Almost sure scattering for the 4D energy-critical defocusing nonlinear wave equation with radial data

Dodson, Benjamin; Lührmann, Jonas; Mendelson, Dana

doi:10.1353/ajm.2020.0013

Cited by 35 publications

(61 citation statements)

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“…The first equivalence in (26) is a direct consequence of the definition of the H s x -norm. Now, we prove the bound in (26). From Minkowski's integral inequality, Khintchine's inequality, and Lemma 2.9, we have for all N ě 2 that…”

Section: Harmonic Analysismentioning

confidence: 84%

“…Using the deterministic well-posedness theorem and stability theory, it can be shown (cf. [26,44]) that the solution to (6) exists as long as the energy of v remains bounded. Of course, due to the forcing term in (6), the energy is no longer conserved.…”

Section: #´Bmentioning

confidence: 99%

“…The bound (8), however, is not sufficient to obtain global control on the energy of v, and hence does not prove almost sure scattering. Assuming the regularity condition s ą 1 2 and that the (deterministic) data pf 0 , f 1 q is spherically symmetric, Dodson, Lührmann, and Mendelson proved almost sure scattering in [26]. In their argument, the energy increment is estimated byˇˇż…”

Section: #´Bmentioning

confidence: 99%

“…Under the bootstrap hypothesis, one can then control the second factor in (9) by the square-root of the energy, and this eventually leads to a global bound. The method of [26] has since been used in several related works. In [27], Dodson, Lührmann, and Mendelson used local energy decay to improve the regularity condition to s ą 0.…”

Section: Figure 1: Partions Of Phase Spacementioning

confidence: 99%

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Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation

Bringmann

2020

International Mathematics Research Notices

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We study the defocusing energy-critical nonlinear wave equation in four dimensions. Our main result proves the stability of the scattering mechanism under random pertubations of the initial data. The random pertubation is defined through a microlocal randomization, which is based on a unit-scale decomposition in physical and frequency space. In contrast to the previous literature, we do not require the spherical symmetry of the pertubation. The main novelty lies in a wave packet decomposition of the random linear evolution. Through this decomposition, we can adaptively estimate the interaction between the rough and regular components of the solution. Our argument relies on techniques from restriction theory, such as Bourgain's bush argument and Wolff's induction on scales.Contents MSC2010 : 35L05, 42B20, 42B37

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Section: Harmonic Analysismentioning

confidence: 84%

Section: #´Bmentioning

confidence: 99%

Section: #´Bmentioning

confidence: 99%

Section: Figure 1: Partions Of Phase Spacementioning

confidence: 99%

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Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation

Bringmann

2020

International Mathematics Research Notices

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“…A key difference in this setting, however, is that we will first need to treat an equation for the projection of the solution onto the negative eigenvalue. As in [24,25], we will work with the framework of a generalized forced nonlinear wave equation, but in this case we will only use this framework for the component of the solution projected onto the absolutely continuous subspace of the linearized operator H a , see Section 7 as well as the system of equations (7.8) -(7.10) for more details.…”

Section: Satisfiesmentioning

confidence: 99%

The Focusing Energy-Critical Nonlinear Wave Equation With Random Initial Data

Kenig

Mendelson

2019

International Mathematics Research Notices

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We consider the focusing energy-critical quintic nonlinear wave equation in three dimensional Euclidean space. It is known that this equation admits a one-parameter family of radial stationary solutions, called solitons, which can be viewed as a curve inḢ s, for any s > 1/2. By randomizing radial initial data inḢ sfor s > 5/6, which also satisfy a certain weighted Sobolev condition, we produce with high probability a family of radial perturbations of the soliton which give rise to global forward-in-time solutions of the focusing nonlinear wave equation that scatter after subtracting a dynamically modulated soliton. Our proof relies on a new randomization procedure using distorted Fourier projections associated to the linearized operator around a fixed soliton. To our knowledge, this is the first long-time random data existence result for a focusing wave or dispersive equation on Euclidean space outside the small data regime.1.1. Statement of the main theorem and discussion of methods. We consider the stationary solutions φ a and we make the following ansatz:We take the scaling parameter a(t) to be time dependent, and as remarked in [40], this leads to analysis similar to modulation theory, but distinct since resonances are not associated with L 2 projections. The resonance ϕ a(t) := ∂ a φ a(t)from the definition of P ac , see (3.3) below, we see that the condition in the X s norm is weaker than requiring the same bounds for the function f itself. Moreover, this definition highlights the fact that the Lorentz condition will only be required for the low-frequency component, see (7.4).Let s ∈ R and definefor some ε < ε 0 , where ε 0 , C, c > 0 are determined in Theorem 1.5 below.Remark 1.3. In the case of finite energy initial data, the orthogonality conditionensures that the second variation of the energy restricted to a certain codimension one Lipschitz submanifold is non-negative at φ a , see the discussion following [40, Theorem 1]. Although this does not apply to our (infinite energy) initial data, more concretely, we use this condition to avoid growth when solving an ODE for the pure point component of the solution, see (7.12).

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