Almost optimal local well-posedness for the (3+1)-dimensional Maxwell–Klein–Gordon equations

Machedon, Matei; Sterbenz, Jacob

doi:10.1090/s0894-0347-03-00445-4

Cited by 48 publications

(67 citation statements)

References 34 publications

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“…10 Using smallness of energy, it was shown that the parametrix obeys the desired N → S 1 bound and also that the error in (3.6) is perturbative. We remark that in the error estimate, not only the primary but also the secondary (trilinear) null structure, analogous to that in Maxwell-Klein-Gordon discovered in [23], are crucial.…”

Section: Review Of the Small Energy Case: Perturbative Nonlinearitiesmentioning

confidence: 63%

Untitled

2019

Bull. Amer. Math. Soc.

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This article is devoted to the energy critical hyperbolic Yang-Mills equation in the (4 + 1)-dimensional Minkowski space, which is considered by the authors in a sequence of four papers. The final outcome of these papers is twofold: (i) the Threshold Theorem, which asserts that global well-posedness and scattering hold for all topologically trivial initial data with energy below twice the ground state energy; and (ii) the Dichotomy Theorem, which for larger data in arbitrary topological classes provides a choice of two outcomes, either a global scattering solution or a soliton bubbling off. In the last case, the bubbling-off phenomena can happen in one of two ways: (a) in finite time, triggering a finite time blowup; or (b) in infinite time. Our goal here is to first describe the equation and the results, and then to provide an overview of the flow of ideas within their proofs in the above-mentioned four papers. Contents 1. Introduction 171 2. The caloric gauge 183 3. Energy dispersed caloric Yang-Mills waves 192 4. Large data, causality, and the temporal gauge 197 5. To bubble or not to bubble 201 About the authors 208 References 208

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Section: Review Of the Small Energy Case: Perturbative Nonlinearitiesmentioning

confidence: 63%

Untitled

2019

Bull. Amer. Math. Soc.

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“…Klainerman and Machedon [5] proved global well-posedness in energy space in Coulomb gauge and temporal gauge. Local well-posedness in Coulomb gauge for data for φ in the Sobolev space H s and for A in H r with r = s > 1/2, i.e., almost down to the critical space with repect to scaling, was shown by Machedon and Sterbenz [6]. In Lorenz gauge the global well-posedness result in energy space is due to Selberg and Teshafun [12].…”

Section: Hartmut Pechermentioning

confidence: 89%

Low regularity solutions for the (2+1)-dimensional Maxwell-Klein-Gordon equations in temporal gauge

Pecher¹

2016

CPAA

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The Maxwell-Klein-Gordon equations in 2+1 dimensions in temporal gauge are locally well-posed for low regularity data even below energy level. The corresponding (3+1)-dimensional case was considered by Yuan. Fundamental for the proof is a partial null structure in the nonlinearity which allows to rely on bilinear estimates in wave-Sobolev spaces by d'Ancona, Foschi and Selberg, on an (L p x L q t ) -estimate for the solution of the wave equation, and on the proof of a related result for the Yang-Mills equations by Tao.

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“…However, for generic equations, this intuition fails, see [39]; in particular, one may have to impose s > s 1 > s c . It is correct for geometric wave equations with null-structure such as WM [24], YM [31], MKG [40].…”

Section: Well-posedness Typology For Nonlinear Wave Equationsmentioning

confidence: 99%

Global regularity and singularity development for wave maps

Krieger

2007

Surveys in Differential Geometry

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Almost optimal local well-posedness for the (3+1)-dimensional Maxwell–Klein–Gordon equations

Cited by 48 publications

References 34 publications

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Untitled

Low regularity solutions for the (2+1)-dimensional Maxwell-Klein-Gordon equations in temporal gauge

Global regularity and singularity development for wave maps

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