On the stochastic nonlinear Schrödinger equation

Transcription

1 On the stochastic nonlinear Schrödinger equation Annie Millet collaboration with Z. Brzezniak SAMM, Paris 1 and PMA Workshop Women in Applied Mathematics, Heraklion - May 3 211

2 Outline 1 The NL Shrödinger equation on R d / on compact manifolds 2 Maximal solution 3 Global solution

3 Outline 1 The NL Shrödinger equation on R d / on compact manifolds 2 Maximal solution 3 Global solution

4 Outline 1 The NL Shrödinger equation on R d / on compact manifolds 2 Maximal solution 3 Global solution

5 The deterministic nonlinear Schrödinger equation in R d Find Z : [, T ] R d C solution to: Z() = z H 1 := H 1,2 i t Z(t) [ Z(t)dt + λ Z(t) 2σ Z(t) ] dt =, Equation modelling the propagation of nonlinear dispersive waves used in many areas in physics, such as: hydrodynamics, plasma physics, non-linear optics, molecular biology the sign of λ describes the focusing/defocusing cases under a small non linearity (sucritical case ) they are robust under large non linearity, they are unstable (blow up occurs)

6 The deterministic nonlinear Schrödinger equation in R d Find Z : [, T ] R d C solution to: Z() = z H 1 := H 1,2 i t Z(t) [ Z(t)dt + λ Z(t) 2σ Z(t) ] dt =, Equation modelling the propagation of nonlinear dispersive waves used in many areas in physics, such as: hydrodynamics, plasma physics, non-linear optics, molecular biology the sign of λ describes the focusing/defocusing cases under a small non linearity (sucritical case ) they are robust under large non linearity, they are unstable (blow up occurs)

7 The deterministic nonlinear Schrödinger equation in R d Find Z : [, T ] R d C solution to: Z() = z H 1 := H 1,2 i t Z(t) [ Z(t)dt + λ Z(t) 2σ Z(t) ] dt =, Equation modelling the propagation of nonlinear dispersive waves used in many areas in physics, such as: hydrodynamics, plasma physics, non-linear optics, molecular biology the sign of λ describes the focusing/defocusing cases under a small non linearity (sucritical case ) they are robust under large non linearity, they are unstable (blow up occurs)

8 The deterministic nonlinear Schrödinger equation in R d Find Z : [, T ] R d C solution to: Z() = z H 1 := H 1,2 i t Z(t) [ Z(t)dt + λ Z(t) 2σ Z(t) ] dt =, Equation modelling the propagation of nonlinear dispersive waves used in many areas in physics, such as: hydrodynamics, plasma physics, non-linear optics, molecular biology the sign of λ describes the focusing/defocusing cases under a small non linearity (sucritical case ) they are robust under large non linearity, they are unstable (blow up occurs)

9 The NL Schrödinger equation on compact manifolds M compact Riemanian manifold of dimension d without boundary Z : [, T ] M C solution to i t Z(t) + Z(t)dt = f (Z(t))dt, Z() = z H 1 := H 1,2 (M) Burq, Gérard and Tzvetkov (24) proved the following: Dimension d = 2 f (z) = F ( z 2 )z for some polynomial F with real coefficients s.t. F (r) + as r + (or F (r) = ar δ with a > or a < and δ < 2) then the NLS equation has a unique solution in C([, T ], H 1 ) If Z H s := H s,2 with s > 1, then Z. C([, T ], H s ) Dimension d = 3 the same result holds for F (r) = r i.e. i t Z(t) + Z(t)dt = Z(t) 2 Z(t)dt

10 The Strichartz inequalities in R d (S(t), t R) group S(t) = exp(it ) acting on L 2 (R d ) (unitary group) p 2, q < be an admissible pair 2 p + d q = d 2 S(.)Z L p (,T ;L q ) C Z L 2 known to be sharp γ, r be another admissible pair 2 γ = d( r γ + 1 γ = 1 and 1 r + 1 r = 1, and φ Lγ (, T ; L r (R d )) S φ(t) = t S(t r)φ(r)dr ( T S φ(t) p L q (R d ) dt ) 1 p ) C(T ) φ L γ (,T ;L r (R d ))

