Almost Sure Well-Posedness of Cubic NLS on the Torus Below L 2 J. Colliander University of Toronto Sapporo, 23 November 2009 joint work with Tadahiro Oh (U. Toronto)
1 Introduction: Background, Motivation, New Results 2 Canonical Gaussian Measures 3 Wick Ordered Cubic NLS 4 Almost Sure Local Well-Posedness 5 Almost Sure Global Well-Posedness Bourgain s High-Low Fourier Truncation Adaptation for Almost Sure GWP Proof Measure Zero Issue?
1. Introduction: Background, Motivation, New Results
Cubic Nonlinear Schrödinger Equation Consider the following Cauchy problem: { iu t u xx ± u u 2 = 0 u t=0 = u 0, x T = R/2πZ (NLS ± 3 (T)) with random initial data below L 2 (T). Main Goals: Establish almost sure LWP with initial data u 0 of the form u 0 (x) = u ω 0 (x) = n Z g n (ω) n α einx, = (1 + 2 ) 1 2 {g n } n Z = standard C-valued Gaussians on (Ω, F, P). Extend local-in-time solutions to global-in-time solutions without an available invariant measure.
Well-Posedness vs. Ill-Posedness Threshold Heuristics Dilation Symmetry and Scaling Invariant Sobolev Norm NLS 3 (R): Any solution u spawns a family of solutions u λ (t, x) = 1 λ u( t λ 2, x λ ). D s xu λ (t) L 2 = λ s+ 1 2 = D s x u(t) L 2. The dilation invariant Sobolev index s c = 1 2. We expect NLS 3 (T) is ill-posed for s < 1 2. Galilean Invariance and Galilean Invariant Sobolev Norm The galilean symmetry leaves the L 2 norm invariant. We expect that NLS 3 (T) is ill-posed for s < 0.
Well-Posedness Speculations on NLS ± 3 (T) below L2 Known Results: NLS 3 (T) is globally well-posed in L 2 (T). [B93] Data-solution map H s u 0 u(t) H s not uniformly continuous for s < 0. [BGT02], [CCT03], [M08]. Data-solution map unbounded on H s for s < 1 2. [CCT03] Norm Inflation A priori (local-in-time) bound on u(t) H s (T) AND weak solutions without uniqueness for s 1 6 (CHT09). (Similar prior work on NLS 3 (R) [CCT06], [KT06].) Speculations? 1 No norm inflation for NLS 3 (R) and NLS 3 (T) in H s for s > 1 2? 2 LWP (merely continuous dependence on data) in H s for s > 1 2?
(Finite Dimensional) Invariant Gibbs Measures A function H : R n p R n q R induces Hamiltonian flow on R 2n : { ṗ = q H (Hamilton s Equation) q = p H The vector field X = ( q H, p H) is divergence free: div R 2nX = ( p, q ) ( q H, p H) = 0. Thus, Lebesgue measure n j=1 dp jdq j is invariant under the Hamiltonian flow. (Liouville s Theorem) H(p(t), q(t)) is invariant under the flow: d dt H(p(t), q(t)) = ph ṗ + q H q = 0. (Hamiltonian Conservation)
(Finite Dimensional) Invariant Gibbs Measures We can combine Hamiltonian conservation and Lebesgue measure invariance to build other flow-invariant measures: dµ f (p, q) = f (H(p, q)) n dp j dq j. The Gibbs measure arises when we choose f to be a Gaussian and normalize it to have total measure 1: j=1 n dµ = Z 1 e H(p,q) dp j dq j. The Gibbs measure can be shown to be well-defined in infinite dimensions even though the Lebesgue measure can t be. j=1
Invariant Gibbs Measures for NLS 3 (T) Time Invariant Quantities for NLS ± 3 flow: Mass = u(t, x) 2 dx. T 1 Energy = H[u(t)] = 2 u(t) 2 dx± 1 4 u(t) 4 dx. The Gibbs measure associated to NLS ± 3 (T), dµ = Z 1 e 1 2 T u 2 dx 1 2 ux 2 dx 1 4 u 4 dx x T du(x), (with an appropriate L 2 cutoff in the focusing case) is normalizable and invariant under NLS 3 (T) flow. [B94] Gibbs measure is absolutely cts. w.r.t. Wiener Measure dρ 1 = Z 1 1 e 1 2 1 x u 2 dx x T du(x).
