Stochastic mSQG equations with multiplicative transport noises: White noise solutions and scaling limit

Luo, Dejun; Zhu, Rongchan

doi:10.1016/j.spa.2021.06.013

Cited by 5 publications

(4 citation statements)

References 24 publications

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“…On the one hand, it was shown in [17] that, under a suitable scaling of the noise, the white noise solutions of stochastic 2D Euler equations converge weakly to the unique stationary solution of the 2D Navier-Stokes equation driven by space-time white noise; the latter has been studied intensively in the last three decades, cf [1,2,14]. On the other hand, in the more regular L 2 -regime, we proved in [15] that the limit equation is the vorticity form of the deterministic 2D Navier-Stokes equations; see [19] for an earlier work on linear transport equations and [25,26] for similar scaling limits in both regimes on the stochastic modified surface quasi-geostrophic equations. A common feature in the above limit results is that the uniqueness of solutions to approximating nonlinear equations is unknown, while the limit equations are uniquely solvable in suitable sense.…”

Section: Introductionmentioning

confidence: 79%

Convergence of stochastic 2D inviscid Boussinesq equations with transport noise to a deterministic viscous system

Luo

2021

Nonlinearity

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The inviscid 2D Boussinesq system with thermal diffusivity and multiplicative noise of transport type is studied in the L 2-setting. It is shown that, under a suitable scaling of the noise, weak solutions to the stochastic 2D Boussinesq equations converge weakly to the unique solution of the deterministic viscous Boussinesq system. Consequently, the transport noise asymptotically regularises the inviscid 2D Boussinesq system and enhances dissipation in the limit.

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Section: Introductionmentioning

confidence: 79%

Convergence of stochastic 2D inviscid Boussinesq equations with transport noise to a deterministic viscous system

Luo

2021

Nonlinearity

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“…First we recall the similar result on the torus. Recalling the Theorem 1 of [9] (also Theorem 1.1 of [20] by letting θ " 0), the following theorem was proved Theorem 4.1 (Existence). Let ε P p0, 1q.…”

Section: Main Theorem Of the Papermentioning

confidence: 99%

“…Similarly to Euler equations, there are also some results via point-vortex model to approximate mSQG equations, such as [9], [19], [20], [10](for more general models), [11], and [26]. In [9], the point-vortex approximation is used to show the existence of white noise solutions of the weak formulation of mSQG equations on the torus (see Definition 3.1 for the definition of white noise solutions).…”

Section: Introductionmentioning

confidence: 99%

On White Noise Solutions of mSQG Equations on $\mathbb{R}^{2}$

Liang¹

2022

Preprint

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“…Moreover, such a noise also appears in a scaling limit of point vortex approximation and the vorticity form of the 2D Euler equations perturbed by a certain transport type noise (cf. [FL20,FL21,LZ21]). In fact, the scaling limit is given by the vorticity form of the 2D Navier-Stokes system driven by the curl of space-time white noise, which in the velocity-pressure variables reads as 2D Navier-Stokes equations driven by space-time white noise.…”

Section: Introductionmentioning

confidence: 99%

Global existence and non-uniqueness for 3D Navier--Stokes equations with space-time white noise

Hofmanová¹,

Zhu²,

Zhu³

2021

Preprint

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We establish global-in-time existence and non-uniqueness of probabilistically strong solutions to the three dimensional Navier-Stokes system driven by space-time white noise. In this setting, solutions are expected to have space regularity at most −1/2 − κ for any κ > 0. Consequently, the convective term is ill-defined analytically and probabilistic renormalization is required. Up to now, only local well-posedness has been known. With the help of paracontrolled calculus we decompose the system in a way which makes it amenable to convex integration. By a careful analysis of the regularity of each term, we develop an iterative procedure which yields global non-unique probabilistically strong paracontrolled solutions. Our result applies to any divergence free initial condition inand implies also non-uniqueness in law. Contents 1. Introduction 2 1.1. Singular SPDEs 3 1.2. Convex integration 5 1.3. Decomposition 6 1.4. Final remarks 8 2. Preliminaries 9 2.1. Function spaces 9 2.2. Paraproducts, commutators and localizers 10 3. More regular stochastic perturbations 12 3.1. The case of γ ∈ (1/2, 3/2] 13 3.2. The case of γ ∈ (1/6, 1/2] 13 3.3. The case of γ ∈ (0, 1/6] 14 4. Paracontrolled solutions 15 4.1. Stochastic objects 15 Date: December 30, 2021. 2010 Mathematics Subject Classification. 60H15; 35R60; 35Q30. Key words and phrases. stochastic Navier-Stokes equations, probabilistically strong solutions, paracontrolled calculus, non-uniqueness in law, convex integration.This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 949981). The financial support by the DFG through the CRC 1283 "Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications" is greatly acknowledged. R.Z. is grateful to the financial supports of the NSFC (No. 11922103).

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Stochastic mSQG equations with multiplicative transport noises: White noise solutions and scaling limit

Cited by 5 publications

References 24 publications

Convergence of stochastic 2D inviscid Boussinesq equations with transport noise to a deterministic viscous system

Convergence of stochastic 2D inviscid Boussinesq equations with transport noise to a deterministic viscous system

On White Noise Solutions of mSQG Equations on $\mathbb{R}^{2}$

Global existence and non-uniqueness for 3D Navier--Stokes equations with space-time white noise

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