11 The Strichartz inequalities on manifolds Laplace Beltrami operator H s,q := Dom(( q ) s 2 ) for 2 q < and s > shorthand notation H s,2 = H s (S(t), t R) group S(t) = exp(it ) acting on L 2 (M) (unitary group) Strichartz inequalities p 2, q < be an admissible pair 2 p + d q = d 2 Let s ; for Z H s+ 1 p one has S(.)Z L p (,T ;H s,q ) C(T ) Z H s+ 1 p and for φ L 1 (, T, H s+ 1 p ) if S φ(t) = t S(t r)φ(r)dr ( T ) S φ(t) p 1 p H dt s,q T C(T ) φ(t) s+ 1 dt H p

12 Compact Manifold / Full space R d Same constraints on admissible pairs: 2 p + d q = d 2 In R d better Strichartz inequality: no gain/loss of regularity S(.)Z L p (,T ;L q ) C Z L 2 known to be sharp Compact manifold: loss of regularity unavoidable (L p spaces are included in one another) S(.)Z L p (,T ;H s,q ) C(T ) Z H s+ 1 p Is the regularity loss of 1/p sharp? No clear answer in general No on the sphere of dimension d 2, S(.)Z L 4 ([,T ] M) C(T ) Z H s with s > s(d) and s(2) = 1 8, s(d) = d is sharp

13 Key ingredients of the proof (manifold) Fixed point argument Let F be of degree δ, s > d 2 1 max(δ 1,2) and Z H s R For T small enough, equation i t Z(t) + Z(t)dt = f (Z(t))dt, Z() = z H 1 := H 1,2 (M) has a unique solution in C([, T ]; H s ) L p (, T ; L ) Maximal solution There exists T such that the solution to NLS exists in C([, T ); H s ) L p (, T ; L ) and as t T with t < T, the norm converges to Global solutions - Conservation laws f (z) = F ( z 2 )z and F real There exists C > such that for every t [, T ) Z(t, x) 2 dx = Z (x) 2 dx M M Z(t, x) 2 dx + M M F ( Z(t, x) 2 )dx = C

14 Proofs of conservation laws Z 1, Z 2 = Z 1 Z 2 ; F real-valued. First conservation law d t Z(t, x) 2 dx = 2 Re Z(t, x), d t Z(t, x) dx M M = 2 Re ( i Z(t, x), Z(t, x) F ( Z(t, x) 2 )Z(t, x) ) dx = M Second conservation law d t Z(t, x) 2 dx + d t F ( Z(t, x) 2 )dx = M M [ 2 Re Z(t, x), dt Z(t, x) + F ( Z(t, x) 2 )Re Z(t, x), d t Z(t, x) ] dx M = 2 Re Z(t, x) + F ( Z(t, x) 2 Z(t, x), d t Z(t, x) dx M = 2 Re ( i d t Z(t, x), d t Z(t, x) ) dx = M

15 Key ingredients of the proof (continued) d = 2 prove that T = in two cases; second conservation law Case 1 F (r) = ar δ + γ<δ b γr γ with a > Z(t) a Z(t) 2δ 2δ C + γ<δ For 1 γ δ 1; Hölder s inequality yields Z(t) 2γ 2γ ɛ Z(t) 2δ 2δ + C Z 2 2/ɛ b γ Z(t) 2γ 2γ Case 2 F (r) = ar δ with a <, apply the Gagliardo Nirenberg inequality Z(t) 2δ 2δ C Z(t) (δ 1)d 2 Z(t) 2δ (δ 1)d 2 ɛ Z(t) (C/ɛ) Z n(δ) 2 if δ < 2

16 The stochastic problem Add a random perturbation driven by a functional real-valued Wiener process W with diffusion coefficient g(z s ) Try to prove the existence and uniqueness of a solution Z C([, T ], H 1 ) Solved by de Bouard and Debussche on R d and in the particular case f (z) = λ z 2σ z (i.e., F (r) = r σ+1 ) and g(z) = z In their setting W is L 2 (R d )-valued W (t, w, ω) = k φe k (x)β k (t, ω) where β k, k are independent real-valued Brownians, (e k ) is an ONB of L 2 (R d ) and φ is an operator such that φφ is trace class.