Invariant Gibbs Measures for NLS + 3 (T2 ) LWP for NLS + 3 (T2 ) is known for H 0+ (T 2 ) (not L 2 ).[B93] For T 2, Wiener measure is supported on H 0 (T 2 ) \ L 2 (T 2 ). Nevertheless, the Gibbs measure for the defocusing Wick ordered cubic NLS on T 2 iu t u + (u u 2 2u u 2 dx) = 0 (WNLS(T 2 )) was normalized and proved to be flow-invariant. [B96] WNLS(T 2 ) is GWP on support of Gibbs measure! [B96] Questions: Is NLS + 3 (T2 ) ill-posed on L 2? Or on any space between the Gibbs measure support and H 0+?
Invariant Gibbs Measures for NLS + 3 (T2 ) Commun. Math. Phys. 176, 421-445 (1996) Communications in Mathematical 9 Springer-Verlag 1996 Invariant Measures for the 2D-Defocusing Nonlinear Schrfdinger Equation Jean Bourgain School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA Received: 29 August 1994/in revised form: 23 May 1995
Random Data Cauchy Theory Consider the cubic NLW on T 3 : { u + u 3 = 0 (u(0), t u(0)) = (f 0, f 1 ) H s H s 1. (NLW) This problem is Ḣ 1 2 -critical. Ill-posedness known for s < 1 2. Well-posedness is known for s 1 2. Consider the Gaussian randomization map: Ω H s (ω, f ) f ω (x) = n Z 3 f (n)gn (ω)e in x. NLW is almost surely well-posed for randomized supercritical data (f ω 0, f ω 1 ) Hs H s 1, s 1 4. [BTz]
Random Data Cauchy Theory Invent. math. 173, 449 475 (2008) DOI: 10.1007/s00222-008-0124-z Random data Cauchy theory for supercritical wave equations I: local theory Nicolas Burq 1,2, Nikolay Tzvetkov 3 1 Département de Mathématiques, Université Paris XI, 91 405 Orsay Cedex, France (e-mail: nicolas.burq@math.u-psud.fr) 2 Institut Universitaire de France, Paris, France 3 Département de Mathématiques, Université Lille I, 59 655 Villeneuve d Ascq Cedex, France (e-mail: nikolay.tzvetkov@math.univ-lille1.fr) Oblatum 17-IX-2007 & 19-II-2008 Published online: 4 April 2008 Springer-Verlag 2008
New Results (joint work with Tadahiro Oh) Theorem (Almost Sure Local Well-Posedness for WNLS(T)) WNLS(T) is almost surely locally well-posed (with respect to a canonical Gaussian measure) on H s (T) for s > 1 3. Theorem (Almost Sure Global Well-Posedness for WNLS(T)) WNLS(T) is almost surely globally well-posed (with respect to a canonical Gaussian measure) on H s (T) for s > 1 8.
2. Canonical Gaussian Measures
Canonical Gaussian Measures on Sobolev spaces We can regard u ω 0 (x) = n Z g n (ω) n α einx as a typical element in the support of the Gaussian measure where D = 2 x. dρ α = Z 1 α e 1 2 u 2 dx 1 2 D α u 2 dx x T du(x), We will make sense of dρ α using the Gaussian weight even though the Lebesgue measure does not make sense.
Construction of ρ α Define the Gaussian measure ρ α,n on C 2N+1 by dρ α,n = Z 1 1 N e 2 n N (1+ n 2α ) û n 2 n N dû n. We may view ρ α,n as induced probability measure on C 2N+1 under the map { } gn (ω) ω (1 + n 2α ) 1 2 n N Question: As N, where does the limit ρ α = lim N ρ α,n make sense as a countably additive probability measure?
