On time splitting for NLS in the semiclassical regime

On time splitting for NLS in the semiclassical regime Rémi Carles CNRS & Univ. Montpellier Rémi Carles (Montpellier) Splitting for semiclassical NLS 1 / 21

Splitting for NLS i t u + 1 2 u = f ( u 2) u, t 0, x R d, with u : [0, T ] R d C, and f : R + R. Splitting: solve successively two parts of the equation. 1 ODE: 1 i t u + 2 u = f ( u 2) u. 2 Linear PDE: i t u + 1 2 u = f ( u 2) u 0. Interest: two equations which are easy to solve. Rémi Carles (Montpellier) Splitting for semiclassical NLS 2 / 21

Solving the equations i t u + 1 2 u = f ( u 2) u. 1 ODE: i t u = f ( u 2) u. It is a linear equation! Indeed, t ( u 2 ) = 0 since f : R + R. 2 Linear PDE: i t u + 1 u = 0. 2 Same thing, thanks to Fourier (in space): i t û ξ 2 2 û = 0. explicit formula for the ODE, and FFT for the PDE. Rémi Carles (Montpellier) Splitting for semiclassical NLS 3 / 21

Splitting scheme(s) Denote by X t the linear flow: X t u 0 = u(t), where i t u + 1 2 u = 0 ; u t=0 = u 0, and by Y t the nonlinear flow: Y t u 0 = u(t), where i t u = f ( u 2) u ; u t=0 = u 0. Lie-Trotter: Z t L = Y t X t or Z t L = X t Y t. Strang: Z t S = X t/2 Y t X t/2 or (... ). Higher order... Rémi Carles (Montpellier) Splitting for semiclassical NLS 4 / 21

Convergence of the approximation i t u + 1 2 u = u 2 u ; u t=0 = u 0. Theorem (Besse-Bidégaray-Descombes 02; Lubich 08) Case d 2: for u 0 H 2 (R d ) and all T > 0, C, h 0 such as if t ]0, h 0 ], n N with n t [0, T ], ( Z t L ) n u0 u(n t) L 2 C (m 2, T ) t, with m j = max 0 t T u(t) H j (R d ). If d = 3 and u 0 H 4 (R d ), ( Z t S ) n u0 u(n t) L 2 C (m 4, T ) ( t) 2. Rémi Carles (Montpellier) Splitting for semiclassical NLS 5 / 21

Semiclassical regime iε t u ε + ε2 2 uε = f ( u ε 2) u ε, and ε 0. Initial datum of WKB type: u ε 0 (0, x) = a 0(x)e iφ 0(x)/ε. Conserved quantities: Mass: Energy: d dt uε (t) 2 L = 0. ( 2 d ε u ε (t) 2 L + F ( u ε (t, x) 2) ) dx = 0. dt 2 R d u ε H 1 ε 1. More generally, m j = O(ε j ) (sharp). The splitting error estimates become useless in the limit ε 0. Rémi Carles (Montpellier) Splitting for semiclassical NLS 6 / 21

Hydrodynamics iε t u ε + ε2 2 uε = f ( u ε 2) u ε ; u ε 0(0, x) = a 0 (x)e iφ 0(x)/ε. WKB type approximation: u ε (t, x) a(t, x)e iφ(t,x)/ε. Position density: ρ ε (t, x) = u ε (t, x) 2. Current density: J ε (t, x) = ε Im (u ε (t, x) u ε (t, x)). Formally (justifications exist), ρ ε and J ε converge to: t ρ + div J = 0 ; ρ t=0 = a 0 2, ( ) J J t J + div + ρ f (ρ) = 0 ; J ρ t=0 = a 0 2 φ 0. Identifying terms: ρ = a 2, J = a 2 φ. Rémi Carles (Montpellier) Splitting for semiclassical NLS 7 / 21

