Global Regularity for the Yang-Mills Equations on High Dimensional Minkowski Space
Global Regularity for the Yang-Mills Equations on High Dimensional Minkowski Space

Author: Joachim Krieger

Publisher: American Mathematical Soc.

Published: 2013-04-22

Total Pages: 111

ISBN-13: 082184489X

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This monograph contains a study of the global Cauchy problem for the Yang-Mills equations on $(6+1)$ and higher dimensional Minkowski space, when the initial data sets are small in the critical gauge covariant Sobolev space $\dot{H}_A^{(n-4)/{2}}$. Regularity is obtained through a certain ``microlocal geometric renormalization'' of the equations which is implemented via a family of approximate null Cronstrom gauge transformations. The argument is then reduced to controlling some degenerate elliptic equations in high index and non-isotropic $L^p$ spaces, and also proving some bilinear estimates in specially constructed square-function spaces.

Global and Local Regularity of Fourier Integral Operators on Weighted and Unweighted Spaces
Global and Local Regularity of Fourier Integral Operators on Weighted and Unweighted Spaces

Author: David Dos Santos Ferreira

Publisher: American Mathematical Soc.

Published: 2014-04-07

Total Pages: 86

ISBN-13: 0821891197

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The authors investigate the global continuity on spaces with of Fourier integral operators with smooth and rough amplitudes and/or phase functions subject to certain necessary non-degeneracy conditions. In this context they prove the optimal global boundedness result for Fourier integral operators with non-degenerate phase functions and the most general smooth Hörmander class amplitudes i.e. those in with . They also prove the very first results concerning the continuity of smooth and rough Fourier integral operators on weighted spaces, with and (i.e. the Muckenhoupt weights) for operators with rough and smooth amplitudes and phase functions satisfying a suitable rank condition.

On the Regularity of the Composition of Diffeomorphisms
On the Regularity of the Composition of Diffeomorphisms

Author: H. Inci

Publisher: American Mathematical Soc.

Published: 2013-10-23

Total Pages: 72

ISBN-13: 0821887416

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For M a closed manifold or the Euclidean space Rn we present a detailed proof of regularity properties of the composition of Hs-regular diffeomorphisms of M for s > 12dim⁡M+1.

Near Soliton Evolution for Equivariant Schrodinger Maps in Two Spatial Dimensions
Near Soliton Evolution for Equivariant Schrodinger Maps in Two Spatial Dimensions

Author: Ioan Bejenaru

Publisher: American Mathematical Soc.

Published: 2014-03-05

Total Pages: 120

ISBN-13: 0821892150

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The authors consider the Schrödinger Map equation in 2+1 dimensions, with values into \mathbb{S}^2. This admits a lowest energy steady state Q, namely the stereographic projection, which extends to a two dimensional family of steady states by scaling and rotation. The authors prove that Q is unstable in the energy space \dot H^1. However, in the process of proving this they also show that within the equivariant class Q is stable in a stronger topology X \subset \dot H^1.

Strange Attractors for Periodically Forced Parabolic Equations
Strange Attractors for Periodically Forced Parabolic Equations

Author: Kening Lu

Publisher: American Mathematical Soc.

Published: 2013-06-28

Total Pages: 97

ISBN-13: 0821884840

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The authors prove that in systems undergoing Hopf bifurcations, the effects of periodic forcing can be amplified by the shearing in the system to create sustained chaotic behavior. Specifically, strange attractors with SRB measures are shown to exist. The analysis is carried out for infinite dimensional systems, and the results are applicable to partial differential equations. Application of the general results to a concrete equation, namely the Brusselator, is given.

The Sine-Gordon Equation in the Semiclassical Limit: Dynamics of Fluxon Condensates
The Sine-Gordon Equation in the Semiclassical Limit: Dynamics of Fluxon Condensates

Author: Robert J. Buckingham

Publisher: American Mathematical Soc.

Published: 2013-08-23

Total Pages: 148

ISBN-13: 0821885456

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The authors study the Cauchy problem for the sine-Gordon equation in the semiclassical limit with pure-impulse initial data of sufficient strength to generate both high-frequency rotational motion near the peak of the impulse profile and also high-frequency librational motion in the tails. They show that for small times independent of the semiclassical scaling parameter, both types of motion are accurately described by explicit formulae involving elliptic functions. These formulae demonstrate consistency with predictions of Whitham's formal modulation theory in both the hyperbolic (modulationally stable) and elliptic (modulationally unstable) cases.

Weighted Bergman Spaces Induced by Rapidly Increasing Weights
Weighted Bergman Spaces Induced by Rapidly Increasing Weights

Author: Jose Angel Pelaez

Publisher: American Mathematical Soc.

Published: 2014-01-08

Total Pages: 136

ISBN-13: 0821888021

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This monograph is devoted to the study of the weighted Bergman space $A^p_\omega$ of the unit disc $\mathbb{D}$ that is induced by a radial continuous weight $\omega$ satisfying $\lim_{r\to 1^-}\frac{\int_r^1\omega(s)\,ds}{\omega(r)(1-r)}=\infty.$ Every such $A^p_\omega$ lies between the Hardy space $H^p$ and every classical weighted Bergman space $A^p_\alpha$. Even if it is well known that $H^p$ is the limit of $A^p_\alpha$, as $\alpha\to-1$, in many respects, it is shown that $A^p_\omega$ lies ``closer'' to $H^p$ than any $A^p_\alpha$, and that several finer function-theoretic properties of $A^p_\alpha$ do not carry over to $A^p_\omega$.