We establish new results for the radial nonlinear wave and Schr\"odinger
equations on the ball in $\Bbb R^2$ and $\Bbb R^3$, for random initial data.
More precisely, a well-defined and unique dynamics is obtained on the support
of the corresponding Gibbs measure. This complements results from
\cite{B-T1,B-T2} and \cite {T1,T2}.Comment: Research announcement, 6 page

Bourgain, Jean

Bulut, Aynur

Comptes Rendus Mathematique

English

arXiv

Jean Bourgain

Aynur Bulut

Crossref

Gibbs measure evolution in radial nonlinear wave and Schrödinger equations on the ball

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 350 (2012) 571–575Contents lists available at SciVerse ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comOrdinary Differential EquationsGibbs measure evolution in radial nonlinear wave and Schrödingerequations on the ball ✩Mesures de Gibbs et équations non-linéaires des ondes et de Schrödinger sur la bouleJean Bourgain, Aynur BulutSchool of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USAa r t i c l e i n f o a b s t r a c tArticle history:Received 6 May 2012Accepted 14 May 2012Available online 8 June 2012Presented by Jean BourgainWe establish new results for the radial nonlinear wave and Schrödinger equations onthe ball in R2 and R3, for random initial data. More precisely, a well-defined andunique dynamics is obtained on the support of the corresponding Gibbs measure. Thiscomplements results from Burq and Tzvetkov (2008) [8,9] and Tzvetkov (2006, 2008) [10,11].© 2012 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éOn démontre des résultats nouveaux sur les solutions radiales de l’équation des ondeset l’équation de Schrödinger sur la boule B dans R2 et R3 pour des conditions initialesaléatoires. Plus exactement, on établit une dynamique bien définie et unique sur le supportde la mesure de Gibbs. Ceci complète des résultats de Burq et Tzvetkov (2008) [8,9] etTzvetkov (2006, 2008) [10,11].© 2012 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Version française abrégéeOn considère les équations non linéaires (radiales et défocalisantes) des ondes (NLW) et Schrödinger (NLS) sur la boule Bdans R2 et R3 :(∂2t − )w + |w|α w = 0 (NLW)(i∂t + )u − |u|αu = 0 (NLS)ainsi que leurs versions tronquées (en introduisant un projecteur P N sur [e1, . . . , eN ], où les en sont les fonctions propres deDirichlet sur B) et les mesures de Gibbs correspondantes. On établit des estimées espace-temps et une dynamique unique,quand N → ∞, dans les modèles (NLW) en dimension 3 pour α < 4 (le cas α < 3 étant traité dans [8,9]), et (NLS) endimension 2, α arbitraire (voir [11] pour le cas α < 4) et en dimension 3 pour α = 2.✩ The research of J.B. was partially supported by NSF grants DMS-0808042 and DMS-0835373 and the research of A.B. was supported by NSF underagreement Nos. DMS-0635607 and DMS-0808042.E-mail addresses: bourgain@math.ias.edu (J. Bourgain), abulut@math.ias.edu (A. Bulut).1631-073X/$ – see front matter © 2012 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.http://dx.doi.org/10.1016/j.crma.2012.05.006572 J. Bourgain, A. Bulut / C. R. Acad. Sci. Paris, Ser. I 350 (2012) 571–5751. The equations and the Gibbs measureDenote B = Bd the unit ball in Rd . We consider the defocusing nonlinear wave (NLW) and nonlinear Schrödinger (NLS)equation{ (∂2t − )w + |w|α w = 0(w, ∂t w)|t=0 = ( f1, f2) (1){(i∂t + )u − |u|αu = 0u|t=0 = φ (2)on the spatial domain B with Dirichlet boundary conditions and with radial initial data. Thus ( f1, f2) is real valued andradial in (1), φ is a radial complex-valued function in (2). It is convenient to rewrite (1) as a first order equation in t ,introducing the complex function u = w + i(√−)−1∂t w . Then (1) turns into the equation{(i∂t −√−)u + (√−)−1(|Re u|α Re u) = 0u|t=0 = φ = f1 + i(√−)−1 f2. (3)Both (2), (3) are Hamiltonian equations taking the respective forms iut = ∂ H∂ ū and iut = (√−)−1 ∂ H∂ ū with HamiltoniansH(φ) =∫B|∇φ|2 + 22 + α∫B|φ|α+2 (4)andH(φ) =∫B|∇φ|2 + 22 + α∫B|Re φ|α+2. (5)Denote {en}n1 the radial Dirichlet eigenfunctions of B andP Nφ =N∑n=1φnenthe projection operator. The ‘truncated’ equations(i∂t + )u − P N(|u|αu) = 0 (6)(i∂t −√−)u − P N((√−)−1(|Re u|α Re u)) = 0 (7)where u(t) = ∑Nn=1 un(t)en are globally well-posed in time and correspond to finite dimensional Hamiltonian models. TheGibbs measureμ(N)G (dφ) = e−H(φ)N∏1d2φ (8)is invariant under their respective flow.2. Statement of the main resultsOur results are the continuation of those obtained in [8,9] and [10,11], as we address various cases that were not treatedin these papers.We consider random initial data given by a Gaussian processφω =N∑n=1gn(ω)nπen (9)with {gn}n1 independent normalized complex Gaussian random variables. The free measure μ(N)F induced by the mapω → φω allows to re-express the Gibbs measure asμ(N)G = e−1α+2∫ |φ|α+2μ(N)F . (10)Thus, the Gibbs measure is a weighted version of the free measure and has the advantage of being preserved under theflow. This fact is crucial in the papers cited above and also in the results discussed here. Note that φω ∈ H 12 −(B) almostJ. Bourgain, A. Bulut / C. R. Acad. Sci. Paris, Ser. I 350 (2012) 571–575 573surely (a.s.). Fixing φ = φω and considering the truncated solutions uNφ = uN , uN |t=0 = P Nφ (which are well-defined globallyin time), there are two natural issues. The first is to establish space–time regularity estimates on uN that are uniform in N .The second is to prove that for N → ∞, the sequence {uN} converges to a unique limit. Of course, these properties are onlyvalid a.s. in ω.Theorem 2.1 (3D NLW). Let α < 4. For almost all ω, the solutions uN of (7), uN |t=0 = P N (φω) satisfysupN∥∥uN(t) − eit√−(P Nφ)∥∥Hsx < ∞ (11)for all s < 5−α2 and t ∈R.Moreover, considering uN as random variables in ω, the sequence {uN} converges in mean in the space Ct<T Hsx for s < 12 , T < ∞arbitrary.The case α < 3 is covered by Theorem 1 in [9].Theorem 2.2 (2D NLS). Let α ∈ 2Z+ be arbitrary and uN the solutions of (6), uN |t=0 = P N (φω). Then the sequence {uN } convergesin the mean in the space Ct<T Hsx for s <12 , T < ∞.The assumption α ∈ 2Z+ is not essential, and more general sufficiently smooth defocusing nonlinearities may be handledas well. The subquintic case was treated in [11].Theorem 2.3 (3D NLS). Let d = 3 and consider Eq. (6) with α = 2. The solutions uN , uN |t=0 = P N (φω) converge in the mean in thespace Ct<T Hsx, s <12 , T < ∞.3. Comments on the proofsAs in the many earlier works, the arguments are a combination of probabilistic and harmonic analysis techniques; seefor instance the classical works [3–5] on this topic and the survey of these results in [6]. We only comment on the proof ofTheorem 2.3, which is by far the most delicate.The starting point is Duhamel’s formula on a fixed time interval [0, T ]uN(t) = u(t) = eit(P Nφ) + it∫0ei(t−τ ) P N(u|u|2)(τ )dτ . (12)The spaces Xs,b = Xs,b([0, T ]) are defined in the usual way, see also [1,2] where they were first introduced. Let s  0 andb  0. For functions f on B3 × [0, T ], admitting a representation of the formf (x, t) =∞∑n=1[ ∞∫−∞fn,λ e2π iλt dλ]en(x) for x ∈ B3, 0  t  T (13)where(∑n∫n2s(1 + ∣∣n2 − λ∣∣2b)| fn,λ|2 dλ) 12< ∞ (14)we define ‖ f ‖s,b as the inf (14) over all representations (13).With these notations, it follows that for 12 < b < 1∥∥∥∥t∫0ei(t−τ ) f (τ )dτ∥∥∥∥s,b C(∑n∫n2s| fn,λ|2(1 + |n2 − λ|2)1−b dλ) 12. (15)The inclusions X0, 14 + ⊂ L3−x L2t and X0+, 12 + ⊂ L3x L4t imply that X0+,b1 ⊂ L3x L43−4b1t for14 < b1 <12 . It follows by duality that for12 < b <34(15) ∥∥(√−)s+ f ∥∥ 3245−4b. (16)Lx Lt574 J. Bourgain, A. Bulut / C. R. Acad. Sci. Paris, Ser. I 350 (2012) 571–575From the Gibbs measure conservation under the flow, one derives the a priori inequality (on finite time intervals)∥∥(√−)su∥∥L px Lqt < C for p < 61 + 2s , q < ∞. (17)Using (12), (16), (17), it follows that‖u‖s,b < C for s < 12 , b <34. (18)Recall that u = uφ , u|t=0 = φ and statements such as (17), (18) require exclusion of small-measure