Splitting in the semiclassical limit Idea: as long as the solution to the exact equation writes u ε (t, x)=a ε (t, x)e iφε (t,x)/ε, (a ε, φ ε uniformly bounded H s ), then so does the numerical solution obtained by splitting. ODE: iε t u ε = f ( u ε 2 )u ε, u ε t=0 = aε 0e iφε 0 /ε. u ε (t, x) = a0 ε(x)eiφε 0 (x)/ε itf ( aε 0 (x) 2)/ε. Amounts to considering the system: { t φ ε = f ( a ε 2) ; φ ε t=0 = φε 0, t a ε = 0 ; a ε t=0 = aε 0. Rémi Carles (Montpellier) Splitting for semiclassical NLS 8 / 21

Linear PDE: iε t u ε + ε2 2 uε = 0, u t=0 ε = aε 0e iφε 0 /ε. The solution can be written as u ε = a ε e iφε /ε, with t φ ε + 1 2 φε 2 = 0 ; φ ε t=0 = φε 0, t a ε + φ ε a ε + 1 2 aε φ ε = i ε 2 aε ; a t=0 ε = aε 0. The system is decoupled: φ ε solves Burgers. Before singularity formation, solve the first equation (φ ε uniformly bounded H s ), then the second is a linear PDE with bounded coefficients (a ε uniformly bounded H s 2 ). Rémi Carles (Montpellier) Splitting for semiclassical NLS 9 / 21

Outcome The numerical solution is written as a ε e iφε /ε, by solving successively { t φ ε = f ( a ε 2), t a ε = 0. and t φ ε + 1 2 φε 2 = 0, t a ε + φ ε a ε + 1 2 aε φ ε = i ε 2 aε. Amounts to do some splitting on t φ ε + 1 2 φε 2 = f ( a ε 2), t a ε + φ ε a ε + 1 2 aε φ ε = i ε 2 aε. Rémi Carles (Montpellier) Splitting for semiclassical NLS 10 / 21

Déjà vu t φ ε + 1 2 φε 2 = f ( a ε 2), t a ε + φ ε a ε + 1 2 aε φ ε = i ε 2 aε. Without i ε 2 aε, for f (ρ) = +ρ γ 1, symmetrised version of isentropic Euler. With i ε 2 aε, system introduced by Emmanuel Grenier (f > 0). Generalizations: WKB regime for other equations (f 0, Schrödinger-Poisson, f (ρ) = λρ σ ). Rémi Carles (Montpellier) Splitting for semiclassical NLS 11 / 21

Propagating Sobolev regularity: linear PDE t φ ε + 1 2 φε 2 = 0, t a ε + φ ε a ε + 1 2 aε φ ε = i ε 2 aε. The linear PDE has been replaced by a nonlinear system. φ ε solves (multid) Burgers: local resolution, propagation of H s regularity (s > d/2 + 1), tame estimates. If φ ε L ([0, τ]; H s ), s > d/2 + 1, one cannot hope better than a ε L ([0, τ]; H s 1 ). Rémi Carles (Montpellier) Splitting for semiclassical NLS 12 / 21

Propagating Sobolev regularity: ODE { t φ ε = f ( a ε 2), t a ε = 0. If f is local (f (ρ) = ρ γ 1 ), then for a ε L ([0, τ]; H σ ), σ > d/2, φ ε L ([0, τ]; H σ ) and not better: the numerical scheme does not preserve the regularity. The issue of loss of regularity vanishes if a Poisson type nonlinearity is considered. Other way to overcome the loss of regularity (linear equation): work in time dependent analytic regularity (joint work with C. Gallo). Rémi Carles (Montpellier) Splitting for semiclassical NLS 13 / 21

Sobolev regularity Hypothesis f (ρ) = K ρ, where the Fourier transform of K, 1 K(ξ) = (2π) d/2 e ix ξ K(x)dx, R d satisfies: Example If d 2, sup ξ R d (1 + ξ 2 ) K(ξ) < ; If d 3, sup ξ R d ξ 2 K(ξ) <. If d 3, Schrödinger-Poisson: f ( u 2 )u = V p u, where V p = λ u 2, λ R. Rémi Carles (Montpellier) Splitting for semiclassical NLS 14 / 21

