Concentration Compactness for the Critical Maxwell-Klein-Gordon Equation

Krieger, Joachim; Lührmann, Jonas

doi:10.1007/s40818-015-0004-y

Cited by 26 publications

(46 citation statements)

References 49 publications

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“…Finally, we note that an independent proof of global well-posedness and scattering of (MKG) has recently been announced by Krieger-Lührmann [17], following a version of the Bahouri-Gérard nonlinear profile decomposition [1] and the Kenig-Merle concentration compactness/rigidity scheme [13,14], developed by Krieger-Schlag [18] for the energy critical wave maps problem.…”

Section: Other Work On the Maxwell-klein-gordon Equationsmentioning

confidence: 90%

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

Tataru

2016

Ann. PDE

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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-KleinGordon 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.

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Section: Other Work On the Maxwell-klein-gordon Equationsmentioning

confidence: 90%

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

Tataru

2016

Ann. PDE

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“…We now turn to the details of the decomposition of the essentially singular sequence of data {φ n [0]} n≥1 into frequency atoms. Here we follow relatively closely Section 9.1 and Section 9.2 in [17] as well as Section 7.2 in [16], which in turn partially mimic Section III.1 in Bahouri-Gérard [1]. First, we need to introduce some terminology from [1].…”

Section: Concentration Compactness Stepmentioning

confidence: 99%

“…Here we note that the minimal blowup solution will merely have energy class regularity but that a strong local well-posedness theory for (WM) is only available at sub-critical regularities. For this reason we actually have to introduce a concept of energy class radial wave maps, which we achieve in Subsection 7.2 by regularization and reduction to the small energy case via finite speed of propagation, analogously to the procedures in [17] and [16].…”

Section: Conclusion Of the Induction On Frequency Processmentioning

confidence: 99%

“…Our goal is now to rule out the existence of such an essentially singular sequence of wave maps {φ n } n≥1 , hence proving Theorem 1.1. To this end we follow the general philosophy of the concentration compactness/rigidity method introduced by Kenig-Merle [6,7], but more precisely we shall follow the implementation of this strategy for energy critical wave maps into the hyperbolic plane as in [17] as well as for the energy critical Maxwell-Klein-Gordon equation as in [16]. In this section we carry out a "twisted" Bahouri-Gérard type profile decomposition that takes into account the strong low-high interactions in the wave maps nonlinearity.…”

Section: Concentration Compactness Stepmentioning

confidence: 99%

“…However, since the local well-posedness theory [8][9][10]12, 13] only pertains to data of regularity H 1+ x (R 2 ) × H 0+ x (R 2 ), we first of all have to introduce a notion of the wave maps evolution of radially symmetric energy class data. We shall achieve this analogously to the procedures in [17] and [16] by regularization and reduction to the small energy case via finite speed of propagation. We begin with the following "high-frequency perturbation" lemma.…”

Section: 2mentioning

confidence: 99%

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Concentration Compactness for Critical Radial Wave Maps

2018

Self Cite

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Abstract. We consider radially symmetric, energy critical wave maps from (1 + 2)-dimensional Minkowski space into the unit sphere S m , m ≥ 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-Tahvildar-Zadeh [3, 4] and Struwe [31, 33] as well as of Nahas [22] 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 7] and a "twisted" Bahouri-Gérard type profile decomposition [1], following the implementation of this strategy by the second author and Schlag [17] for energy critical wave maps into the hyperbolic plane as well as by the last two authors [16] for the energy critical Maxwell-Klein-Gordon equation.

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Global Solution for Massive Maxwell‐Klein‐Gordon Equations

Klainerman

Wang

Yang

2019

Comm Pure Appl Math

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We derive the asymptotic properties of the mMKG system (Maxwell coupled with a massive Klein‐Gordon scalar field) in the exterior of the domain of influence of a compact set. This complements the previous well‐known results, restricted to compactly supported initial conditions, based on the so‐called hyperboloidal method. That method takes advantage of the commutation properties of the Maxwell and Klein‐Gordon equations with the generators of the Poincaré group to resolve the difficulties caused by the fact that they have, separately, different asymptotic properties. Though the hyperboloidal method is very robust and applies well to other related systems, it has the well‐known drawback of requiring compactly supported data. In this paper we remove this limitation based on a further extension of the vector field method adapted to the exterior region. Our method applies, in particular, to nontrivial charges. The full problem can then be treated by patching together the new estimates in the exterior with the hyperboloidal ones in the interior. This purely physical space approach introduced here maintains the robust properties of the old method and can thus be applied to other situations such as the coupled Einstein Klein‐Gordon equation. © 2019 Wiley Periodicals, Inc.

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Concentration Compactness for the Critical Maxwell-Klein-Gordon Equation

Cited by 26 publications

References 49 publications

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

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

Concentration Compactness for Critical Radial Wave Maps

Global Solution for Massive Maxwell‐Klein‐Gordon Equations

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