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

Röckner and Zhang in [31] proved the existence of a unique strong solution to a stochastic tamed 3D Navier-Stokes equation in the whole space and for the periodic boundary case using a result from [36]. In the latter case, they also proved the existence of an invariant measure. In this paper, we improve their results (but for a slightly simplified system) using a self-contained approach. In particular, we generalise their result about an estimate on the L 4 -norm of the solution from the torus to R 3 , see Lemma 5.1 and thus establish the existence of an invariant measure on R 3 for a time-homogeneous damped tamed 3D Navier-Stokes equation, given by (6.1). (2010). Primary 60H15; Secondary 35R60, 35Q30, 76D05. Mathematics Subject ClassificationKeywords. Stochastic tamed Navier-Stokes equations, stochastic damped Navier-Stokes equations, invariant measures. 1 i.e. the equation is satisfied in the weak sense and the solution u belongs to the space L ∞ (0, T ; H) ∩ L 2 (0, T ; V) ∩ L β+1 (0, T ; L β+1 ). 2 A pair of function (u, p) is a strong solution iff it is a weak solution and u ∈ L ∞ (0, T ; V) ∩ L 2 (0, T ; H 2 ) ∩ L ∞ (0, T ; L β+1 ).

show abstract

“…Proof. The lemma can be proved by following the steps of the proof of Lemma 4.1 [7] for our choice of functional spaces.…”

Section: Compactnessmentioning

confidence: 99%

Stochastic Tamed Navier–Stokes Equations on $${\mathbb {R}}^3$$: The Existence and the Uniqueness of Solutions and the Existence of an Invariant Measure

Brzeźniak

Dhariwal

2020

J. Math. Fluid Mech.

Self Cite

View full text Add to dashboard Cite

show abstract

“…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

View full text Add to dashboard Cite

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.

show abstract

“…In their joint papers [BHW17] and [BHW18] together with Lutz Weis, the first and second named author developed a different approach to the stochastic NLS with Gaussian noise. By complementing the classical Faedo-Galerkin approximation with methods from spectral theory and particularly, a general version of the Littlewood-Paley decomposition, they were able to prove the existence of a martingale solution.…”

mentioning

confidence: 99%

“…Subsequently, the authors concentrated on the special case of 2D manifolds with bounded geometry and 3D compact manifolds and proved pathwise uniqueness using appropriate Strichartz estimates from [BGT04] and [BS14]. For a slight generalization of the existence result from [BHW17] allowing a certain class of non-conservative nonlinear noise, we refer to the PhD thesis [Hor18a] of the second author.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Weak martingale solutions for the stochastic nonlinear Schrödinger equation driven by pure jump noise

Brzeźniak

Hornung

Manna

2019

Stoch PDE: Anal Comp

Self Cite

View full text Add to dashboard Cite

We construct a martingale solution of the stochastic nonlinear Schrödinger equation with a multiplicative noise of jump type in the Marcus canonical form. The problem is formulated in a general framework that covers the subcritical focusing and defocusing stochastic NLS in H 1 on compact manifolds and on bounded domains with various boundary conditions. The proof is based on a variant of the Faedo-Galerkin method. In the formulation of the approximated equations, finite dimensional operators derived from the Littlewood-Paley decomposition complement the classical orthogonal projections to guarantee uniform estimates. Further ingredients of the construction are tightness criteria in certain spaces of càdlàg functions and Jakubowski's generalization of the Skorohod-Theorem to nonmetric spaces.

show abstract

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

Cited by 22 publications

References 53 publications

Stochastic Tamed Navier–Stokes Equations on $${\mathbb {R}}^3$$: The Existence and the Uniqueness of Solutions and the Existence of an Invariant Measure

Stochastic Tamed Navier–Stokes Equations on $${\mathbb {R}}^3$$: The Existence and the Uniqueness of Solutions and the Existence of an Invariant Measure

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

Weak martingale solutions for the stochastic nonlinear Schrödinger equation driven by pure jump noise

Contact Info

Product

Resources

About