Gaussian Measures on Hilbert Spaces H, real separable Hilbert space B : H H, linear, positive, self-adjoint operator {e n } n=1, eigenvectors of B forming an O.N. basis of H {λ n } n=1, corresponding eigenvalues where Define ρ N (M) = (2π) N 2 ( N n=1 ) λ 1 2 n e 1 2 F N n=1 λ 1 n x 2 n N dx n n=1 { M = {x H : (x 1,, x N ) F}, F, Borel set in R N x n = x, e n = jth coordinate of x. Thus, ρ N is a Gaussian measure on E N = span{e 1,, e N }. Now, define ρ on H by ρ EN = ρ N.
Gaussian Measures on Hilbert Spaces Fact (1) ρ is countably additive if and only if B is of trace class, i.e. λn < (2) If (1) holds, then ρ N ρ as N Now, let B = diag(1 + n 2σ : n 0). Then, we have 1 2 (1 + n 2α ) û n 2 1 2 (1 + n 2σ )(1 + n 2(σ+α) ) û n 2 n N n N = 1 2 B 1 û n, û n H σ+α. B is of trace class iff n Z n 2σ < iff σ < 1 2. Thus ρ α = lim N ρ α,n defines countably additive measure on H α 1 2 = s<α 1 H s \ H α 1 2. 2
Typical Element as Gaussian Random Fourier Series Typical elements in the support of ρ α may be represented u ω 0 (x) = n Z g n (ω)e n (x), where e n (x) = 1 n α einx Given another O.N. basis {ẽ n } of H α (T), we have u ω 0 = n g n (ω)ẽ n, where { g n } is another family of i.i.d. standard Gaussians. u 0 can be regarded as Gaussian randomization of the function φ with Fourier coefficient φ n = 1 n α. φ H s, s < α 1 2 but not in Hα 1 2. Gaussian randomization of the Fourier coefficients does not give any smoothing a.s. (in the Sobolev scale.) Hence, supp(ρ α ) s<α 1 H s \ H α 1 2. 2
3. Wick Ordered Cubic NLS
WNLS instead of NLS WNLS is equivalent to NLS 3 (T) when u 0 L 2 (T). Note that µ = u 2 dx = 1 2π u 2 dx is conserved. Now set v e ±2iµt u to see that NLS ± 3 Wick ordered cubic NLS: { ivt v xx ± (v v 2 2v v 2 dx) = 0 v t=0 = u 0 = g n(ω) n Z n e inx H α 1 α 2. is equivalent to the WNLS is inequivalent to NLS 3 for u 0 H s with s < 0. Indeed, we can t define the phase µ in this case. (WNLS) We choose to consider WNLS for α < 1 2 (below L2 (T)) instead of NLS 3.
Ill-Posedness of WNLS below L 2 (T) The [BGTz] example u N,a (x, t) = ae i(nx+n2 t a 2 t), with a C and N N, still solves WNLS. Thus, uniform continuity of the flow map for WNLS fails in H s, s < 0. Molinet s ill-posedness result does not apply to WNLS. Local-in-time solutions to (WNLS) in FL p L 2 (T), 2 < p <, by a power series method. [Christ07] Uniqueness is unknown.
4. Almost Sure Local Well-Posedness
Nonlinear smoothing under randomization Duhamel formulation of WNLS: u(t) = Γu(t) := S(t)u 0 ± i t 0 S(t t )N (u)(t )dt where S(t) = e i 2 xt and N (u) = u u 2 2u u 2. S(t)u 0 has the same regularity as u 0 for each fixed t R. i.e. S(t)u ω 0 Hα 1 2 \ H α 1 2 a.s., below L 2 for α 1 2. Main Observation: By nonlinear smoothing, Duhamel term t 0 S(t t )N (u)(t )dt L 2 (T) even when α 1 2. This permits us to run a contraction argument in X s,b.