The exact solution If for s > d/2 + 1, φ 0 L, ( φ 0, a 0 ) H s+1 H s, then there exists T > 0 such that the system t φ ε + 1 2 φε 2 = f ( a ε 2) ; φ ε 0 = φ 0, t a ε + φ ε a ε + 1 2 aε φ ε = i ε 2 aε ; a0 ε = a 0, has a unique solution (φ ε, a ε ) C([0, T ]; L H s ) such that φ ε C([0, T ]; H s+1 ). In addition, the bounds are uniform in ε ]0, 1]. We can take T = T max δ, where T max is the lifespan of the Euler-Poisson system (if T max < ). WKB form preserved for the exact solution, on [0, T ]. Rémi Carles (Montpellier) Splitting for semiclassical NLS 15 / 21

Main result Theorem There exist ε 0 > 0 and C, c 0 independent of ε ]0, ε 0 ] such that for all t (0, c 0 ], for all n N such that t n = n t [0, T ], we have: 1. There exist φ ε and a ε with sup t [0,T ] ( a ε (t) H s (R d ) + φ ε (t) H s+1 (R d ) + φ ε (t) L (R d ) such that u ε = a ε e iφε /ε on [0, T ] R d. 2. There exist φ ε n and a ε n with a ε n H s (R d ) + φ ε n H s+1 (R d ) + φ ε n L (R d ) C, ) C, such that (Zε t ) n ( a 0 e iφ 0/ε ) = ane ε iφn/ε, and the following error estimate holds: a ε n a ε (t n ) H s 1 + φ ε n φ ε (t n ) H s + φ ε n φ ε (t n ) L C t. Rémi Carles (Montpellier) Splitting for semiclassical NLS 16 / 21

Back to initial unknowns Corollary Under the previous assumptions, with previous notations: (Z t ) n u0 ε S tn ε ε u0 ε L 2 (R d ) C t ε. Main quadratic observables: (Zε t ) n u0 ε 2 L u ε (t n ) 2 C t, 1 (R d ) L (R d ) ( ) Im ε(zε t ) n u0 ε t (Zε ) n u0 ε J ε (t n ) L 1 (R d ) L (R d ) C t. Remark In agreement with numerical experiments by Bao-Jin-Markowich 03. Rémi Carles (Montpellier) Splitting for semiclassical NLS 17 / 21

Scheme of the proof Remark Regularity estimates on the exact solution. (Local) regularity estimates on the numerical scheme. Local error estimate (after Descombes-Thalhammer). From local to global: Lady Windermere s fan and induction (after Holden-Lubich-Risebro). Working in phase/amplitude representation yields L bounds independent of ε ]0, 1], which were not known. Rémi Carles (Montpellier) Splitting for semiclassical NLS 18 / 21

Analytic regularity The phase φ ε and the amplitude a ε belong to H l ρ, l > d/2 + 1, where ψ 2 H l ρ = R d ξ 2l e 2ρ ξ ˆψ(ξ) 2 dξ, with a time dependent weight ρ. Inspired by the analysis of Ginibre & Velo 01. Requirements: ρ(t) δ > 0 on [0, T ], ρ(t) 1. For instance, ρ(t) = M 0 Mt, with M 0, M 1. Advantages: The previous loss of regularity issue disappears, No symmetry needed in the hydrodynamical form (unlike in Grenier s approach). Typically, we can consider f ( u 2 )u = λ u 2σ u, σ N, λ R. Rémi Carles (Montpellier) Splitting for semiclassical NLS 19 / 21

Parabolization of the Euler system The time dependent analytic norm implies the general property ψ 2 H l ρ = R d ξ 2l e 2ρ ξ ˆψ(ξ) 2 dξ d dt ψ 2 H = 2 tψ, ψ ρ l H l ρ + 2 ρ ψ 2. Hρ l+1/2 Last term: as if a parabolic term (of order 1) had been added ( ρ < 0). Implicit dependence of M = ρ in the computations: assume that the initial data a 0 and φ 0 satisfy ( 1+δ â 0 (ξ) 2 + ˆφ 0 (ξ) 2) dξ <, R d e ξ for some δ > 0 (e.g.: Gaussian data, or compact support on Fourier side). Same error estimate as before (in all H s, with T > 0 independent of ε). Rémi Carles (Montpellier) Splitting for semiclassical NLS 20 / 21

