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

Oh, Sung-Jin; Tataru, Daniel

doi:10.1007/s40818-016-0006-4

Cited by 24 publications

(19 citation statements)

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“…We have elected not to give a detailed proof, as this statement is not used in the present paper. See [28] for the analysis of the case of the divergence equation ∂ l R l = U .…”

Section: Main Results For the Symmetric Divergence Equationmentioning

confidence: 99%

On Nonperiodic Euler Flows with Hölder Regularity

Isett

2016

Arch Rational Mech Anal

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In (Isett, Regularity in time along the coarse scale flow for the Euler equations, 2013), the first author proposed a strengthening of Onsager's conjecture on the failure of energy conservation for incompressible Euler flows with Hölder regularity not exceeding 1/3. This stronger form of the conjecture implies that anomalous dissipation will fail for a generic Euler flow with regularity below the Onsager critical space L ∞ t B 1/3 3,∞ due to low regularity of the energy profile. This paper is the first and main paper in a series of two, the results of which may be viewed as first steps towards establishing the conjectured failure of energy regularity for generic solutions with Hölder exponent less than 1/5. The main result of the present paper shows that any given smooth Euler flow can be perturbed in C 1/5−ε t,x on any pre-compact subset of R × R 3 to violate energy conservation. Furthermore, the perturbed solution is no smoother than C. As a corollary of this theorem, we show the existence of nonzero C 1/5−ε t,x solutions to Euler with compact space-time support, generalizing previous work of the first author (Isett, Hölder continuous Euler flows in three dimensions with compact support in time, 2012) to the nonperiodic setting.

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“…We have elected not to give a detailed proof, as this statement is not used in the present paper. See [28] for the analysis of the case of the divergence equation ∂ l R l = U .…”

Section: Main Results For the Symmetric Divergence Equationmentioning

confidence: 99%

On Nonperiodic Euler Flows with Hölder Regularity

Isett

2016

Arch Rational Mech Anal

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“…Recently, a proof of the global regularity and scattering affirmations in the preceding theorem was obtained by Oh-Tataru [32][33][34], following the method developed by Sterbenz-Tataru [40,41] in the context of critical wave maps. Our conclusions were reached before the appearance of their work and our methods are completely independent.…”

Section: Theorem 12 Consider the Evolution Problem (Mkg-mentioning

confidence: 99%

Concentration Compactness for the Critical Maxwell-Klein-Gordon Equation

Krieger

Lührmann

2015

Ann. PDE

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“…As we will explain in Section 1.2, [18] may be regarded as one of the direct predecessors of the present work. In the energy critical case d = 4, global wellposedness of (MKG) for arbitrary finite energy data was recently established by the second author and Tataru [22,23,24], and independently by Krieger-Lührmann [17]. In contrast, although (MD) is also energy critical on R 1+4 , the energy for (MD) is not coercive; whether our Theorem 1.1 may be extended to the large data case is therefore unclear.…”

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confidence: 86%

Global Well-Posedness of High Dimensional Maxwell–Dirac for Small Critical Data

Gavrus

2020

Memoirs of the AMS

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In this paper, we prove global well-posedness of the massless Maxwell-Dirac equation in Coulomb gauge on R 1+d (d ≥ 4) for data with small scale-critical Sobolev norm, as well as modified scattering of the solutions. Main components of our proof are A) uncovering null structure of Maxwell-Dirac in the Coulomb gauge, and B) proving solvability of the underlying covariant Dirac equation. A key step for achieving both is to exploit (and justify) a deep analogy between Maxwell-Dirac and Maxwell-Klein-Gordon (for which an analogous result was proved earlier by Krieger-Sterbenz-Tataru [18]), which says that the most difficult part of Maxwell-Dirac takes essentially the same form as Maxwell-Klein-Gordon.The authors thank Daniel Tataru for many fruitful conversations. C. Gavrus was supported in part by the NSF grant DMS-1266182. S.-J. Oh is a Miller Research Fellow, and acknowledges support from the Miller Institute. Part of this work was carried out during the trimester program 'Harmonic Analysis and PDEs' at the Hausdorff Institute of Mathematics in Bonn.such that the gauge transformÃ(0) = A(0)−dχ obeys the Coulomb gauge condition (4.7). Moreover, the small data condition (1.4) is preserved up to multiplication by a universal constant for ǫ * small enough. Such a gauge transformation can be found by solving the Poisson equation ∆χ = div x A j (0).We remark that our method do not apply to the case of nonzero mass m = 0, although the observations made in this paper suggest that it would likely follow from a corresponding result for the massive Maxwell-Klein-Gordon equations; see 'Parallelism with Maxwell-Klein-Gordon' in Section 1.2. The physically interesting case of d = 3, with or without mass, remains open. 1.1. Previous work. A brief survey of previous results on (MD) and related equations is in order. After early work on local well-posedness of (MD) on R 1+3 by Gross [13] and Bournaveas [4], D'Ancona-Foschi-Selberg [8] established local well-posedness of (MD) on R 1+3 in the Lorenz gauge ∂ µ A µ = 0 for data SMALL CRITICAL DATA GWP FOR MAXWELL-DIRAC 5Another type of null structure that arise in our work is spinorial null forms. These are bilinear forms with the symbol Π ± (ξ)Π ∓ (η), which were first uncovered by D'Ancona, Foschi, Selberg for the Dirac-Klein-Gordon system in [7]. These authors further investigated the spinorial null forms in the study of the Maxwell-Dirac equation on R 1+3 in the Lorenz gauge (in [8]; see also [9]). In the work of Bejenaru-Herr [3,2] and Bournaveas-Candy [5], these null forms were used in the proof of global well-posedness of the cubic Dirac equation for small critical data.A more detailed exposition of the null structure of MD-CG is given in Section 7.4 below. At this point we simply note that the null structure alone is insufficient to close the proof of Theorem 1.1 due to the presence of nonperturbative nonlinearity, which is the next topic of discussion.Presence of nonperturbative nonlinearity. As in many previous works on low regularity well-posedness, we take a paradifferentia...

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

Cited by 24 publications

References 34 publications

On Nonperiodic Euler Flows with Hölder Regularity

On Nonperiodic Euler Flows with Hölder Regularity

Concentration Compactness for the Critical Maxwell-Klein-Gordon Equation

Global Well-Posedness of High Dimensional Maxwell–Dirac for Small Critical Data

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