Gursky, C. Kelleher, and J. Streets, A conformally invariant gap theorem in Yang-Mills theory , preprint , arXiv Translated from the French original; With a foreword by James Eells. MR [10] Carlos E. Kenig and Frank Merle , Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation , Acta Math.

Klainerman and M. Machedon , Space-time estimates for null forms and the local existence theorem , Comm. Machedon, Finite energy solutions of the Yang-Mills equations in , Ann. Krieger , W.

Schlag , and D. Tataru , Renormalization and blow up for the critical Yang-Mills problem , Adv. PDE 1 , no. Small energy , Ann. Hyperbolic Differ. Oh and D. Tataru, Global well-posedness and scattering of the - dimensional Maxwell-Klein-Gordon equation , Invent. Tataru, Local well-posedness of the -dimensional Maxwell-Klein-Gordon equation at energy regularity , Ann. PDE 2 , no. Tataru, The Yang-Mills heat flow and the caloric gauge , preprint , arXiv Tataru, The hyperbolic Yang-Mills equation in the caloric gauge. Local well-posedness and control of energy dispersed solutions , preprint , arXiv Global Anal.

JEMS 18 , no. Sibner , R. In the last case, the bubbling off phenomena can happen either a in finite time, triggering a finite time blow-up, or b in infinite time. Our goal here is to describe these results, and to provide an overview of the flow of ideas within their proofs.

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Unlike the previous approaches, the new gauge is applicable for large data, while the special analytic structure of the Yang-Mills equations is still manifest. These are classical results first proved by S. Machedon using the method of local Coulomb gauges, which had been difficult to extend to other settings.

Curriculum Vitae pdf. AP] Abstract: In this article, we prove various illposedness results for the Cauchy problem for the incompressible Hall- and electron-magnetohydrodynamic MHD equations without resistivity. The proof is completed by showing that the latter solutions do not exist. Jan 20, May 5, Notes 1 Lecture notes on linear wave equation.

On the Cauchy problem for the Hall and electron magnetohydrodynamic equations without resistivity I: illposedness near degenerate stationary solutions , with I. Lawrie , J. Luehrmann and S. The following four papers constitute a series, whose overview is provided in the summary below. The Yang-Mills heat flow and the caloric gauge , with D.

The hyperbolic Yang-Mills equation in the caloric gauge. Local well-posedness and control of energy dispersed solutions , with D. The hyperbolic Yang-Mills equation for connections in an arbitrary topological class , with D. Tataru , to appear in Comm. The threshold conjecture for the energy critical hyperbolic Yang-Mills equation , with D. The following two papers constitute a series; for an overview, see Section 1. Strong cosmic censorship in spherical symmetry for two-ended asymptotically flat initial data I.

The interior of the black hole region , with J. Luk , to appear in Ann. Strong cosmic censorship in spherical symmetry for two-ended asymptotically flat initial data II.

## Analysis Seminar

The exterior of the black hole region , with J. Dynamical black holes with prescribed masses in spherical symmetry , with J. Luk and S. Solutions to the Einstein-scalar-field system in spherical symmetry with large bounded variation norms , with J.

Yang , Ann. Global well-posedness of high dimensional Maxwell-Dirac for small critical data , with C. Gavrus, to appear in Mem. Small data global existence and decay for relativistic Chern-Simons equations , with M.

- Local well-posedness of Yangâ€“Mills equations in Lorenz gauge below the energy norm | SpringerLink.
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Lawrie and S. Shahshahani , Int. IMRN , no. The following two papers are parts of the preprint arXiv Isett , Arch.

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Rational Mech. ARMA Vol. Shahshahani , Math. AP] Abstract: In this paper we continue the analysis of equivariant wave maps from 2-dimensional hyperbolic space into surfaces of revolution that was initiated in earlier papers. The following three papers constitute a series; for an overview, see Sections of Paper Tataru , Ann. Tataru , Amer. Tataru , Invent. Lawrie , Comm. AP] Abstract: The recently established threshold theorem for energy critical wave maps states that wave maps with energy less than that of the ground state i.

Shahshahani , J. AP] Abstract: In this paper we study k-equivariant wave maps from the hyperbolic plane into the 2-sphere as well as the energy critical equivariant SU 2 Yang-Mills problem on 4-dimensional hyperbolic space.

## Yang-Mills equations - DispersiveWiki

Luk , Duke Math. Profile decomposition for wave equations on hyperbolic space with applications , with A. AP] Abstract: The goal for this paper is twofold. Stability of stationary equivariant wave maps from the hyperbolic plane , with A. Shahshahani , Amer. AP] Abstract: In this paper we initiate the study of equivariant wave maps from 2d hyperbolic space into rotationally symmetric surfaces.

Quantitative decay rates for dispersive solutions to the Einstein-scalar field system in spherical symmetry , with J. One has. The dynamics has a constraint and we need a gauge condition to get the equations of motion. This can also be seen in the corresponding Lagrangian formulation, after a Legendre transform, where a Lagrange multiplier indeed appears. Now, one can write down the corresponding Poisson brackets obtaining.

## Global regularity for the Yang-Mills equations on high dimensional Minkowski space (ebook)

We note that these are operator equations and so we need to build a proper state space to give them a meaning. Presently, a rigorous proof of existence of all this construction does not exist yet and would be a proof of existence for the Yang-Mills theory itself. The best one can do is to perform a small perturbation theory of these equations on a Fock space built on the solutions of the leading order equations. Also this construction has not a rigorous proof but is common practice between physics community with a considerable success. But, in order to perform such kind of computations, a different approach is used that heavily relies on path integration.

No sound foundation for path integrals in Minkowski spaces exists yet. Jump to: navigation , search. Categories : Equations Geometry Wave. Personal tools Log in. Critical regularity. Lorentzian , gauge , conformal.