17 The stochastic problem (continued) prove existence of a maximal solution by some fixed point argument prove existence of a global solution prove the first (L 2 ) conservation law Problem: Itô s formula gives an extra quadratic variation term. Stratonovich integrals Z(t) dw t allow to prove the first conservation law Z(t) 2 = Z a.s. get a global solution Z C([, T ]; H 1 ) for f (z) = λ z 2σ z if either λ > or σ < 2/d and other assumptions. Rewrite in terms of an Itô integral (drift correction). Hamiltonian similar to that of the second conservation law reduces to the stochastic term de Bouard and Debussche required more assumptions on the noise which is a true Wiener process on H 1,α H 1 for α > 2d

18 The stochastic problem (continued) prove existence of a maximal solution by some fixed point argument prove existence of a global solution prove the first (L 2 ) conservation law Problem: Itô s formula gives an extra quadratic variation term. Stratonovich integrals Z(t) dw t allow to prove the first conservation law Z(t) 2 = Z a.s. get a global solution Z C([, T ]; H 1 ) for f (z) = λ z 2σ z if either λ > or σ < 2/d and other assumptions. Rewrite in terms of an Itô integral (drift correction). Hamiltonian similar to that of the second conservation law reduces to the stochastic term de Bouard and Debussche required more assumptions on the noise which is a true Wiener process on H 1,α H 1 for α > 2d

19 The stochastic problem (continued) prove existence of a maximal solution by some fixed point argument prove existence of a global solution prove the first (L 2 ) conservation law Problem: Itô s formula gives an extra quadratic variation term. Stratonovich integrals Z(t) dw t allow to prove the first conservation law Z(t) 2 = Z a.s. get a global solution Z C([, T ]; H 1 ) for f (z) = λ z 2σ z if either λ > or σ < 2/d and other assumptions. Rewrite in terms of an Itô integral (drift correction). Hamiltonian similar to that of the second conservation law reduces to the stochastic term de Bouard and Debussche required more assumptions on the noise which is a true Wiener process on H 1,α H 1 for α > 2d

20 Stochastic equation Find Z : [, T ] R d C solution to: Z() = z H 1 := H 1,2 i t Z(t) ( Z(t)dt + λ Z(t) 2σ Z(t))dt = Z(t) dw (t), dw Stratonovich integral where W is a space regular infinite dimensional real Brownian Problem solved by A. de Bouard and A. Debussche (23) Theorem Suppose that 1) σ > for d = 1, 2, < σ < 2 for d = 3 and either σ < 1 d 1 or 1 2 σ < 2 d 2 for d 4 2) σ < 2 d or λ < 3) W takes values in H 1,2 H 1,α with α > 2d Then the stochastic NLSE has a unique global solution in C([, T ]; H 1 )

21 The framework for compact manifolds Back to a compact manifold M of dimension d = 2 Given Z() = z H 1 (M); find a maximal solution to i t Z(t) + Z(t)dt = f (Z(t))dt + g(z(t))dw (t) f is polynomial of degree β (previous case β = 2δ 1) 2 p + 2 q = 1, p > max(β 1, 2), s > 1 1 p, ŝ = s 1 p > 1 q W cylindrical Brownian motion on Hilbert space K Let G(Z) = g Z; suppose G : H s Hŝ,q L (2) (K, H s ) where L (2) (K, H s ) denotes the set of Hilbert-Schmidt operators

22 Small time solution Proposition Suppose that G : H s Hŝ,q L (2) (K, H s ) is such that for Z, ζ H s Hŝ,q and some a > : (i) G(Z) L(2) (K,H s ) C(1 + Z a H ) (1 + Z s H ŝ,q) (ii) [ G(Z) G(ζ) L(2) (K,H s ) C Z ζ H ŝ,q 1 + Z a H s + ζ a ] H [ s + C Z ζ H s 1 + Z H ŝ,q + ζ H ŝ,q] [ 1 + Z a 1 H + ζ a 1 ] s H s Then if s > d 2 1 p, for small T there exists a unique (mild) solution in C([, T ]; H s ) L p (, T ; Hŝ,q ) to the stochastic NLS equation dz(t) = i Z(t)dt if (Z(t))dt ig(z(t))dw (t), Z() H s