Contraction around random linear solution Let Γ denote the right hand side of Duhamel s formula. For each small δ > 0, we construct Ω δ Ω with ρ α (Ω C δ ) < e 1 δ c so that Γ defines a contraction on S(t)u ω 0 + B on [ δ, δ] for all ω Ω δ, where B denotes the ball of radius 1 in the Bourgain space X 0, 1 2 +,δ. Recall the Bourgain space X s,b (T R) defined by u X s,b (T R) = n s τ n 2 b û(n, τ) L 2 n L 2 τ (Z R) and its local-in-time version X s,b,δ on the time interval [δ, δ]. For ω Ω δ with P(Ω c δ ) < e 1 δ c, it suffices to prove N (u) X 0, 1 2 +,δ δθ, θ > 0
Theorem (Almost Sure LWP for WNLS) Let α > 1 6. Then, WNLS is LWP almost surely in Hα 1 2 (T). More precisely, there exist c > 0 such that for each δ 1, there exists a set Ω T F with the following properties: (i) P(Ω c T ) = ρ α u 0 (Ω c T ) < e 1 δ c, where u 0 : Ω H α 1 2 (T). (ii) ω Ω T a unique solution u of WNLS in S(t)u ω 0 + C([ δ, δ]; L 2 (T)) C([ δ, δ]; H α 1 2 (T)). In particular, we have almost sure LWP with respect to the Gaussian measure ρ α supported on H s (T) for each s > 1 3.
Invariance of White Noise under WNLS flow? When α = 0, ρ 0 corresponds to the white noise dρ 0 = Z 1 0 e 1 2 u 2 dx x T du(x) which is supported on H 1 2 and is formally invariant. Our results are partial results in this direction. White noise invariance for KdV has been established. [QV], [OQV]
Decompose Nonlinearity: Wick Order Cancellation We write the nonlinearity N (u) as N (u) = u u 2 2u u 2 = N 1 (u, u, u) N 2 (u, u, u) where N 1 (u 1, u 2, u 3 )(x) = n 2 n 1,n 3 û 1 (n 1 )û 2 (n 2 )û 3 (n 3 )e i(n 1 n 2 +n 3 )x N 2 (u 1, u 2, u 3 )(x) = n û1(n)û 2 (n)û 3 (n)e inx. Main Goal: Prove the trilinear estimates (j = 1, 2): N j (u 1, u 2, u 3 ) X 0, 1 2 +,δ δθ, θ > 0.
Random-Linear vs. Deterministic-Smooth We analyze the N 1, N 2 contributions leading to N j (u 1, u 2, u 3 ) X 0, 1 2 +,δ δθ, θ > 0, by assuming u j is one of the following forms: Type I (random, linear, rough): u j (x, t) = n g n (ω) n α ei(nx+n2 t) H α 1 2 Type II (deterministic, smooth): u j with u j X 0, 2 1 +,δ 1. This is done using a case-by-case analysis. (similar to [B96]) Trilinear terms: I, I, I; I, I, II; I, II, II; etc.
Inputs Used in Trilinear Analysis The X s,b machinery, dyadic decomposition, standard stuff. Algebraic identity, divisor estimate: For n = n 1 n 2 + n 3, n 2 (n 2 1 n2 2 + n2 3 ) = 2(n 2 n 1 )(n 2 n 3 ). Strichartz controls on deterministed Type II terms. The periodic L 4 -Strichartz u L 4 x,t u X 0, 3 8. and interpolations with the trivial estimate u L 2 x,t = u X 0,0. Probabilistic estimates on Type I terms. Randomizing Fourier coefficients does not lead to more smoothness but does lead to improved L p properties.
Large Deviation Estimate Lemma Let f ω (x, t) = c n g n (ω)e i(nx+n2 t), where {g n } is a family of complex valued standard i.i.d. Gaussian random variables. Then, for p 2, there exists δ, T 0 > 0 such that for T T 0. P( f ω L p (T [ T,T]) > C c n l 2 n ) < e c T δ This gives good L p control on the Type I terms.
5. Almost Sure Global Well-Posedness We establish almost sure global well-posedness for WNLS by adapting Bourgain s high/low Fourier truncation method. Overview of Discussion: 1 Precise statement of almost sure GWP result. 2 Describe high/low Fourier truncation method for GWP. 3 Explain adaptation to prove almost sure GWP.
