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

Kenig, Carlos E.; Mendelson, Dana

doi:10.1093/imrn/rnz174

Cited by 9 publications

(5 citation statements)

References 60 publications

(68 reference statements)

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“…They first noticed that the ansatz = (0) + R allows one to put the remainder R in a higher regularity space than the linear term (0) . This idea has since been applied to many different situations (see, e.g., [5,8,11,15,21,33,38]), though most of these works either involve only the first-order expansion (i.e., = 0) or involve higher-order expansions with only suboptimal bounds (e.g., [2]). To the best of our knowledge, the present paper is the first work where the sharp bounds for these ( ) terms are obtained to arbitrarily high order (at least in the dispersive setting).…”

Section: The Scheme and Probabilistic Criticalitymentioning

confidence: 99%

On the derivation of the wave kinetic equation for NLS

Deng

Hani

2021

Forum of Mathematics, Pi

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A fundamental question in wave turbulence theory is to understand how the wave kinetic equation describes the long-time dynamics of its associated nonlinear dispersive equation. Formal derivations in the physics literature, dating back to the work of Peierls in 1928, suggest that such a kinetic description should hold (for well-prepared random data) at a large kinetic time scale $T_{\mathrm {kin}} \gg 1$ and in a limiting regime where the size L of the domain goes to infinity and the strength $\alpha $ of the nonlinearity goes to $0$ (weak nonlinearity). For the cubic nonlinear Schrödinger equation, $T_{\mathrm {kin}}=O\left (\alpha ^{-2}\right )$ and $\alpha $ is related to the conserved mass $\lambda $ of the solution via $\alpha =\lambda ^2 L^{-d}$ . In this paper, we study the rigorous justification of this monumental statement and show that the answer seems to depend on the particular scaling law in which the $(\alpha , L)$ limit is taken, in a spirit similar to how the Boltzmann–Grad scaling law is imposed in the derivation of Boltzmann’s equation. In particular, there appear to be two favourable scaling laws: when $\alpha $ approaches $0$ like $L^{-\varepsilon +}$ or like $L^{-1-\frac {\varepsilon }{2}+}$ (for arbitrary small $\varepsilon $ ), we exhibit the wave kinetic equation up to time scales $O(T_{\mathrm {kin}}L^{-\varepsilon })$ , by showing that the relevant Feynman-diagram expansions converge absolutely (as a sum over paired trees). For the other scaling laws, we justify the onset of the kinetic description at time scales $T_*\ll T_{\mathrm {kin}}$ and identify specific interactions that become very large for times beyond $T_*$ . In particular, the relevant tree expansion diverges absolutely there. In light of those interactions, extending the kinetic description beyond $T_*$ toward $T_{\mathrm {kin}}$ for such scaling laws seems to require new methods and ideas.

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Section: The Scheme and Probabilistic Criticalitymentioning

confidence: 99%

On the derivation of the wave kinetic equation for NLS

Deng

Hani

2021

Forum of Mathematics, Pi

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“…Still, Kenig and Mendelson [37] recently addressed the problem of asymptotic stability of large solitons for the quintic focusing wave equation with randomized radial initial data. They introduced a randomization procedure based on the distorded Fourier transform adapted to the linearized operator around a soliton, and derived some intricate kernel estimates due to the presence of a resonance at zero for this operator.…”

Section: Probabilistic Well-posedness Theorymentioning

confidence: 99%

“…We refer the interested reader to the seminal work of Agmon [1], grounded on the previous works [2] and [35]. In the probabilistic context, we shall follow the strategy used by Kenig and Mendelson for the quintic focusing NLW [37] to provide a randomization procedure based on a distorded frequency decomposition. This randomization procedure commutes with the flow e −itH and is therefore suitable for the underlying linear dynamic of (NLS).…”

Section: Schrödinger Equation With a Short-range Potentialmentioning

confidence: 99%

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Asymptotic stability of small ground states for NLS under random perturbations

Camps¹

2020

Preprint

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We consider the cubic Schrödinger equation on the euclidean space perturbed by a short-range potential V . The presence of a negative simple eigenvalue for −∆ + V gives rise to a curve of small and localized nonlinear ground states that yield some time-periodic solutions known to be asymptotically stable in the energy space. We study the persistence of these coherent states under rough perturbations. We shall construct a large measure set of small scaling-supercritical solutions below the energy space that display some asymptotic stability behavior. The main difficulty is the need to handle the interactions of localized and dispersive terms in the modulation equations. To do so, we use a critical-weighted strategy to combine probabilistic nonlinear estimates in critical spaces based on U p , V q (controlling higher order terms) with some local energy decay estimates (controlling lower order terms). We also revisit in the perturbed setting the analysis of [4] on the probabilistic global well-posedness and scattering for small supercritical initial data. We use a distorded Fourier transform and semiclassical functional calculus to generalize probabilistic and bilinear Strichartz estimates.

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“…Stable manifolds for the critical wave equation in 3 spatial dimensions were constructed in [13] (studied further in [16]), [1], and [15]. Modulated soliton solutions emerging from randomized initial data were studied in [10].…”

Section: Introductionmentioning

confidence: 99%

Global, Non-scattering solutions to the quintic, focusing semilinear wave equation on $\mathbb{R}^{1+3}$

Pillai¹

2021

Preprint

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We consider the quintic, focusing semilinear wave equation on R 1`3 , in the radially symmetric setting, and construct infinite time blow-up, relaxation, and intermediate types of solutions. More precisely, we first define an admissible class of time-dependent length scales, which includes a symbol class of functions. Then, we construct solutions which can be decomposed, for all sufficiently large time, into an Aubin-Talentini (soliton) solution, re-scaled by an admissible length scale, plus radiation (which solves the free 3 dimensional wave equation), plus corrections which decay as time approaches infinity. The solutions include infinite time blow-up and relaxation with rates including, but not limited to, positive and negative powers of time, with exponents sufficiently small in absolute value. We also obtain solutions whose soliton component has oscillatory length scales, including ones which converge to zero along one sequence of times approaching infinity, but which diverge to infinity along another such sequence of times. The method of proof is similar to a recent wave maps work of the author, which is itself inspired by matched asymptotic expansions.

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

Cited by 9 publications

References 60 publications

On the derivation of the wave kinetic equation for NLS

On the derivation of the wave kinetic equation for NLS

Asymptotic stability of small ground states for NLS under random perturbations

Global, Non-scattering solutions to the quintic, focusing semilinear wave equation on $\mathbb{R}^{1+3}$

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