Profiles for the radial focusing energy-critical wave equation in odd dimensions

Rodriguez, Casey

doi:10.48550/arxiv.1412.1388

Cited by 8 publications

(14 citation statements)

References 19 publications

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“…To use this to prove Theorem 3.3 for ℓ = 1 and d = 5 see the argument after [11,Proposition 3.2]. For ℓ ≥ 2 we have d ≥ 7 and one can use the more delicate arguments from [5] and the harmonic analysis machinery on exterior domains from [14]; see also [16]. As a heuristic we note that using (2.8) with d = 2ℓ + 3 together with the Strauss estimate applied to the nonlinearity…”

Section: Small Data Theory and Concentration Compactnessmentioning

confidence: 99%

Stable soliton resolution for exterior wave maps in all equivariance classes

Kenig

Lawrie

Liu

et al. 2015

Advances in Mathematics

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In this paper we consider finite energy ℓ-equivariant wave maps from R 1+3t,x \(R × B(0, 1)) → S 3 with a Dirichlet boundary condition at r = 1, and for all ℓ ∈ N. Each such ℓ-equivariant wave map has a fixed integer-valued topological degree, and in each degree class there is a unique harmonic map, which minimizes the energy for maps of the same degree. We prove that an arbitrary ℓ-equivariant exterior wave map with finite energy scatters to the unique harmonic map in its degree class, i.e., soliton resolution. This extends the recent results of the first, second, and fourth authors on the 1-equivariant equation to higher equivariance classes, and thus completely resolves a conjecture of Bizoń, Chmaj and Maliborski, who observed this asymptotic behavior numerically. The proof relies crucially on exterior energy estimates for the free radial wave equation in dimension d = 2ℓ + 3, which are established in the companion paper [10]. * with compact support under the first norm on the right-hand side of (1.2). For

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Section: Small Data Theory and Concentration Compactnessmentioning

confidence: 99%

Stable soliton resolution for exterior wave maps in all equivariance classes

Kenig

Lawrie

Liu

et al. 2015

Advances in Mathematics

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“…The soliton resolution conjecture was proved in [21] by the first, third and fourth authors, for d = 3. For other dimensions (still in the radial case), soliton resolution is only known along a sequence of times, see [8], [67] and [41].…”

Section: Introductionmentioning

confidence: 99%

“…x is not a Strichartz space anymore, and the local Cauchy theory must be modified. These problems are dealt with in [3] (see also [67]), where an appropriate local Cauchy theory, based on the following norms (introduced in [45]):…”

Section: Introductionmentioning

confidence: 99%

Soliton resolution along a sequence of times for the focusing energy critical wave equation

et al. 2017

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In this paper, we prove that any solution of the energy-critical wave equation in space dimensions 3, 4 or 5, which is bounded in the energy space decouples asymptotically, for a sequence of times going to its maximal time of existence, as a finite sum of modulated solitons and a dispersive term. This is an important step towards the full soliton resolution in the nonradial case and without any size restrictions. The proof uses a Morawetz estimate very similar to the one known for energy-critical wave maps, a virial type identity and a new channels of energy argument based on a lower bound of the exterior energy for well-prepared initial data.

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“…Another major problem in the field is the Soliton Resolution Conjecture, which predicts that a bounded (in an appropriate sense) solution decomposes asymptotically into a sum of energy bubbles at different scales and a radiation term (a solution of the linear wave equation). This was proved for the radial energy-critical wave equation in dimension N = 3 by Duyckaerts, Kenig and Merle [15], following the earlier work of the same authors [14], where such a decomposition was proved only for a sequence of times (this last result was generalized to any odd dimension by Rodriguez [35]).…”

mentioning

confidence: 82%

“…Profile decomposition and consequences. For details about the nonlinear profile decomposition for the critical wave equation we refer to [2] (the defocusing case), [13] (dimension N = 3) and [35] (any dimension). For the reader's convenience we recall the following result [35, Proposition 2.3].…”

Section: Coercivity Recall That Fmentioning

confidence: 99%

Construction of two-bubble solutions for energy-critical wave equations

Jendrej¹

2019

American Journal of Mathematics

108

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We construct pure two-bubbles for some energy-critical wave equations, that is solutions which in one time direction approach a superposition of two stationary states both centered at the origin, but asymptotically decoupled in scale. Our solution exists globally, with one bubble at a fixed scale and the other concentrating in infinite time, with an error tending to 0 in the energy space. We treat the cases of the power nonlinearity in space dimension 6, the radial Yang-Mills equation and the equivariant wave map equation with equivariance class k ≥ 3. The concentration speed of the second bubble is exponential for the first two models and a power function in the last case.Remark 1.1. More precisely, we will prove thatRemark 1.2. We construct here pure two-bubbles, that is the solution approaches a superposition of two stationary states, with no energy transformed into radiation. By the conservation of energy and the decoupling of the two bubbles, we necessarily have E(u(t)) = 2E(W ). Pure one-bubbles cannot concentrate and are completely classified, see [16].

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Profiles for the radial focusing energy-critical wave equation in odd dimensions

Cited by 8 publications

References 19 publications

Stable soliton resolution for exterior wave maps in all equivariance classes

Stable soliton resolution for exterior wave maps in all equivariance classes

Soliton resolution along a sequence of times for the focusing energy critical wave equation

Construction of two-bubble solutions for energy-critical wave equations

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