Renormalization and blow up for charge one equivariant critical wave maps

Krieger, Joachim; Schlag, Wilhelm; Tataru, Daniel

doi:10.1007/s00222-007-0089-3

Cited by 199 publications

(512 citation statements)

References 25 publications

(9 reference statements)

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“…In fact, a secondary goal here is to show that the method used in both papers is flexible and applies to quite distinct scenarios. The main differences between this paper and [10] are as follows:…”

Section: +1mentioning

confidence: 96%

See 1 more Smart Citation

Slow blow-up solutions for the H1(R3) critical focusing semilinear wave equation

Krieger

Schlag

Tataru³

2009

Duke Math. J.

157

315

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Abstract. Given ν > 1 2 and δ > 0 arbitrary, we prove the existence of energy solutions ofin R 3+1 that blow up exactly at r = t = 0 as t → 0−. These solutions are radial and of the form u = λ(t)2 is the stationary solution of (0.1), and η is a radiation term with ZOutside of the light-cone there is the energy bound Zfor all small t > 0. The regularity of u increases with ν. As in our accompanying paper on wave-maps [10], the argument is based on a renormalization method for the 'soliton profile' W (r).

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Section: +1mentioning

confidence: 96%

“…• In contrast to [10], the linearized operator exhibits negative spectrum. This produces exponential instability of the linearized wave flow.…”

Section: +1mentioning

confidence: 98%

Slow blow-up solutions for the H1(R3) critical focusing semilinear wave equation

Krieger

Schlag

Tataru³

2009

Duke Math. J.

157

315

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“…A main objective in this respect was to prove small data global existence for various target manifolds and space dimensions, see e.g., [32], [15], [14], [41], [42], [37], [38], [16], [29], [43], [21], [22], [23], [24]. On the other hand, for large data and in the energy critical case, there are newer results on blow up, e.g., [34], [20], [26], [25], [4] and, very recently, also on global existence [19], [39], [40], [36]. We will comment below in more detail on some of those works which are most relevant for us.…”

Section: Introductionmentioning

confidence: 99%

“…This beautiful result holds for a large class of targets and, by ruling out the existence of finite energy harmonic maps, it can be used to show global existence. In the case of the two-sphere as a target, there do exist finite energy harmonic maps and indeed, blow up solutions for this model have been constructed in [20], [26], [25].…”

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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“…Recently, an analogous threshold conjecture was established for wave maps (see [Krieger et al 2008;Rodnianski and Sterbenz 2010;Sterbenz and Tataru 2010a;2010b] and, for hyperbolic space, [Krieger and Schlag 2012;Tao 2008a;2008b;2008c;2009a;2009b]). When ᏹ is a hyperbolic space, or, as in [Sterbenz and Tataru 2010a;2010b], a generic compact manifold, we may define the associated energy threshold E crit = E crit (ᏹ) as follows.…”

Section: Introductionmentioning

confidence: 88%

Untitled

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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Renormalization and blow up for charge one equivariant critical wave maps

Cited by 199 publications

References 25 publications

Slow blow-up solutions for the H1(R3) critical focusing semilinear wave equation

Slow blow-up solutions for the H1(R3) critical focusing semilinear wave equation

On stable self-similar blowup for equivariant wave maps

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