23 The fixed point argument Let Z H s, S(t) = exp(it ) be the Schrödinger group t t Z(t) = S(t)(Z ) + S(t τ)f (Z(τ))dτ + S(t τ)g(z(τ))dw τ For n 1, set θ n (.) = θ( ṅ ) where θ : [, ) [, 1] be of class C, non increasing function inf x R θ (x) 1, θ(x) = 1 iff x [, 1] and θ(x) = iff x [2, ). Find Z n Y T s.t. Z n (t) = S(t)(Z ) + Φ T,n (Z n )(t) + Ψ T,n (Z n )(t), Φ T,n (Z n )(t) = Ψ T,n (Z n )(t) = t t S(t τ) [ θ n ( Z n Yτ )f (Z n (τ)) ] dτ S(t τ) [ θ n ( Z n Yτ )g(z n (τ)) ] dw (τ)

24 The (deterministic) Strichartz inequality Y t = C([, t]; H s ) L p (, t; Hŝ,q ), ŝ = s 1/p > 2/q for Φ T,n (Z)(t) = t S(t τ)[ θ n ( Z Yτ )f (Z(τ)) ] dτ and T [, T ], Φ T,n (u 2 ) Φ T,n (u 1 ) YT C n (T )T β u 1 u 2 YT with β >. S(t) = e it is unitary on H s ; Strichartz inequality ( ŝ = s 1 p and 2 p + 2 q = 1) S(.)Z L p (,T ;Hŝ,q ) C(T ) Z H s implies ( T ) S φ(t) p 1 p dt Hŝ,q T C(T ) φ(t) H s dt

25 The stochastic Strichartz inequality L p (Y T ) := { ( prog. meas. u : E sup t T Set η(t) = θ n ( Z n Yt )g(z n (t)); then Ψ T,n (Z)(t) = t S(t τ)η(τ)dw (τ) Lemma T )} u(t) p H + u(t) p dt s Hŝ,q For every progressively measurable process η with (ŝ = s 1 p ) E T η(t) p L (2) (K,H s ) dt < and Jη(t) = t S(t τ)η(τ)dw (τ), then T T E Jη(t) p dt C T E η(t) p Hŝ,q L (2) (K,H s ) dt Y = Hŝ,q is NOT Hilbert. Extend stochastic calculus to non Hilbert spaces

26 Stochastic integrals for Radonifyng operators Definition Let K be a Hilbert space, Y a martingale type 2 Banach space; a linear operator L : K Y is Radonifying if for any ONB (e k ) of K and any sequence (β k ) of iid N(,1) random variables, the series k 1 β kle k converges in L 2 P ( Ω; Y) (or P a.s.) and L 2 R(K,Y) := Ẽ k β kke k 2 Y Then if Y = Hŝ,q (or W ŝ,q ) for q [2, ) and if (X t ) is predictable with X L 2 (, T ; R(K, Y)), the stochastic integral t X r dw (r) can be defined as an element of L 2 (, T ; Y) (extended from step processes to L 2 (, T ; R(K, Y)) Results from Dettweiler, Neidhardt, Brzezniak, Peszat, Ondrejat Burkholder-Davies-Gundy inequality ( E sup τ t τ X r dw (r) p Y ) ( t ) p C p (Y)E X r 2 R(K,Y) dr 2

27 Proof of the stochastic Strichartz inequality T T ( E Jξ(t) p t ) p/2 Y dt C p E S(t r)ξ(r) 2 R(K,Y) dr dt T ( C p T p/2 1 T E S(t r)ξ(r) p R(K,Y) )dr dt r T ( C p T p/2 1 T E S(t)ξ(r) p R(K,Y) )dr dt Let Λ R(K, H s ) = L (2) (K, H s ) (e j OBN of K, β j iid N(, 1) on ( Ω, F, P)); Strichartz ineq. (Y = Hŝ,q ) and Kahane-Khintchine ineq. T T S(t)Λ p R(K,Y) dt C Ẽ p β j S(t)Λe j dt Y j CẼ S(.) p β j Λe j CẼ p β j Λe j L j p (,T ;W ŝ,q ) H C Λ p j s R(K,H s )

28 The solution to the truncated equation Recall that Z n (t) = S(t)(Z ) + Φ T,n (Z n )(t) + Ψ T,n (Z n )(t) { ( L p (Y T ) = prog. meas. u : E u(t) p H + s sup t T T )} u(t) p dt Hŝ,q Fix n > ; for T small enough, the map Φ T,n + Ψ T,n is a contraction of L p (Y T ) There exists a stopping time T n T such that for τ [, T n ), the solution Z n is unique in C([, τ], H s ) with some control of the L p (, τ; W ŝ,q ) norm The sequence T n is non decreasing and due to uniqueness, Z n (t) = Z n+1 (t) on [, T n ).