φ-sets. We do notelaborate on the quantitative aspects of these matters here.Next, in order to establish convergence properties for N → ∞, let N  N0 and estimate using (12) and the preceding∥∥uN − uN0∥∥0,b ∥∥∥∥t∫0ei(t−τ )(P N0 uN∣∣P N0 uN ∣∣2 − uN0 ∣∣uN0 ∣∣2)(τ )dτ∥∥∥∥0,b+ N−140 . (19)Denoting u1 = uN0 − P N0 uN and u2, u3 factors uN0 , P N0 uN , the integrand in (19) leads to trilinear expressions of theform∑n,n1,n2,n3[ t∫0û1(τ )(n1)û2(τ )(n2)û3(τ )(n3)e(−n2τ )dτ](∫enen1 en2 en3 dx)ene(n2t)(20)where∣∣∣∣∫enen1 en2 en3∣∣∣∣  C min(n,n1,n2,n3). (21)Our analysis of (20) is based on arguments closely related to those in [4]. We first break up (20) in dyadic regions n ∼ N ,ni ∼ Ni (i = 1,2,3) and distinguish the contributions∣∣n2 − n21 + n22 − n23∣∣  min(N, N1, N2 + N3) 1100 (22)and ∣∣n2 − n21 + n22 − n23∣∣ < min(N, N1, N2 + N3) 1100 . (23)The contribution (22) is handled using Xs,b-spaces and inequalities of the type∫∫v̄u1ū2u3 dx dt  ‖v‖σ1,1−b−‖u1‖σ2, 12 −‖u2‖σ3, 34 −‖u3‖L6−x Lqt (24)with σ1, σ2  0, σ1 + σ2 > 0, σ3 = 12 − or σ1 = σ2 = 0, σ3 > 12 and q > 43−4b .Contributions from (23) are evaluated using further probabilistic considerations, in the spirit of [4] and exploiting therandom nature of u2, u3.The most significant terms are∑n[ t∫0û1(τ )(n)(∑m∣∣û(τ )(m)∣∣2(∫ e2ne2m))e(−n2τ )]ene(n2t). (25)Replacing the inner sum u by the free solution eitφ = ∑ gn(ω)n ene(n2t) leads to an expression of the form∑n<N0logn[ t∫0û1(τ )(n)e(−n2τ )dτ]ene(n2t). (26)In order to obtain a contractive estimate in u1, the presence of the logn factors requires to restrict t ∈ [0, T ], with T ∼ 1log N0 .Moreover, the norm ‖ · ‖0,b has to be slightly weakened to a norm |‖ · |‖0,b by allowing in addition to (13), (14) alsoexpressionsf1(x, t) = ψ(t)[∑bnene(n2t)]nJ. Bourgain, A. Bulut / C. R. Acad. Sci. Paris, Ser. I 350 (2012) 571–575 575where|‖ f1|‖0,b =(‖ψ‖∞ + ‖ψ‖H1/2)(∑|bn|2) 12< ∞.Note that the logarithmic divergency above is barely compatible with the error term in (19).Remark. An alternative approach of interest would be to apply the normal forms approach on finite time intervals (cf. [7])in order to make reductions of the Hamiltonian by suitable symplectic transformations.References[1] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equa-tions, Geom. Funct. Anal. 3 (2) (1993) 107–156.[2] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV equation,Geom. Funct. Anal. 3 (3) (1993) 209–262.[3] J. Bourgain, Periodic nonlinear Schrödinger equation in invariant measures, CMP 166 (1994) 1–26.[4] J. Bourgain, Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, CMP 176 (1996) 421–445.[5] J. Bourgain, Invariant measures for the Gross–Piatevskii equation, J. Math. Pures Appl. (9) 76 (8) (1997) 649–702.[6] J. Bourgain, Nonlinear Schrödinger Equations, IAS/Park City Math. Ser., vol. 5, American Mathematical Society, Providence, RI, 1999.[7] J. Bourgain, A remark on normal forms and the ‘I-method’ for periodic NLS, J. Anal. Math. 94 (2004) 125–157.[8] N. Burq, N. Tzvetkov, Random data Cauchy theory for supercritical wave equations, I (local theory), Invent. Math. 173 (3) (2008) 449–475.[9] N. Burq, N. Tzvetkov, Random data Cauchy theory for supercritical wave equations, II (a global existence result), Invent. Math. 173 (3) (2008) 977–996.[10] N. Tzvetkov, Invariant measures for the nonlinear Schrödinger equation on the disc, Dyn. Partial Differ. Equ. 3 (2006) 111–160.[11] N. Tzvetkov, Invariant measures for the defocusing nonlinear Schrödinger equation, Ann. Inst. Fourier (Grenoble) 58 (7) (2008) 2543–2609.

Comptes Rendus Mathématique

We establish new results for the radial nonlinear wave and Schrödinger equations on the ball in R2 and R3, for random initial data. More precisely, a well-defined and unique dynamics is obtained on the support of the corresponding Gibbs measure. This complements results from Burq and Tzvetkov (2008) [8,9] and Tzvetkov (2006, 2008) [10,11]. © 2012 Académie des sciences

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