The nonlinear stochastic Schrödinger equation via stochastic Strichartz estimates

Hornung, Fabian

doi:10.1007/s00028-018-0433-7

Cited by 39 publications

(32 citation statements)

References 23 publications

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“…Using the additional structure in the special case of the d-dimensional torus M = T d and algebraic nonlinearities, i.e. α = 2k + 1 for some k ∈ N, the authors employed a fixed point argument based on multilinear Strichartz estimates and an estimate of the stochastic convolution in Bourgain spaces X s,b combined with the truncation method from [14], [15] and [20]. As a result, they solved the NLS with multiplicative noise in L 2 (Ω, C([0, τ ], H s (T d )) ∩ X s,b ([0, τ ])) for all s > s crit := d 2 − 2 α−1 and some b < 1 2 as well as some stopping time τ > 0.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Martingale solutions for the stochastic nonlinear Schrödinger equation in the energy space

Brzeźniak

Hornung

Weis

2018

Probab. Theory Relat. Fields

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Martingale solutions for the stochastic nonlinear Schrödinger equation in the energy space

Brzeźniak

Hornung

Weis

2018

Probab. Theory Relat. Fields

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“…For the global well-posedness of (1.1) in the mass-subcritical case (i.e., 1 < α < 1+ 4 d ), see [5,6,18,19,34,36]. See also [15] for the compact manifold setting, and [12,13,14,20,21] for the study of martingale solutions.…”

Section: Introductionmentioning

confidence: 99%

Optimal bilinear control of stochastic nonlinear Schrödinger equations: mass-(sub)critical case

Deng

2020

Probab. Theory Relat. Fields

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We study optimal bilinear control problems for stochastic nonlinear Schrödinger equations in both the mass subcritical and critical case. For general initial data of the minimal L 2 regularity, we prove the existence and first order Lagrange condition of an open loop control. Furthermore, we obtain uniform estimates of (backward) stochastic solutions in new spaces of type U 2 and V 2 , adapted to evolution operators related to linear Schrödinger equations with lower order perturbations. In particular, we obtain a new temporal regularity of rescaled (backward) stochastic solutions, which is the key ingredient in the proof of tightness of approximating controls induced by Ekeland's variational principle.

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“…They proved global existence and uniqueness of mild solutions in (i) L 2 (R) for the one-dimensional cubic SNLS and (ii) H 1 (R d ) for defocusing energysubcritical SNLS. Other works related to SNLS on R d include the works by Barbu, Röckner, and Zhang [1,2] and by Hornung [24].…”

Section: Introductionmentioning

confidence: 99%

Stochastic nonlinear Schrödinger equations on tori

Cheung

Mosincat

2018

Stoch PDE: Anal Comp

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We consider the stochastic nonlinear Schrödinger equations (SNLS) posed on d-dimensional tori with either additive or multiplicative stochastic forcing. In particular, for the one-dimensional cubic SNLS, we prove global well-posedness in L 2 (T). As for other power-type nonlinearities, namely (i) (super)quintic when d = 1 and (ii) (super)cubic when d ≥ 2, we prove local well-posedness in all scaling-subcritical Sobolev spaces and global well-posedness in the energy space for the defocusing, energy-subcritical problems.Date: March 8, 2018. 2010 Mathematics Subject Classification. 60H15. Key words and phrases. stochastic nonlinear Schrödinger equations; well-posedness. 1. The multiplicative noise given by the Stratonovich product u • φξ with real-valued ξ is relevant in physical applications, as it conserves the mass of u (i.e. t → u(t) 2 L 2x (T d ) is constant) almost surely. Our analysis can handle either the Itô or the Stratonovich product, and we choose to work with the former for the sake of simpler exposition.Here, P ≤N is the Littlewood-Paley projection onto frequencies {n ∈ Z d : |n| ≤ N }, p ≥ 2(d+2) d , and ε > 0 is an arbitrarily small quantity 3 . However, such Strichartz estimates are not strong enough for a fixed point argument in mixed Lebesgue spaces for the deterministic NLS on T d . To overcome this problem, we shall employ the Fourier restriction norm method by means of X s,b -spaces defined via the norms(1.7)The indices s, b ∈ R measure the spatial and temporal regularities of functions u ∈ X s,b , and F t,x denotes Fourier transform of functions defined on R × T d . This harmonic analytic 2. Here, W s,r (T d ) denotes the L r -based Sobolev space defined by the Bessel potential norm:, where n := 1 + |n| 2 . When r = 2, we have H s (T d ) = W s,2 (T d ).3. More recently, Killip and Vişan [25] removed the arbitrarily small loss of ε derivatives in (1.6) when p > 2(d+2) d . However, we do not need this scale-invariant improvement in our results.

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The nonlinear stochastic Schrödinger equation via stochastic Strichartz estimates

Cited by 39 publications

References 23 publications

Martingale solutions for the stochastic nonlinear Schrödinger equation in the energy space

Martingale solutions for the stochastic nonlinear Schrödinger equation in the energy space

Optimal bilinear control of stochastic nonlinear Schrödinger equations: mass-(sub)critical case

Stochastic nonlinear Schrödinger equations on tori

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