Random data Cauchy theory for supercritical wave equations II: a global existence result

Burq, Nicolas; Tzvetkov, Nikolay

doi:10.1007/s00222-008-0123-0

Cited by 192 publications

(189 citation statements)

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“…Such solution is unique when d = 2. Similar well-posededness results for randomized data were obtained for the supercritical nonlinear Schrödinger equation by Bourgain [1] and for super-critical nonlinear wave equations by Burq and Tzvetkov in [2,3,4]. The approach of Burq and Tzvetkov was applied in the context of the Navier-Stokes in order to obtain local in time solutions to the corresponding integral equation for randomized initial data in L 2 (T 3 ), as well as global in time solutions to the corresponding integral equation for randomized initial data that are small in L 2 (T 3 ) by Zhang and Fang [30] and by Deng and Cui [10].…”

Section: Introductionsupporting

confidence: 78%

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

Nahmod¹,

Pavlović²,

Staffilani³

2013

SIAM J. Math. Anal.

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ABSTRACT. In this paper we show that after suitable data randomization there exists a large set of super-critical periodic initial data, in H −α (T d ) for some α(d) > 0, for both 2d and 3d Navier-Stokes equations for which global energy bounds hold. As a consequence, we obtain almost sure large data super-critical global weak solutions. We also show that in 2d these global weak solutions are unique.

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Section: Introductionsupporting

confidence: 78%

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

Nahmod¹,

Pavlović²,

Staffilani³

2013

SIAM J. Math. Anal.

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“…In particular, they can be seen as a sort of "conservation laws", and therefore can be sometimes used to extend to global times local solutions; what is more, they may be used to prove some results of well-posedness for Cauchy problems with initial data below the critical regularity. In [5] and [6] the authors proved indeed almost sure supercritical well posedness (i.e. existence of a solution for initial data belonging to a subset of full measure of a set of functions of regularity below the scaling critical one) for a class of cubic-nonlinear wave equations on a 3D compact manifold (see also [10] for cubic NLS, [16], [17] for derivative NLS and [18] for quintic NLS).…”

Section: Introduction and Main Resultsmentioning

confidence: 97%

Stability of invariant measures and continuity of the KdV flow

Cacciafesta

2016

Bull Braz Math Soc, New Series

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“…Recently, Richards [42] treated the case of the quartic KdV (p = 5). There have been papers in this direction by Bourgain [4,6,7,8,9] and and other mathematicians that followed his idea [44,45,12,14,33,36,46,31,19,20,21]. In the following, we set β = 1 for simplicity.…”

Section: 1mentioning

confidence: 99%

On the Cameron–Martin theorem and almost-sure global existence

Oh¹,

Quastel²

2015

Proceedings of the Edinburgh Mathematical Society

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Abstract. In this note, we discuss various aspects of invariant measures for nonlinear Hamiltonian PDEs. In particular, we show almost sure global existence for some Hamiltonian PDEs with initial data of the form: "a smooth deterministic function + a rough random perturbation", as a corollary to Cameron-Martin Theorem and known almost sure global existence results with respect to Gaussian measures on spaces of functions. Main results1.1. Introduction. In this note, we discuss almost sure global existence results for some nonlinear Hamiltonian partial differential equations (PDEs) as corollaries to Cameron-Martin Theorem [16]. In particular, we show almost sure global existence with initial data of the form u 0 (x; ω) = v 0 (x) + φ(x; ω),(1.1)where v 0 is a deterministic smooth function and φ(ω) is a random function of low regularity. On T, the function φ is given as where {g n } n∈Z is a sequence of independent standard complex-valued Gaussian random variables on a probability space (Ω, F, P ). For both (1.2) and (1.3), we easily see that φ lies almost surely in H 1 We drop the factor of 2π throughout the paper, when it plays no important role. 2) with α = 0 corresponds to the mean-zero Gaussian white noise on T:Let us describe one of the motivations for studying the Cauchy problems with initial data of the form (1.1), namely a smooth deterministic function + a rough random perturbation. (1.5) Given smooth physical data in an ideal situation, we may introduce rough and random perturbations to these data due to the limitations of accuracy in physical observations and storage of such data. Hence, we believe that it is important to study Cauchy problems with initial data of the form (1.5). Initial data (1.1) with (1.2) or (1.3) are the simplest models for (1.5) with rough Gaussian perturbations. One typical random noise we introduce in this kind of situation is the white noise, which appears ubiquitously in the physics literature. The white noise, however, is very rough and we can handle a smooth initial condition perturbed by the white noise only in a limited case.1.2. Invariant Gibbs measures for Hamiltonian PDEs. Given a Hamiltonian flow on R 2n :with Hamiltonian H(p, q) = H(p 1 , · · · , p n , q 1 , · · · , q n ), Liouville's theorem states that the Lebesgue measure n j=1 dp j dq j on R 2n is invariant under the flow. Then, it follows from the conservation of the Hamiltonian H that the Gibbs measures e −βH(p,q) n j=1 dp j dq j are invariant under the dynamics of (1.6), where β > 0 is the reciprocal temperature.In the context of the nonlinear Schrödinger equations (NLS) on T:with the Hamiltonian: |φx| 2 dx dφ.(1.9)2 Throughout the paper, Z denotes various normalizing constants. ON CAMERON-MARTIN THEOREM AND ALMOST SURE GLOBAL EXISTENCE 3Here, dφ denotes the non-existent Lebesgue measure on the infinite dimensional phase space of functions on T, and thus the expression (1.9) is merely formal at this point.Noting that eφx| 2 dx dφ is the Wiener measure on T with variance β −1 , Lebowitz-Rose-Speer showed that such...

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Random data Cauchy theory for supercritical wave equations II: a global existence result

Cited by 192 publications

References 10 publications

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

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

Stability of invariant measures and continuity of the KdV flow

On the Cameron–Martin theorem and almost-sure global existence

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