives of the data obey this improved decay (see the definition of M, E). For the other derivatives, the weights is merely (1 + r) 1+γ0 . This makes the analysis more delicate. Moreover, as the solution decays weaker initially, our decay rate is weaker than that in [16] (only decay rate in τ , the decay in r is the same). However if we assume the same decay of the initial data as in [16], then we are able to obtain the same decay for the solution.We use a new approach developed in [30] to study the asymptotic behavior of solutions of (MKG). This new method was originally int...

Sharp Asymptotic Behavior of Solutions of the 3d Vlasov–Maxwell System with Small Data

“…The study of non linear systems with a presence of charge was initiated by [20] in the context of the Maxwell-Klein Gordon equations. The first complete proof of such a result was given by Lindblad and Sterbenz in [18] and improved later by Yang (see [23]). Let us also mention the work of [2].…”

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“…We use the r p -weighted energy estimates which was first introduced by Dafermos-Rodnianski in [8] for the study of decay of linear waves on black hole spacetimes. The method has also been used in the first author's works on MKG equations, see [26,25], where p < 2. The new point in the current work is that we have to deal with the end point case p = 2 to get the sharp peeling estimates.…”

“…However the decay estimates in [22] are too weak to see the effect of the charges while the latter work in addition affirmatively answered a conjecture of Shu in [23] that the nonzero charge can only affect the asymptotic behaviours of the solutions outside a forward light cone. Another consequence of the method used in [26] allowed the author to improve the small data results to data merely small on the scalar field while the Maxwell field can be arbitrarily large, see details in [25]. This result can be interpreted as the global nonlinear stability of linear Maxwell fields.…”

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“…The goal of this paper consists mainly two folds: Firstly we are trying to construct global large solutions to RVM with asymptotic decay properties. This is partially inspired by the work [20] of Rein and the global stability of Maxwell field coupled to scalar fields [25], [4]. We show that linear Maxwell field is also stable under perturbation of Vlasov field, that is, smallness is only required on the density distribution and the Maxwell field can be arbitrarily large.…”