Almost Sure Existence of Global Weak Solutions for Supercritical Navier--Stokes Equations

Nahmod, Andrea R.; Pavlović, Nataša; Staffilani, Gigliola

doi:10.1137/120882184

Cited by 56 publications

(60 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since µ " N´1 2 #A m ě N´1 2 , the bound on prt 0 , t 0`N sˆR d qz r K N t 0 ,x 0 easily follows from the decay estimate (42). Thus, we now control the contribution on r K N t 0 ,x 0 .…”

Section: Remark 44mentioning

confidence: 99%

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

Bringmann

2020

International Mathematics Research Notices

View full text Add to dashboard Cite

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

show abstract

Section: Remark 44mentioning

confidence: 99%

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

Bringmann

2020

International Mathematics Research Notices

View full text Add to dashboard Cite

show abstract

“…By taking an appropriate union over sets of this type (with T ↓ 0), he obtained local well-posedness almost surely for the Wick-ordered cubic NLS below L 2 (T 2 ). For other works that have used nonlinear smoothing to establish local dynamics in the support of measures on phase space, see (for example) BurqTzvetkov [5,6,7], Oh [26], Colliander-Oh [10], and Nahmod-Pavlović-Staffilani [21].…”

Section: 2mentioning

confidence: 99%

Invariance of the Gibbs measure for the periodic quartic gKdV

Richards

2016

Ann. Inst. H. Poincaré C Anal. Non Linéaire

View full text Add to dashboard Cite

Abstract. We prove invariance of the Gibbs measure for the (gauge transformed) periodic quartic gKdV. The Gibbs measure is supported on H s (T) for s < 1 2 , and the quartic gKdV is analytically ill-posed in this range. In order to consider the flow in the support of the Gibbs measure, we combine a probabilistic argument and the second iteration and construct local-in-time solutions to the (gauge transformed) quartic gKdV almost surely in the support of the Gibbs measure. Then, we use Bourgain's idea to extend these local solutions to global solutions, and prove the invariance of the Gibbs measure under the flow. Finally, Inverting the gauge, we construct almost sure global solutions to the (ungauged) quartic gKdV below H 1 2 (T).

show abstract

“…This approach was initiated by Bourgain [8,9] for the periodic nonlinear Schrödinger equation in one and two space dimensions, building upon the constructions of invariant measures by Glimm-Jaffe [32] and Lebowitz-Rose-Speer [45], and by Burq-Tzvetkov [15,16] in the context of the cubic nonlinear wave equation on a three-dimensional compact Riemannian manifold. There has since been a vast and fascinating body of research, using probabilistic tools to study many nonlinear dispersive or hyperbolic equations in scaling super-critical regimes, see for example [66,25,49,28,17,27,50,51,47,12,7,6,56,29] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation

Dodson

Lührmann

Mendelson

2019

Advances in Mathematics

View full text Add to dashboard Cite

We consider the Cauchy problem for the defocusing cubic nonlinear Schrödinger equation in four space dimensions and establish almost sure local well-posedness and conditional almost sure scattering for random initial data in H s x (R 4 ) with 1 3 < s < 1. The main ingredient in the proofs is the introduction of a functional framework for the study of the associated forced cubic nonlinear Schrödinger equation, which is inspired by certain function spaces used in the study of the Schrödinger maps problem, and is based on Strichartz spaces as well as variants of local smoothing, inhomogeneous local smoothing, and maximal function spaces. Additionally, we prove an almost sure scattering result for randomized radially symmetric initial data in H s x (R 4 ) with 1 2 < s < 1.

show abstract

Almost Sure Existence of Global Weak Solutions for Supercritical Navier--Stokes Equations

Cited by 56 publications

References 27 publications

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

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

Invariance of the Gibbs measure for the periodic quartic gKdV

Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation

Contact Info

Product

Resources

About