Fractionally Dissipative Stochastic Quasi-Geostrophic Type Equations on ${\mathbb{R}}^{d}$

Brzeźniak, Zdzisław; Motyl, Elżbieta

doi:10.1137/17m1111589

Cited by 10 publications

(20 citation statements)

References 28 publications

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“…where N ∈ N. It is clear that {τ N } N ∈N is an increasing sequence. Furthermore, since θ (1) and θ (2) are global martingale solutions, we deduce that lim n→∞ τ N = ∞, P -a.s. Hence, the desired result can be obtained if we prove that for any N, T > 0,…”

Section: Suppose That For All Stopping Timesmentioning

confidence: 68%

“…For the long time behavior of solutions to the 2D stochastic quasi-geostrophic equation, we refer the reader to [17,33]. Very recently, the existence of martingale solutions to a stochastic fractionally quasi-geostrophic equation on R d has been proved in [1]. It is worth mentioning that, all of these problems were considered in L 2 .…”

Section: Tongtong Liang and Yejuan Wangmentioning

confidence: 99%

“…Lemma 4. Let p ≥ 2 and α ∈ [0, 1 2 ) be given. Then for any progressively measurable process G ∈ L p (Ω × [0, T ]; L 2 (U, X)), there exists a constant C p,α > 0 independent of G such that…”

Section: 2mentioning

confidence: 99%

“…Assume that, in addition to the conditions imposed in Theorem 11, g ∈ Lipu(C (H s ), H s ) and h ∈ Lipu(C (H s ), L 2 (U, H s )), where s ≥ 2 − 2α and α ∈ ( 1 2 , 1). Let θ (1) and θ (2) be global martingale solutions of (3.1) defined on the same stochastic basis (Ω, F, {F t } t≥0 , P, W ) and starting from the same initial value ϕ. Then P ω : θ (1) (t, ω) = θ (2) (t, ω), ∀t ≥ 0 = 1.…”

Section: Suppose That For All Stopping Timesmentioning

confidence: 99%

See 3 more Smart Citations

Sub-critical and critical stochastic quasi-geostrophic equations with infinite delay

Liang¹,

Wang²

2021

DCDS-B

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In this paper, we investigate a stochastic fractionally dissipative quasi-geostrophic equation driven by a multiplicative white noise, whose external forces contain hereditary characteristics. The existence and uniqueness of both local martingale and local pathwise solutions are established in H s with s ≥ 2 − 2α, where α ∈ ( 1 2 , 1). For the critical case α = 1 2 , we obtain the similar results in H s with s > 1.

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Section: Suppose That For All Stopping Timesmentioning

confidence: 68%

Section: Tongtong Liang and Yejuan Wangmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: Suppose That For All Stopping Timesmentioning

confidence: 99%

See 2 more Smart Citations

Sub-critical and critical stochastic quasi-geostrophic equations with infinite delay

Liang¹,

Wang²

2021

DCDS-B

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“…We consider the following stochastic quasi-geostrophic equation in Here θ : R + × R 2 → R denotes the temperature, v : R 2 → R 2 is the 2D velocity field and θ 0 ∈ L 2 (R 2 , R). We refer to [7,32,42] for the background and more references on this model. Let ψ : R 2 → R be the stream function which satisfies (−∆)…”

Section: Application To Stochastic 2d Quasi-geostrophic Equationsmentioning

confidence: 99%

Maximal Inequalities and Exponential Estimates for Stochastic Convolutions Driven by Lévy-type Processes in Banach Spaces with Application to Stochastic Quasi-Geostrophic Equations

Zhu¹,

Brzeźniak²,

Liu³

2019

SIAM J. Math. Anal.

Self Cite

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We present remarkably simple proofs of Burkholder-Davis-Gundy inequalities for stochastic integrals and maximal inequalities for stochastic convolutions in Banach spaces driven by Lévy-type processes. Exponential estimates for stochastic convolutions are obtained and two versions of Itô's formula in Banach spaces are also derived. Based on the obtained maximal inequality, the existence and uniqueness of mild solutions of stochastic quasi-geostrophic equation with Lévy noise is established.

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Well-posedness for a stochastic Camassa–Holm type equation with higher order nonlinearities

Miao

Rohde

Tang

2023

Stoch PDE: Anal Comp

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This paper aims at studying a generalized Camassa–Holm equation under random perturbation. We establish a local well-posedness result in the sense of Hadamard, i.e., existence, uniqueness and continuous dependence on initial data, as well as blow-up criteria for pathwise solutions in the Sobolev spaces $$H^s$$ H s with $$s>3/2$$ s > 3 / 2 for $$x\in \mathbb R$$ x ∈ R . The analysis on continuous dependence on initial data for nonlinear stochastic partial differential equations has gained less attention in the literature so far. In this work, we first show that the solution map is continuous. Then we introduce a notion of stability of exiting time. We provide an example showing that one cannot improve the stability of the exiting time and simultaneously improve the continuity of the dependence on initial data. Finally, we analyze the regularization effect of nonlinear noise in preventing blow-up. Precisely, we demonstrate that global existence holds true almost surely provided that the noise is strong enough.

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Fractionally Dissipative Stochastic Quasi-Geostrophic Type Equations on ${\mathbb{R}}^{d}$

Cited by 10 publications

References 28 publications

Sub-critical and critical stochastic quasi-geostrophic equations with infinite delay

Sub-critical and critical stochastic quasi-geostrophic equations with infinite delay

Maximal Inequalities and Exponential Estimates for Stochastic Convolutions Driven by Lévy-type Processes in Banach Spaces with Application to Stochastic Quasi-Geostrophic Equations

Well-posedness for a stochastic Camassa–Holm type equation with higher order nonlinearities

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