Global Schrödinger maps in dimensions d&gt;=2: Small data in the critical Sobolev spaces

Bejenaru, Ioan; Ionescu, Alexandru D.; Kenig, Carlos E.; Tataru, Daniel

doi:10.4007/annals.2011.173.3.5

Cited by 138 publications

(264 citation statements)

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“…Blow-up dynamics for equivariant critical Schrödinger maps are studied in [Perelman 2012]. Global well-posedness given data with small critical Sobolev norm (in all dimensions d ≥ 2) is shown in [Bejenaru et al 2011c]. Recent work of the author [Smith 2012b] extends the result of Bejenaru et al and the present conditional result to global regularity (in d = 2) assuming small critical Besov normḂ 1 2,∞ .…”

Section: Introductionsupporting

confidence: 48%

“…Theorem 1.1 yields short-time existence and uniqueness as well as a blowup criterion; as such it is central to the continuity arguments used for global results. In the small-energy setting, global regularity (and more) was proved for (1-1) by Bejenaru, Ionescu, Kenig, and Tataru [Bejenaru et al 2011c]. We now state a special case of their main result, omitting for the sake of brevity the consideration of higher spatial dimensions and continuity of the solution map.…”

Section: Introductionmentioning

confidence: 99%

“…The small-energy (take d = 2) theory for (1-1) is now well-understood: building upon previous work (see below or [Bejenaru et al 2011c, §1] for a brief history), global well-posedness and global-in-time bounds on certain Sobolev norms are shown in [Bejenaru et al 2011c] given initial data with sufficiently small energy. The high-energy theory, however, is still very much in development.…”

Section: Introductionmentioning

confidence: 99%

“…For global results in the d = 1 setting, see [Chang et al 2000;]. For d ≥ 3, see [Bejenaru 2008a;2008b;Bejenaru et al 2007;2011c;Ionescu and Kenig 2006;2007b]. Concerning the related modified Schrödinger map system, see [Kato 2005;Kato and Koch 2007;Nahmod et al 2007].…”

Section: Introductionmentioning

confidence: 99%

“…In order to go beyond the small-energy results of [Bejenaru et al 2011c], we introduce physical-space proofs of local smoothing and bilinear Strichartz estimates, in the spirit of [Planchon and Vega 2009;Planchon 2012Planchon , p. 1042Tao 2010], that do not heavily depend upon perturbative methods. The local smoothing estimate that we establish is a nonlinear analogue of that shown in [Ionescu and Kenig 2006].…”

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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Section: Introductionsupporting

confidence: 48%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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2013

Anal. PDE

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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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Global existence of smooth solutions for the incompressible Landau–Lifshitz–Gilbert flow with Dzyaloshinskii–Moriya interaction in two‐dimensional torus and ℝ²

Wang,

Guo

2024

Math Methods in App Sciences

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This paper focuses on establishing the existence of smooth solutions for the incompressible Landau–Lifshitz–Gilbert equation with Dzyaloshinskii–Moriya (DM) interaction and V flow in two‐dimensional domains, including the torus and Euclidean space. The primary objective is to establish global existence results by investigating the conditions under which a smooth solution exists for all time, provided that the norm of the initial magnetization gradient is sufficiently small. Rigorous mathematical proofs for these global existence results are provided by combining analytical techniques and energy estimates. These findings enhance our understanding of solution behavior and regularity in the Landau–Lifshitz–Gilbert equation with DM interaction, shedding light on its dynamics in different spatial domains.

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

Cited by 138 publications

References 49 publications

Untitled

Untitled

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

Global existence of smooth solutions for the incompressible Landau–Lifshitz–Gilbert flow with Dzyaloshinskii–Moriya interaction in two‐dimensional torus and ℝ²

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

Cited by 138 publications

References 49 publications

Untitled

Untitled

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

Global existence of smooth solutions for the incompressible Landau–Lifshitz–Gilbert flow with Dzyaloshinskii–Moriya interaction in two‐dimensional torus and ℝ2

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Global existence of smooth solutions for the incompressible Landau–Lifshitz–Gilbert flow with Dzyaloshinskii–Moriya interaction in two‐dimensional torus and ℝ²