Wave maps on (1+2)-dimensional curved
spacetimes

Gavrus, Cristian; Jao, Casey; Tataru, Daniel

doi:10.2140/apde.2021.14.985

Cited by 2 publications

(2 citation statements)

References 37 publications

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“…Their works exploit the delicate smoothing effects of Schrödinger equations. On the other hand, for wave maps on small perturbations of Euclidean spaces, Lawrie [24] studied small data global theory for high dimensions, and Gavrus-Jao-Tataru [13] established optimal local well-posedness in the energy critical case. We also mention the works of [32,35,36] where we proved asymptotic stability of harmonic maps under the wave map between hyperbolic planes.…”

Section: Introductionmentioning

confidence: 99%

On global dynamics of Schrödinger map flows on hyperbolic planes near harmonic maps

Li¹

2020

Preprint

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The results of this paper are twofold: In the first part, we prove that for Schrödinger map flows from hyperbolic planes to Riemannian surfaces with non-positive sectional curvatures, the harmonic maps which are holomorphic or anti-holomorphic of arbitrary size are asymptotically stable. In the second part, we prove that for Schrödinger map flows from hyperbolic planes into Kähler manifolds, the admissible harmonic maps of small size are asymptotically stable. The asymptotic stability results stated here contain two types: one is the convergence in L ∞x as the previous works, the other is convergence to harmonic maps plus radiation terms in the energy space, which is new in literature of Schrödinger map flows without symmetry assumptions.

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Section: Introductionmentioning

confidence: 99%

On global dynamics of Schrödinger map flows on hyperbolic planes near harmonic maps

Li¹

2020

Preprint

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“…The above stability problem then reduces to the stability problem of these particular solutions. For this reason, we mention particularly the results of global well-posedness of the wave maps [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21] etc. in the critical dimension 2+1.…”

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confidence: 99%

Nonlinear stability of the totally geodesic wave maps in non-isotropic manifolds

Senhao¹,

Ma²,

Zhang³

2022

Preprint

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In this article we investigate a type of totally geodesic map which has its image being a geodesic in an anisotropic Riemannian manifold. We consider its nonlinear stability among the family of wave maps. We first establish the factorization property and then formulate the stability problem into a PDE system in a specially constructed chart of geodesic normal coordinates. With a generalization of the hyperboloidal foliation, we establish the global existence result associate to small initial data for this PDE system, which leads to the geometric stability.

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Wave maps on (1+2)-dimensional curved spacetimes

Cited by 2 publications

References 37 publications

On global dynamics of Schrödinger map flows on hyperbolic planes near harmonic maps

On global dynamics of Schrödinger map flows on hyperbolic planes near harmonic maps

Nonlinear stability of the totally geodesic wave maps in non-isotropic manifolds

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