On global dynamics of the Maxwell–Klein–Gordon equations

Yang, Shiwu; Yu, Pin

doi:10.4310/cjm.2019.v7.n4.a1

Cited by 15 publications

(32 citation statements)

References 25 publications

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“…On the other hand, fixing the residual gauge freedom if necessary and using the relations (7.2), for the Yang-Mills potential we deduce |A l | cu −2 + |Âl| cu −2 + , |An| cv −2 + |Ân| cv −2 + , and |A|s 2 cΩ cu −1 + v −1 + . The above decay rates reproduce the decay rates of Yang and Yu [21], requiring one fewer order of differentiability in the data. However, our results do not apply to the case of arbitrary charge at spatial infinity.…”

Section: Minkowski Spacesupporting

confidence: 55%

Large data decay of Yang–Mills–Higgs fields on Minkowski and de Sitter spacetimes

Grigalius

2019

Journal of Mathematical Physics

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In this article we extend Eardley and Moncrief's L ∞ estimates [5] for the conformally invariant Yang-Mills-Higgs equations to the Einstein cylinder. Our method is to first work on Minkowski space and localize their estimates, and then carry them to the Einstein cylinder by a conformal transformation. By patching local estimates together, we deduce global L ∞ estimates on the cylinder, and extend Choquet-Bruhat and Christodoulou's [1] small data well-posedness result to large data. Finally, by employing another conformal transformation, we deduce exponential decay rates for Yang-Mills-Higgs fields on de Sitter space, and inverse polynomial decay rates on Minkowski space. the backward lightcone of p. Their key observation is that these lightcone integrals can be estimated by expressions of the formwhich implies, via Grönwall's inequality, that the L ∞ norms cannot blow up in finite time. Part of the trick is to define the L ∞ norms in a gauge-independent manner, and use the Crönstrom gauge in intermediate calculations. Equipped with this estimate, it is then straightforward to show that the H 2 × H 1 norm of the solution does not blow up in finite time. An incarnation of this method has been adapted, for pure Yang-Mills equations, to arbitrary smooth globally hyperbolic four dimensional spacetimes by Chruściel and Shatah [3], by replacing the lightcone integrals with Friedlander's representation formula [6] for the covariant wave equation. However, Chruściel and Shatah require effectively H 3 × H 2 data to deal with a term that causes difficulties in curved space 1 . Though the system has been well-studied, Eardley and Moncrief's method with H 2 × H 1 data for coupled Yang-Mills and Higgs equations does not seem to have been explicitly adapted to curved space, even in the case of the Einstein cylinder. The scalar field part scales differently under a conformal transformation, putting it on unequal footing with the Yang-Mills potential. In particular, this upsets the conformal invariance of the system somewhat, breaking the invariance of the canonical energy-momentum tensor. And although formally the field equations remain conformally invariant, the scalar field introduces a boundary term in the conformal variation of the action that has a non-trivial dependence on the decay of the scalar field. This is expected to be of some importance in path integral formulations of interacting quantum field theories.In this article we extend the L ∞ estimates of Eardley and Moncrief to the Einstein cylinder. Our method is inspired by and combines the techniques of [1,5,9]: we first work on Minkowski space and localize Eardley and Moncrief's estimates, removing the requirement of the global finiteness of the energy. Then, using a conformal transformation, we glue a small conical patch of Minkowski space onto the Einstein cylinder, and show that L ∞ estimates in the Minkowskian patch imply local L ∞ estimates on the cylinder. By patching a finite number of such cones all the way around the Einstein cylinder, we deduce L ∞ bounds ...

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Section: Minkowski Spacesupporting

confidence: 55%

Large data decay of Yang–Mills–Higgs fields on Minkowski and de Sitter spacetimes

Grigalius

2019

Journal of Mathematical Physics

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“…Let's only consider the second order derivatives as the one derivative case should be easier and the corresponding estimate can follow in a similar way. For the first term on the right hand side of the above inequality, we bound |D µ φ| by the pointwise estimates obtained in Proposition 2 and the other term |D X D Y φ| by the bound of energy flux along the outgoing null hypersurface in (31). Indeed from Proposition 2, we first can bound that…”

Section: Energy Decay Estimates For the Maxwell Fieldmentioning

confidence: 92%

“…Here G is a two form. We can bound φ by the pointwise bound in (41) and (L XF ) Y µ by the bound of energy flux in (31). We thus can derive that…”

Section: Energy Decay Estimates For the Maxwell Fieldmentioning

confidence: 99%

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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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“…In flat space the same observation imposes precise decay rates on the eletric field E at spatial infinity i 0 (and in particular implies a non-zero r −2 term), so the source term Im(φD 0 φ) is said to correspond to charge at i 0 . Recent work by Yang and Yu [53] and Candy, Kauffman, and Lindblad [10] quantifies such non-zero charge decay rates of the Maxwell-scalar field system in flat space. In de Sitter space, however, one cannot have any charge since there is no spatial infinity.…”

Section: Strong Coulomb Gaugementioning

confidence: 99%

Conformal scattering of the Maxwell-scalar field system on de Sitter space

Grigalius

2019

J. Hyper. Differential Equations

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We prove small data energy estimates of all orders of differentiability between past null infinity and future null infinity of de Sitter space for the conformally invariant Maxwell-scalar field system and construct bounded invertible nonlinear scattering operators taking past asymptotic data to future asymptotic data. We also deduce exponential decay rates for solutions with data having at least two derivatives. The construction involves a carefully chosen complete gauge fixing condition which allows us to control all components of the Maxwell potential, and a nonlinear Grönwall inequality for higher order estimates. * Electronic address: taujanskas@maths.ox.ac.uk

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On global dynamics of the Maxwell–Klein–Gordon equations

Cited by 15 publications

References 25 publications

Large data decay of Yang–Mills–Higgs fields on Minkowski and de Sitter spacetimes

Large data decay of Yang–Mills–Higgs fields on Minkowski and de Sitter spacetimes

Global Solution for Massive Maxwell‐Klein‐Gordon Equations

Conformal scattering of the Maxwell-scalar field system on de Sitter space

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