Almost sure scattering for the energy critical nonlinear wave equation

Bringmann, Bjoern

doi:10.48550/arxiv.1812.10187

Cited by 8 publications

(39 citation statements)

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“…While the preceding almost sure scattering results all rely on the radial symmetry of the original data, the assumption of radial symmetry for the initial data before randomization was removed in [7] for the defocusing energy-critical wave equation in R 4 .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In fact, all the above references cited in the context of almost sure local and global wellposedness of the energy-critical wave equation and NLS on the full-space except [7] and [8] employ a Wiener randomization, i.e. a randomization based on a unit-scale decomposition of frequency space (see Subsection 1.2 below for details).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Combinations of different randomization techniques were not only used in [9] and [36]. In [7], where almost sure scattering for the defocusing energy-critical wave equation without the radial symmetry assumption was shown, the Wiener randomization was merged with the physical-space randomization to a microlocal randomization.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…For the proof of this result, we follow the same strategy as [21,25,20,7]. Starting from the local wellposedness theory of (1.19), we develop a perturbation theory in the corresponding functional framework, which yields a conditional scattering result: If the energy of v, i.e.…”

Section: Using Polar Coordinates We Develop ĝMmentioning

confidence: 99%

“…Remark 1.3. In [7], almost sure scattering for the cubic nonlinear wave equation in R 4 without radial symmetry was shown using a randomization based on a unit-scale decomposition in physical and frequency space, i.e. (1.17) without the randomization in the angular variable.…”

Section: Using Polar Coordinates We Develop ĝMmentioning

confidence: 99%

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Almost sure local wellposedness and scattering for the energy-critical cubic nonlinear Schrödinger equation with supercritical data

Spitz¹

2021

Preprint

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We study the cubic defocusing nonlinear Schrödinger equation on R 4 with supercritical initial data. For randomized initial data in H s (R 4 ), we prove almost sure local wellposedness for 1 7 < s < 1 and almost sure scattering for 5 7 < s < 1. The randomization is based on a unit-scale decomposition in frequency space, a decomposition in the angular variable, and -for the almost sure scattering result -an additional unit-scale decomposition in physical space. We employ new probabilistic estimates for the linear Schrödinger flow with randomized data, where we effectively combine the advantages of the different decompositions.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Using Polar Coordinates We Develop ĝMmentioning

confidence: 99%

Section: Using Polar Coordinates We Develop ĝMmentioning

confidence: 99%

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Almost sure local wellposedness and scattering for the energy-critical cubic nonlinear Schrödinger equation with supercritical data

Spitz¹

2021

Preprint

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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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Almost Sure Global Well-posedness for the Fourth-order nonlinear Schrodinger Equation with Large Initial Data

Chen,

Zhang

2024

Preprint

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We consider the following fourth-order nonlinear Schr{\”o}dinger equation(4NLS) \begin{align*} (i\partial_t+\varepsilon\Delta+\Delta^2)u=c_1u^m+c_2(\partial u)u^{m-1}+c_3(\partial u)^2u^{m-2}, \end{align*} and establish the conditional almost sure global well-posedness for random initial data in $H^s(\mathbb{R}^d)$ for $s\in (s_c-1/2,\ s_c]$ when $d\geq3$ and $m\geq5$, where $s_c:=d/2-2/(m-1)$ is the scaling critical regularity of 4NLS with the second order derivative nonlinearities. Our proof relies on the nonlinear estimates in a new $M$-norm and the stability theory in the probabilistic setting. Similar supercritical global well-posedness results also hold for $d=2,\ m\geq4$ and $ d\geq3,\ 3\leq m<5$.

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Almost sure scattering for the energy critical nonlinear wave equation

Cited by 8 publications

References 0 publications

Almost sure local wellposedness and scattering for the energy-critical cubic nonlinear Schrödinger equation with supercritical data

Almost sure local wellposedness and scattering for the energy-critical cubic nonlinear Schrödinger equation with supercritical data

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

Almost Sure Global Well-posedness for the Fourth-order nonlinear Schrodinger Equation with Large Initial Data

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