On the asymptotic behavior of spherically symmetric wave maps

Christodoulou, Demetrios; Tahvildar-Zadeh, A. Shadi

doi:10.1215/s0012-7094-93-07103-7

Cited by 73 publications

(123 citation statements)

References 1 publication

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“…On the other hand, Theorem 3.2 below gives actually r^ = z/2 for wave-maps on 2 + 1 dimensional Minkowski space-time. We note that the 2 + 1 dimensional wave map equation is related to the vacuum Einstein equations with cylindrical symmetry {cf eg [8,19,20]). …”

Section: Ee^m^' (I-3)mentioning

confidence: 99%

Polyhomogeneous solutions of wave equations in the radiation regime

Chruściel¹,

Lengard²

2000

Journées Équations Aux Dérivées Partielles

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We study the "hyperboloidal Cauchy problem" for linear and semilinear wave equations on Minkowski space-time, with initial data in weighted Sobolev spaces allowing singular behaviour at the boundary, or with polyhomogeneous initial data. Specifically, we consider nonlinear symmetric hyperbolic systems of a form which includes scalar fields with a λφ p nonlinearity, as well as wave maps, with initial data given on a hyperboloid; several of the results proved apply to general space-times admitting conformal completions at null infinity, as well to a large class of equations with a similar non-linearity structure. We prove existence of solutions with controlled asymptotic behaviour, and asymptotic expansions for solutions when the initial data have such expansions. In particular we prove that polyhomogeneous initial data (satisfying compatibility conditions) lead to solutions which are polyhomogeneous at the conformal boundary I + of the Minkowski space-time.

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Section: Ee^m^' (I-3)mentioning

confidence: 99%

Polyhomogeneous solutions of wave equations in the radiation regime

Chruściel¹,

Lengard²

2000

Journées Équations Aux Dérivées Partielles

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“…A considerable amount of the literature is devoted to problems with certain symmetry properties, such as radial or equivariant maps. We mention for instance [6], [7], [27], [30], [33], [35] where various fundamental aspects like local and global well-posedness, asymptotic behavior or blow up are studied. At the end of the 1990s new techniques were developed, capable of treating the full system without symmetry assumptions.…”

Section: Introductionmentioning

confidence: 99%

On stable self-similar blowup for equivariant wave maps

Donninger

2011

Comm. Pure Appl. Math.

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Abstract. We consider co-rotational wave maps from (3 + 1) Minkowski space into the threesphere. This is an energy supercritical model which is known to exhibit finite time blow up via self-similar solutions. The ground state self-similar solution f0 is known in closed form and based on numerics, it is supposed to describe the generic blow up behavior of the system. We prove that the blow up via f0 is stable under the assumption that f0 does not have unstable modes. This condition is equivalent to a spectral assumption for a linear second order ordinary differential operator. In other words, we reduce the problem of stable blow up to a linear ODE spectral problem. Although we are unable, at the moment, to verify the mode stability of f0 rigorously, it is known that possible unstable eigenvalues are confined to a certain compact region in the complex plane. As a consequence, highly reliable numerical techniques can be applied and all available results strongly suggest the nonexistence of unstable modes, i.e., the assumed mode stability of f0.

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“…Hence by (1.9), we have Γ L 4 (R 4 ) ε. Using the Sobolev embeddingḢ 1 (R 4 ) ֒→ L 4 (R 4 ) and the above inequalities we have…”

Section: Linear Dispersive Estimates For Wave Equations On a Curved Bmentioning

confidence: 87%

“…Also in the appendix, Section 9.1, we show that the "covariant" Sobolev spaceṡ H s (M ; N ), with (M, g) = (R 4 , g) with the metric g as in (1.8)-(1.11) are equivalent to the "flat" spacesḢ s ((R 4 , g 0 ); N ) with the Euclidean metric g 0 on R 4 . Hence in what follows we can, when convenient, ignore the non-Euclidean metric g for the purpose of estimating Sobolev norms, replacing covariant derivatives on M with partial derivatives and the volume form dvol g with the Euclidean volume form.…”

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confidence: 87%

The Cauchy problem for wave maps on a curved background

Lawrie

2011

Calc. Var.

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Abstract. We consider the Cauchy problem for wave maps u : R × M → N , for Riemannian manifolds (M, g) and (N, h). We prove global existence and uniqueness for initial data,, where g is a small perturbation of the Euclidean metric. The proof follows the method introduced by Statah and Struwe in [31] for proving global existence and uniqueness of small data wave maps u : R × R d → N in the critical norm, for d ≥ 4. In our argument we employ the Strichartz estimates for variable coefficient wave equations established by Metcalfe and Tataru in [24].

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On the asymptotic behavior of spherically symmetric wave maps

Cited by 73 publications

References 1 publication

Polyhomogeneous solutions of wave equations in the radiation regime

Polyhomogeneous solutions of wave equations in the radiation regime

On stable self-similar blowup for equivariant wave maps

The Cauchy problem for wave maps on a curved background

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