Dynamics of Threshold Solutions for Energy-Critical Wave Equation

Duyckaerts, Thomas; Merle, Frank

doi:10.1093/imrp/rpn002

Cited by 91 publications

(188 citation statements)

References 15 publications

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“…Fix a small ε > 0. As in the proof of Lemma 3.3 of [DM08], using that λ(t)/t → 0 as t → ±∞, we deduce that there exists a constant C 1 , independent of ε, and a time t 1 = t 1 (ε) such that…”

Section: ])mentioning

confidence: 56%

“…The solution u of (1.1) has threshold energy E(W, 0), is not globally defined and satisfies u 0 ∈ L 2 . By Theorem 2 of [DM08], N = 5 and u has to be the special solution W + constructed in this paper, which satisfies u(t) − W Ḣ 1 ≤ e ct as t → −∞. This contradicts the fact that u has compact support in space, concluding Step 1.…”

Section: ])mentioning

confidence: 83%

“…Using compactness and modulation arguments, we get the following control on λ(t) (see Lemma 3.10 in [DM08] and its proof):…”

Section: ])mentioning

confidence: 99%

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Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation

Duyckaerts¹,

Kenig²,

Merle³

2011

J. Eur. Math. Soc.

160

368

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Abstract. Consider the energy-critical focusing wave equation on the Euclidian space. A blow-up type II solution of this equation is a solution which has finite time of existence but stays bounded in the energy space. The aim of this work is to exhibit universal properties of such solutions.Let W be the unique radial positive stationary solution of the equation. Our main result is that in dimension 3, under an appropriate smallness assumption, any type II blow-up radial solution is essentially the sum of a rescaled W concentrating at the origin and a small remainder which is continuous with respect to the time variable in the energy space. This is coherent with the solutions constructed by Krieger, Schlag and Tataru. One ingredient of our proof is that the unique radial solution which is compact up to scaling is equal to W up to symmetries.

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Section: ])mentioning

confidence: 56%

Section: ])mentioning

confidence: 83%

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Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation

Duyckaerts¹,

Kenig²,

Merle³

2011

J. Eur. Math. Soc.

160

368

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“…The following result (see Proposition 5.5 of [DM07b]) shows in particular that −ω 2 is the only negative eigenvalue of L:…”

Section: Convergence To W and W − Near The Thresholdmentioning

confidence: 95%

“…In Section 2, we show that a sequence of solutions (u n ) such that E(u n (0), ∂ t u n (0)) < E(W, 0), |∇u n (0)| 2 < |∇W | 2 and lim n→+∞ u n S(R) = +∞ must converge to W up to modulation for a well-chosen time sequence. This relies on the compactness argument of [KM06b, Section 4], using the profile decomposition of Bahouri-Gérard [BG99], and on the classification of the solutions of (1.1) at the threshold of energy in our previous work [DM07b]. The second step of the proof is an analysis of the behaviour of solutions whose initial conditions are close to (W, 0), which is carried out in Section 3.…”

Section: Introductionmentioning

confidence: 99%

Scattering norm estimate near the threshold for energy-critical focusing semilinear wave equation

Duyckaerts¹,

Merle²

2009

Indiana Univ. Math. J.

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Abstract. We consider the energy-critical semilinear focusing wave equation in dimension N = 3, 4, 5. An explicit solution W of this equation is known. By the work of C. Kenig and F. Merle, any solution of initial condition (u0, u1) such that E(u0, u1) < E(W, 0) and ∇u0 L 2 < ∇W L 2 is defined globally and has finite L 2(N +1) N −2 t,x -norm, which implies that it scatters. In this note, we show that the supremum of the L 2(N +1) N −2 t,x -norm taken on all scattering solutions at a certain level of energy below E(W, 0) blows-up logarithmically as this level approaches the critical value E(W, 0). We also give a similar result in the case of the radial energy-critical focusing semilinear Schrödinger equation. The proofs rely on the compactness argument of C. Kenig and F. Merle, on a classification result, due to the authors, at the energy level E(W, 0), and on the analysis of the linearized equation around W .

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Uniqueness of Two‐Bubble Wave Maps in High Equivariance Classes

Jendrej

Lawrie

2022

Comm Pure Appl Math

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This is the second part of a two-paper series that establishes the uniqueness and regularity of a threshold energy wave map that does not scatter in both time directions.Consider the -valued equivariant energy critical wave maps equation on , with equivariance class . It is known that every topologically trivial wave map with energy less than twice that of the unique -equivariant harmonic map scatters in both time directions. We study maps with precisely the threshold energy E E . In the first part of the series [15] we gave a refined construction of a threshold wave map that asymptotically decouples into a superposition of two harmonic maps (bubbles), one of which is concentrating in scale. In this paper, we show that this solution is the unique (up to the natural invariances of the equation) two-bubble wave map. Combined with our earlier work [14] we obtain an exact description of every threshold wave map.

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Dynamics of Threshold Solutions for Energy-Critical Wave Equation

Cited by 91 publications

References 15 publications

Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation

Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation

Scattering norm estimate near the threshold for energy-critical focusing semilinear wave equation

Uniqueness of Two‐Bubble Wave Maps in High Equivariance Classes

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