Well-posedness of the 3D stochastic primitive equations with multiplicative and transport noise

Brzeźniak, Zdzisław; Slavík, Jakub

doi:10.1016/j.jde.2021.05.049

Cited by 24 publications

(49 citation statements)

References 33 publications

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“…Compared to previous results for the primitive equations with full viscosity and diffusivity [5,12], our approach allows for full transport noise. The key difference to [12] are the pressure estimates in L q (M ) necessary to deduce L q (M )-estimates for the velocity field.…”

Section: Resultsmentioning

confidence: 97%

“…The assumptions on the noise for global existence in [12] require certain smoothing properties that fail for transport noise. The second author with Brzeźniak [5] established global existence for noise allowing transport by the vertical average of the horizontal velocity v. In the case of additive noise, the existence of a random pull-back attractor is known [28]. Logarithmic moment bounds in H 2 (M ) are obtained in [22] and used to prove the existence of ergodic invariant measures supported in H 1 (M ).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Theorem 1.2 is established combining the (deterministic) estimates, the logarithmic Sobolev inequality and logarithmic Gronwall lemma from [8] and an iterated stopping time argument from [5,12]. In [8], one of the key steps is an estimate on the asymptotic behaviour of v L q /q 1/2 w.r.t.…”

Section: Resultsmentioning

confidence: 99%

“…The disadvantage of this approach is that it requires the solution of the Stokes problem to be rather smooth and transport noise cannot be included for this reason. We follow the approach of [5] where the conditions on the noise allow to obtain a random PDE for the pressure by using a hydrostatic Leray-Helmholtz projection. Using the linear structure of the transport part of the noise, we can go beyond the results from [5] and consider transport noise acting not only on the vertical average of the velocity v but also on the remainder v = v − v.…”

Section: Resultsmentioning

confidence: 99%

“…The above argument from [11] carries over to this case almost completely with the following two exceptions -the a.s. convergence of the deterministic nonlinear term and convergence of gradient-dependent stochastic term. The latter has been established in [5,Sectiom 3.4] and therefore it remains to show only the a.s. convergence.…”

Section: T the Filtration Fnmentioning

confidence: 91%

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Stochastic Primitive Equations with Horizontal Viscosity and Diffusivity

Saal¹,

Slavík²

2021

Preprint

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We establish the existence and uniqueness of pathwise strong solutions to the stochastic 3D primitive equations with only horizontal viscosity and diffusivity driven by transport noise on a cylindrical domain M = (−h, 0) × G, G ⊂ R 2 bounded and smooth, with the physical Dirichlet boundary conditions on the lateral part of the boundary. Compared to the deterministic case where the uniqueness of z-weak solutions holds in L 2 , more regular initial data are necessary to establish uniqueness in the anisotropic space H 1 z L 2 xy so that the existence of local pathwise solutions can be deduced from the Gyöngy-Krylov theorem. Global existence is established using the logarithmic Sobolev embedding, the stochastic Gronwall lemma and an iterated stopping time argument.

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Section: Resultsmentioning

confidence: 97%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Section: T the Filtration Fnmentioning

confidence: 91%

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Stochastic Primitive Equations with Horizontal Viscosity and Diffusivity

Saal¹,

Slavík²

2021

Preprint

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Existence and Uniqueness of Maximal Solutions to a 3D Navier-Stokes Equation with Stochastic Lie Transport

Daniel

2022

Mathematics of Planet Earth

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We present here a criterion to conclude that an abstract SPDE possesses a unique maximal strong solution, which we apply to a three dimensional Stochastic Navier-Stokes Equation. Motivated by the work of Kato and Lai we ask that there is a comparable result here in the stochastic case whilst facilitating a variety of noise structures such as additive, multiplicative and transport. In particular our criterion is designed to fit viscous fluid dynamics models with Stochastic Advection by Lie Transport (SALT) as introduced in Holm (Proc R Soc A: Math Phys Eng Sci 471(2176):20140963, 2015). Our application to the Incompressible Navier-Stokes equation matches the existence and uniqueness result of the deterministic theory. This short work summarises the results and announces two papers (Crisan et al., Existence and uniqueness of maximal strong solutions to nonlinear SPDEs with applications to viscous fluid models, in preparation; Crisan and Goodair, Analytical properties of a 3D stochastic Navier-Stokes equation, 2022, in preparation) which give the full details for the abstract well-posedness arguments and application to the Navier-Stokes Equation respectively.

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On the 3D Navier-Stokes Equations with Stochastic Lie Transport

Goodair,

Crisan

2023

Mathematics of Planet Earth

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We prove the existence and uniqueness of maximal solutions to the 3D SALT (Stochastic Advection by Lie Transport) Navier-Stokes Equation in velocity and vorticity form, on the torus and the bounded domain respectively. In particular we demonstrate the efficacy of Goodair et al. (Existence and Uniqueness of Maximal Solutions to SPDEs with Applications to Viscous Fluid Equations, 2023. Stochastics and Partial Differential Equations: Analysis and Computations, pp.1-64) in showing the well-posedness for both the velocity and vorticity form of the equation, as well as obtaining the first analytically strong existence result for a fluid equation perturbed by Lie transport noise on a bounded domain.

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Well-posedness of the 3D stochastic primitive equations with multiplicative and transport noise

Cited by 24 publications

References 33 publications

Stochastic Primitive Equations with Horizontal Viscosity and Diffusivity

Stochastic Primitive Equations with Horizontal Viscosity and Diffusivity

Existence and Uniqueness of Maximal Solutions to a 3D Navier-Stokes Equation with Stochastic Lie Transport

On the 3D Navier-Stokes Equations with Stochastic Lie Transport

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