Theorem (Almost Sure GWP for WNLS) Let α ( 3 8, 1 2 ]. Then, WNLS is LWP almost surely in Hα 1 2 (T). More precisely, for almost every ω Ω! solution u e it 2 x u ω 0 + C(R; H α 1 2 (T)) of WNLS with initial data given by the random Fourier series u ω 0 (x) = n Z g n (ω) 1 + n α einx. In particular, we have almost sure global well-posedness with respect to the Gaussian measure supported on H s (T), s > 1 8.
2 2 4 4 2 Bourgain s High-Low Fourier Truncation IMRN International Mathematics Research Notices 1998, No. 5 Refinements of Strichartz Inequality and Applications to 2D-NLS with Critical Nonlinearity J. Bourgain Summary Consider the 2D IVP { iut + u + λ u 2 u = 0 u(0) = ϕ L 2 (R 2 ). ( ) The theory on the Cauchy problem asserts a unique maximal solution
Bourgain s High-Low Fourier Truncation Consider the Cauchy problem for defocusing cubic NLS on R 2 : { (i t + )u = + u 2 u u(0, x) = φ 0 (x). (NLS + 3 (R2 )) We describe the first result to give GWP below H 1. NLS + 3 (R2 ) is GWP in H s for s > 2 3 [Bourgain 98]. Proof cuts solution into low and high frequency parts. For u 0 H s, s > 2 3, Proof gives (and crucially exploits), u(t) e it φ 0 H 1 (R 2 x).
Setting up; Decomposing Data Fix a large target time T. Let N = N(T) be large to be determined. Decompose the initial data: φ 0 = φ low + φ high where φ low (x) = Our plan is to evolve: ξ <N e ix ξ φ0 (ξ)dξ. φ 0 = φ low + φ high u(t) = u low (t) + u high (t).
Setting up; Decomposing Data Low Frequency Data Size: Kinetic Energy: φ low 2 L = 2 = ξ <N ξ <N ξ 2 φ 0 (ξ) 2 dx ξ 2(1 s) ξ 2s φ 0 (ξ) 2 dx = N 2(1 s) φ 0 2 H s C 0N 2(1 s). Potential Energy: φ low L 4 x φ low 1/2 φ L 2 low 1/2 L 2 High Frequency Data Size: = H[φ low ] CN 2(1 s). φ high L 2 C 0 N s, φ high H s C 0.
LWP of Low Frequency Evolution along NLS The NLS Cauchy Problem for the low frequency data { (i t + )u low = + u low 2 u low u low (0, x) = φ low (x) is well-posed on [0, T lwp ] with T lwp φ low 2 H 1 N 2(1 s). We obtain, as a consequence of the local theory, that u low L 4 [0,Tlwp ],x 1 100.
LWP of High Frequency Evolution along DE The NLS Cauchy Problem for the low frequency data { (i t + )u high = +2 u low 2 u high + similar + u high 2 u high u high (0, x) = φ high (x) is also well-posed on [0, T lwp ]. Remark: The LWP lifetime of NLS evolution of u low AND the LWP lifetime of the DE evolution of u high are controlled by u low (0) H 1.
Extra Smoothing of Nonlinear Duhamel Term The high frequency evolution may be written u high (t) = e it u high + w. The local theory gives w(t) L 2 N s. Moreover, due to smoothing (obtained via bilinear Strichartz), we have that w H 1, w(t) H 1 N 1 2s+. (SMOOTH!) Let s assume (SMOOTH!).
Nonlinear High Frequency Term Hiding Step! t [0, T lwp ], we have u(t) = u low (t) + e it φ high + w(t). At time T lwp, we define data for the progressive sheme: u(t lwp ) = u low (T lwp ) + w(t lwp ) + e itlwp φ high. for t > T lwp. u(t) = u (2) low (t) + u(2) high (t)
Hamiltonian Increment: φ low (0) u (2) low (T lwp) The Hamiltonian increment due to w(t lwp ) being added to low frequency evolution can be calcluated. Indeed, by Taylor expansion, using the bound (SMOOTH!) and energy conservation of u low evolution, we have using H[u (2) low (T lwp)] = H[u low (0)] + (H[u low (T lwp ) + w(t lwp )] H[u low (T lwp )]) N 2(1 s) + N 2 3s+ N 2(1 s). Moreover, we can accumulate N s increments of size N 2 3s+ before we double the size N 2(1 s) of the Hamiltonian. During the iteration, Hamiltonian of low frequency pieces remains of size N 2(1 s) so the LWP steps are of uniform size N 2(1 s). We advance the solution on a time interval of size: N s N 2(1 s) = N 2+3s. For s > 2 3, we can choose N to go past target time T.