Theorem Suppose that d, σ N, d, σ 1, and λ R. Let φ 0, a 0 such that ( 1+δ ˆφ 0 (ξ) 2 + â 0 (ξ) 2) dξ <, for some δ > 0. R d e ξ T, ε 0, c 0 > 0 and (C k ) k N s. t. ε (0, ε 0 ], the following holds: 1.!u ε = Sεu t 0 ε C([0, T ], sh s ). Moreover, there exist φ ε and a ε with ( ) sup t [0,T ] a ε (t) H k (R d ) + φ ε (t) H k (R d ) C k, k N, such that u ε (t, x) = a ε (t, x)e iφε (t,x)/ε for all (t, x) [0, T ] R d. 2. There exist φ ε n and a ε n with a ε n H k (R d ) + φ ε n H k (R d ) C k, k N such that (Zε t ) n ( a 0 e iφ 0/ε ) = ane ε iφn/ε, and: a ε n a ε (t n ) H k + φ ε n φ ε (t n ) H k C k t. Rémi Carles (Montpellier) Splitting for semiclassical NLS 21 / 21

Local error estimate Let A an operator, and E A its propagator: t E A (t, v) = A (E A (t, v)) ; E A (0, v) = v. Theorem (Descombes-Thalhammer) Suppose that F (u) = A(u) + B(u), and denote S t (u) = E F (t, u) and Z t (u) = E B (t, E A (t, u)) the exact and numerical flows, respectively. The exact formula holds Z t (u) S t (u) = t τ1 0 0 2 E F (t τ 1, Z τ 1 (u)) 2 E B (τ 1 τ 2, E A (τ 1, u)) [B, A] (E B (τ 2, E A (τ 1, u))) dτ 2 dτ 1. NB: 2 E = linearized flow. Rémi Carles (Montpellier) Splitting for semiclassical NLS 1 / 5

Standard framework for NLS: A = i ε 2 ; B(v) = i ε f ( v 2) v; F (v) = A(v) + B(v). Linearized exact flow: 2 E F (t, u)w 0 = w, where iε t w + ε2 2 w = f ( u 2) w + f (uw + uw) u; w t=0 = w 0. Drawback: does not preserve the (monokinetic) WKB structure. If u = ae iφ/ε, then iε t w + ε2 2 w = f ( a 2) w + f For w 0 = b 0 e iϕ 0/ε, in general, there does not hold Remark ( ) ae iφ/ε w + ae iφ/ε w ae iφ/ε. w = b ε e iϕε /ε, b ε, ϕ ε uniformly bounded in H s. More simply, a ε ne iφε n /ε a ε e iφε /ε has no reason to be factored α ε ne iϕε n /ε (with uniform bounds). Rémi Carles (Montpellier) Splitting for semiclassical NLS 2 / 5

That is another reason to work on systems: { t φ ε = f ( a ε 2), t a ε = 0. and t φ ε + 1 2 φε 2 = 0, t a ε + φ ε a ε + 1 2 aε φ ε = i ε 2 aε. Rémi Carles (Montpellier) Splitting for semiclassical NLS 3 / 5

Precised framework ( ) ( φ 1 ) A = 2 φ 2 a φ a 1 2 a φ + i ε 2 a, ( ) ( ( φ f a 2 ) ) B =. a 0 Remark Both operators are nonlinear. [A, B] ( ) φ = a ( ( φ f a 2 ) div f ( a 2 φ ) ) ε div f (Im (a a)) a f ( a 2) + 1 2 a f ( a 2). Rémi Carles (Montpellier) Splitting for semiclassical NLS 4 / 5

Local error Theorem (Local error estimate for WKB states) Let s > d/2 + 1 and µ > 0. Suppose that φ ε H s+1 µ, a ε H s µ. There exist C, c 0 > 0 (depending on µ) independent of ε (0, 1] such that ( ( )) ( ) ( ) ( φ ε L t, a ε := Zε t φ ε a ε Sε t φ ε Ψ a ε = ε ) (t) A ε, (t) where A ε and Ψ ε satisfy Ψ ε (t) H s + A ε (t) H s 1 Ct 2, 0 t c 0. Rémi Carles (Montpellier) Splitting for semiclassical NLS 5 / 5