Carleman Estimate and Unique Continuation Property for the Linear Stochastic Korteweg–de Vries Equation

Gao, Peng

doi:10.1017/s0004972714000276

Cited by 19 publications

(12 citation statements)

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“…Proof of Proposition 1. The main idea in this part comes from [18,15,16]. We consider the stochastic differential equation…”

Section: Appendixmentioning

confidence: 99%

Limiting dynamics for stochastic nonclassical diffusion equations

Gao

2022

DCDS-B

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<p style='text-indent:20px;'>In this paper, we are concerned with the dynamical behavior of the stochastic nonclassical parabolic equation, more precisely, it is shown that the inviscid limits of the stochastic nonclassical diffusion equations reduces to the stochastic heat equations. The key points in the proof of our convergence results are establishing some uniform estimates and the regularity theory for the solutions of the stochastic nonclassical diffusion equations which are independent of the parameter. Based on the uniform estimates, the tightness of distributions of the solutions can be obtained.</p>

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“…Proof of Proposition 1. The main idea in this part comes from [18,15,16]. We consider the stochastic differential equation…”

Section: Appendixmentioning

confidence: 99%

Limiting dynamics for stochastic nonclassical diffusion equations

Gao

2022

DCDS-B

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“…By the same method as in [5, Step 3 of Proof of Theorem 2.1] and [11,Section 3], (12), (13) and (14) allow to pass to the limit in the terms of the approximate problem (6) to deduce that (u, y) satisfies…”

Section: Peng Gaomentioning

confidence: 99%

Carleman estimates and Unique Continuation Property for 1-D viscous Camassa-Holm equation

Gao¹

2017

Discrete &Amp; Continuous Dynamical Systems - A

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This paper is devoted to studying the 1-D viscous Camassa-Holm equation on a bounded interval. We first deduce the existence and uniqueness of strong solution to the viscous Camassa-Holm equation by using Galerkin method. Then we establish an identity for a second order parabolic operator, by applying this identity we obtain two global Carleman estimates for the linear viscous Camassa-Holm operator. Based on these estimates, we obtain two types of Unique Continuation Property for the viscous Camassa-Holm equation.

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“…The key tool to prove the inverse problem in this paper is Carleman estimate, which is a class of weighted energy estimates connected with deterministic/stochastic differential operators. Carleman estimate is applied to many aspects, such as the unique continuation property, 10,11 inverse problem, [12][13][14] and control theory. [15][16][17] There are numerous studies on inverse problems for deterministic hyperbolic equations by Carleman estimate, we refer to literature.…”

Section: Introductionmentioning

confidence: 99%