29 Global solutions Theorem Suppose that d = 2, Z H 1 (i) f (z) = F ( z 2 )z such that * either F (r) = ar δ with a >, or a < and δ < 2 * or F polynomial of degree δ with F (r) + as r + (ii) the noise W (t) takes values in H 1,α for α > 2 (iii) g(z) = g( z 2 )z such that g is of class C 3 with sublinear growth condition and decay of derivatives of order 1 to 3 e.g. g(r) = C 1 + C 2 (r + 1) γ with γ [, 1 ] 2 Then for every T > the NLS equation: Z() = Z, idz(t) + Z(t)dt = F ( Z(t) 2 )Z(t)dt + g( Z(t) 2 )Z(t) dw t has a unique solution in C([, T ], H 1 )

30 Sketch of proof Let Z H 1 and set s = 1. Prove that T = lim n T n = T a.s. * Specific form of the diffusion and drift coefficients: f (z) = F ( z 2 )z and g(z) = g( z 2 )z * Stratonovich formulation of the stochastic integral i t Z t + Z t = f (Z t )dt + g(z t ) dw t Choose p, q admissible pair 2 < q < α, 4γ < p For Z(.) L 2 (, T ; H 1 W 1 1 p,q ) prog. meas., Φ H 1,α, set G(Z)Φ = g( Z 2 )Z Φ H 1,2 W 1 1 p,q. G(Z(t)) dw t = G(Z(t))dW t Trace K ( G (Z(t))G(Z(t)) ) dt Trace K (Λ) = H 1,α Λ(x, x)γ(dx) = k Λ(L(e k), L(e k )) L is the radonifying operator from K to H 1,α related with W t Z t = i Z t i F ( Z t 2 )Z t dt i g( Z t 2 )Z t dw t 1 2 g( Z t 2 ) 2 Z t dt

31 Conservation laws Prove a first conservation law : Z(t) 2 L 2 (M) = Z() 2 L 2 (M) a.s. for t < T Use the Itô formula for Z(t τ) L 2 (M) where Z(t) is a mild solution smoothed by Yosida approximation Prove the second conservation law take advantage of the deterministic one and only deal with extra terms from the stochastic integral

32 Conservation laws Prove a first conservation law : Z(t) 2 L 2 (M) = Z() 2 L 2 (M) a.s. for t < T Use the Itô formula for Z(t τ) L 2 (M) where Z(t) is a mild solution smoothed by Yosida approximation Prove the second conservation law take advantage of the deterministic one and only deal with extra terms from the stochastic integral

33 Concluding remarks in R d Our proof of the stochastic Strichartz inequality extends to R d : for s and 2 p + d q = d 2 and E T η(t) p dt <, then L (2) (K,H s ) E T t S(t r)η(r)dw (r) p dt C pt p/2 E H s T η(t) p L (2) (K,H s ) dt This yields the existence and uniqueness of the solution of the NLS in C([, T ], H 1 ) under weaker conditions on the noise W takes values in L 2 (R d ) W 1,α with α > 2 d. Let g be bounded with some growth conditions on the derivatives of order 1 to 3. Then under some constraints on σ depending on d, Z = z H, i t Z(t) ( Z(t)dt + λ Z(t) 2σ Z(t))dt = g( Z(t) 2 )Z(t) dw (t), has a unique solution in C(, T ; H 1 ). Choose an admissible pair (p, q) with d 2 < q < α; solution belongs to L p (, T ; H 1,q )

34 Work in progress Case of compact manifolds of dimension d = 3 deterministic case proved by approximation of elements of H 1 by elements of H s for s > 1 and if F (r) = r 2 Non linearity: z 2 z Case of a Dirac potential large deviations when the noise is multiplied by a small parameter. multiply the Laplace operator by the noise (and no random forcing) studied by A. de Bouard and A. Debussche for R d