An Approximation Scheme for Schrödinger Maps

McGahagan, H.

doi:10.1080/03605300600856758

Cited by 75 publications

(133 citation statements)

References 17 publications

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“…The use of gauges for equation (1.1) was pioneered in [5], where orthonormal frames were first used in the context of Schrödinger maps. These constructions have been presented by J. Shatah, his students, and collaborators on several occasions (see [32], [33] and the references therein). Unlike in dimensions d ≥ 3, in dimension d = 2 it appears that one cannot use the standard Coulomb gauge, even if the analysis of the Schrödinger equation is combined with elliptic techniques such as the dynamical separation mentioned earlier.…”

Section: ∞ Ementioning

confidence: 99%

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Global Schrödinger maps in dimensions d>=2: Small data in the critical Sobolev spaces

et al. 2011

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Section: ∞ Ementioning

confidence: 99%

“…It is known that sufficiently smooth solutions exist locally in time, even for large data (see, for example, [40], [5], [8], [32] and the references therein). Such theorems for (local in time) smooth solutions are proved using delicate geometric variants of the energy method.…”

Section: Introductionmentioning

confidence: 99%

Global Schrödinger maps in dimensions d>=2: Small data in the critical Sobolev spaces

et al. 2011

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“…Here ⊥ denotes orthogonal projection onto the normal bundle, which, for a given point (x, t), is spanned by u(x, t). For more general analogues of (1-1), e.g., for Kähler targets other than ‫ޓ‬ 2 , see [Ding and Wang 2001;McGahagan 2007;Nahmod et al 2007]. See also [Kenig et al 2000;Kenig and Nahmod 2005;Bejenaru et al 2011b] for connections with other spin systems.…”

Section: Introductionmentioning

confidence: 99%

“…Proof. See [Sulem et al 1986;Chang et al 2000;Ding and Wang 2001;McGahagan 2007] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

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2013

Anal. PDE

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See inside back cover or msp.org/apde for submission instructions.The subscription price for 2013 is US $160/year for the electronic version, and $310/year (+$35, if shipping outside the US) for print and electronic. Subscriptions, requests for back issues from the last three years and changes of subscribers address should be sent to MSP.Analysis & PDE (ISSN 1948(ISSN -206X electronic, 2157 We consider solutions to the linear wave equation on higher dimensional Schwarzschild black hole spacetimes and prove robust nondegenerate energy decay estimates that are in principle required in a nonlinear stability problem. More precisely, it is shown that for solutions to the wave equation g D 0 on the domain of outer communications of the Schwarzschild spacetime manifold .M n m ; g/ (where n 3 is the spatial dimension, and m > 0 is the mass of the black hole) the associated energy flux EOE . † / through a foliation of hypersurfaces † (terminating at future null infinity and to the future of the bifurcation sphere) decays, EOE . † / Ä CD= 2 , where C is a constant depending on n and m, and D < 1 is a suitable higher-order initial energy on † 0 ; moreover we improve the decay rate for the first-order energy to EOE@ t . † R / Ä CD ı = 4 2ı for any ı > 0, where † R denotes the hypersurface † truncated at an arbitrarily large fixed radius R < 1 provided the higher-order energy D ı on † 0 is finite. We conclude our paper by interpolating between these two results to obtain the pointwise estimate j j † R Ä CD 0 ı = 3 2 ı . In this work we follow the new physical-space approach to decay for the wave equation of Dafermos and Rodnianski (2010).

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Asymptotic Stability of Harmonic Maps on the Hyperbolic Plane under the Schrödinger Maps Evolution

Lawrie

Lührmann

et al. 2021

Comm Pure Appl Math

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We consider the Cauchy problem for the Schrödinger maps evolution when the domain is the hyperbolic plane. An interesting feature of this problem compared to the more widely studied case on the Euclidean plane is the existence of a rich new family of finite energy harmonic maps. These are stationary solutions, and thus play an important role in the dynamics of Schrödinger maps. The main result of this article is the asymptotic stability of (some of) such harmonic maps under the Schrödinger maps evolution. More precisely, we prove the nonlinear asymptotic stability of a finite energy equivariant harmonic map Q under the Schrödinger maps evolution with respect to non-equivariant perturbations, provided Q obeys a suitable linearized stability condition. This condition is known to hold for all equivariant harmonic maps with values in the hyperbolic plane and for a subset of those maps taking values in the sphere. One of the main technical ingredients in the paper is a global-in-time local smoothing and Strichartz estimate for the operator obtained by linearization around a harmonic map, proved in the companion paper [36].

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An Approximation Scheme for Schrödinger Maps

Cited by 75 publications

References 17 publications

Global Schrödinger maps in dimensions d>=2: Small data in the critical Sobolev spaces

Global Schrödinger maps in dimensions d>=2: Small data in the critical Sobolev spaces

Untitled

Asymptotic Stability of Harmonic Maps on the Hyperbolic Plane under the Schrödinger Maps Evolution

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