Probabilistic well-posedness for supercritical wave equations with periodic boundary condition on dimension three

Sun, Chenmin; Xia, Bo

doi:10.1215/ijm/1499760018

Cited by 19 publications

(28 citation statements)

References 16 publications

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“…6 In a recent preprint [35], Killip-Murphy-Vişan proved almost sure global well-posedness and scattering below the energy space for the defocusing energy-critical cubic NLS on R 4 in the radial setting, where the Morawetz estimate (among other tools available in the radial setting) played an important role. 7 While the main result in [49] is stated on the three-dimensional torus T 3 , the same result holds on R 3 by the same proof.…”

Section: 2mentioning

confidence: 67%

On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities

Okamoto

Pocovnicu³

2019

Discrete &Amp; Continuous Dynamical Systems - A

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We consider the Cauchy problem for the nonlinear Schrödinger equations (NLS) with non-algebraic nonlinearities on the Euclidean space. In particular, we study the energy-critical NLS on R d , d = 5, 6, and energy-critical NLS without gauge invariance and prove that they are almost surely locally well-posed with respect to randomized initial data below the energy space. We also study the long time behavior of solutions to these equations: (i) we prove almost sure global well-posedness of the (standard) energy-critical NLS on R d , d = 5, 6, in the defocusing case, and (ii) we present a probabilistic construction of finite time blowup solutions to the energy-critical NLS without gauge invariance below the energy space. Contents 2010 Mathematics Subject Classification. 35Q55. Key words and phrases. nonlinear Schrödinger equation; almost sure local well-posedness; almost sure global well-posedness; finite time blowup. c T γ φ 2 H s such that for each ω ∈ Ω T , there exists a unique solution, where S(t) = e it∆ and X 1 T is defined in Section 3 below. Almost sure local well-posedness with respect to the Wiener randomization has been studied in the context of the cubic NLS and the quintic NLS on R d [2, 3, 8] which are energy-critical in dimensions 4 and 3, respectively. Note that when d = 5, 6, the energycritical nonlinearity |u| 4 d−2 u is no longer algebraic, presenting a new difficulty in applying the argument in [2,3,8].

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Section: 2mentioning

confidence: 67%

On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities

Okamoto

Pocovnicu³

2019

Discrete &Amp; Continuous Dynamical Systems - A

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“…The proof of (1) and (2) in this proposition is standard, see for example [24] or [26]. The proof of (3) follows from the similar argument as in Section 5.…”

Section: Polynomial Nonlinearity In the Case Of A General Randomisationmentioning

confidence: 83%

“…The almost sure boundedness of the linear evolution part is guaranteed by the following lemma, see for example Proposition 2.7 in [26]. Lemma 7.2.…”

Section: Polynomial Nonlinearity In the Case Of A General Randomisationmentioning

confidence: 96%

New examples of probabilistic well-posedness for nonlinear wave equations

Sun

Tzvetkov

2020

Journal of Functional Analysis

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We consider fractional wave equations with exponential or arbitrary polynomial nonlinearities. We prove the global well-posedness on the support of the corresponding Gibbs measures. We provide ill-posedness constructions showing that the results are truly super-critical in the considered functional setting. We also present a result in the case of a general randomisation in the spirit of the work by N. Burq and the second author. 1 d .With v = ∂ t u, we rewrite (1.1) as the following first order system:

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“…This equation is H 1 critical and the data is a typical element with respect to µ ∈ M s , s > 1/2. We refer also to [31,43] for extensions of Theorem 2.3 to nonlinearities between cubic and quintic.…”

Section: Extensions To More General Nonlinearitiesmentioning

confidence: 99%

Random Data Wave Equations

Tzvetkov

2019

Lecture Notes in Mathematics

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Nowadays we have many methods allowing to exploit the regularising properties of the linear part of a nonlinear dispersive equation (such as the KdV equation, the nonlinear wave or the nonlinear Schrödinger equations) in order to prove well-posedness in low regularity Sobolev spaces. By well-posedness in low regularity Sobolev spaces we mean that less regularity than the one imposed by the energy methods is required (the energy methods do not exploit the dispersive properties of the linear part of the equation). In many cases these methods to prove well-posedness in low regularity Sobolev spaces lead to optimal results in terms of the regularity of the initial data. By optimal we mean that if one requires slightly less regularity then the corresponding Cauchy problem becomes ill-posed in the Hadamard sense. We call the Sobolev spaces in which these ill-posedness results hold spaces of supercritical regularity. More recently, methods to prove probabilistic well-posedness in Sobolev spaces of supercritical regularity were developed. More precisely, by probabilistic well-posedness we mean that one endows the corresponding Sobolev space of supercritical regularity with a non degenerate probability measure and then one shows that almost surely with respect to this measure one can define a (unique) global flow. However, in most of the cases when the methods to prove probabilistic well-posedness apply, there is no information about the measure transported by the flow. Very recently, a method to prove that the transported measure is absolutely continuous with respect to the initial measure was developed. In such a situation, we have a measure which is quasi-invariant under the corresponding flow.

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Probabilistic well-posedness for supercritical wave equations with periodic boundary condition on dimension three

Cited by 19 publications

References 16 publications

On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities

On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities

New examples of probabilistic well-posedness for nonlinear wave equations

Random Data Wave Equations

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