Why did the scheme progress? Along the the time steps T lwp, 2T lwp,..., N s T lwp, the low and high frequency data have uniform properties: High frequency Duhamel term small in H 1. Low frequency data: Hamiltonian Conservation! High frequency data: Linear! For almost sure GWP result, similar scheme progresses: High frequency Duhamel term small in L 2. Low frequency data: Mass Conservation! High frequency data: Linear! = uniform Gaussian probability bounds.
Adaptation for Almost Sure GWP Proof
Adaptation for Almost Sure GWP Proof We are studying the Cauchy problem for WNLS { ivt v xx ± (v v 2 2v v 2 dx) = 0 v t=0 = u 0 = g n(ω) n Z n e inx H α 1 α 2. (WNLS) Let s = α 1 2 < 0. By a large deviation estimate, P( u ω 0 H s > K) e ck2. Restrict to u ω 0 Ω K = {ω : u ω 0 Hs < K}. Eventually, K. Let φ 0 = P ξ N u ω 0 and φ 0 + ψ 0 = u 0. Low frequency part has φ 0 L 2 N s K.
Low Frequency WNLS Evolution Consider the flow of the low frequency part: { i t u 1 2 xu 1 ± N (u 1 ) = 0 u 1 t=0 = φ 0. This problem is GWP with u 1 (t) L 2 = φ 0 L 2 N s K. Standard local theory = spacetime control: u 1 X 0, 1 2 + [0,δ] φ 0 L 2 N s K, where δ is the time of local existence, i.e. δ = δ(n s K) δ( φ 0 L 2).
High Frequency DE evolution Consider the difference equation for the high-frequency part: { i t v 1 2 xv 1 ± (N (u 1 + v 1 ) N (u 1 )) = 0 v 1 t=0 = ψ 0 = n >N g n(ω) 1+ n α e inx. (DE) Then, u(t) = u 1 (t) + v 1 (t) solves WNLS as long as v 1 solves DE. Probabilistic local theory applies to DE. u 1 is large in the X 0, 1 2 +,δ norm; only quadratic in DE.
High/Low Time Step Iteration By the probabilistic local theory, we can show that DE is LWP on [0, δ] except on a set of measure e 1 δ c. We then obtain v 1 (t) = S(t)ψ 0 + w 1 (t), where the nonlinear Duhamel part w 1 (t) L 2 (T). At time t = δ, we hide w 1 (δ) inside φ 1 : φ 1 := u 1 (δ) + w 1 (δ) L 2 ψ 1 := S(δ)ψ 0 = n N g ne in2 δ n α e inx. Low frequency data φ 1 has (essentially) same L 2 size. High frequency data ψ 1 has same bounds as ψ 0.
Measure Zero Issue?
Measure Zero Issue? Almost sure LWP involves set with nontrivial complement. Almost sure GWP claimed almost everywhere w.r.t Ω. Measure theory clarifies this apparent contradiction. Proposition (GWP off small set) Let α > 1 4. Given T > 0, ε > 0 (unlinked!), Ω T,ε F such that: (i) P(Ω c T,ε ) = ρ α u 0 (Ω c T,ε ) < ε, where u 0 : Ω H α 1 2 (T). (ii) ω Ω T,ε unique solution u of WNLS in S(t)u 0 + C([ T, T]; L 2 (T)) C([ T, T]; H α 1 2 (T))
Measure Zero Issue? For fixed γ > 0: Apply Proposition with T j = 2 j and ε j = 2 j γ to get Ω Tj,ε j. Let Ω γ = j=1 Ω T j,ε j. WNLS is globally well-posed on Ω γ with P(Ω c γ) < γ. Now, let Ω = γ>0 Ω γ. WNLS is GWP on Ω and P( Ω c ) = 0.