Invariant Measure and Statistical Solutions for Non-Autonomous Discrete Klein-Gordon-Schrödinger-Type Equations

Wu, Shang; Huang, Jianhua

doi:10.11948/20190243

Cited by 4 publications

(9 citation statements)

References 21 publications

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“…Let t ∈ R and ε > 0 be arbitrary given numbers. By (33), we have e −αt t −∞ e αs f (s) 2 ds < +∞, from which it can be seen that there exists…”

Section: Preliminariesmentioning

confidence: 96%

“…We remark that our idea originates from [33,37,38,40]. In [38], Zhao, Li and Caraballo first gave some sufficient conditions on the existence of trajectory statistical solutions for autonomous continuous evolution equations, and then proved that the trajectory statistical solution satisfies a Liouville type equation.…”

Section: Introductionmentioning

confidence: 99%

“…Invariant measures of lattice systems are widely investigated by many researchers; see e.g., [17,33,27,34,40]. It is well-known that lattice dynamical systems (LDSs in short) are spatiotemporal systems with discretization in some variables, and widely appear in many different fields such as biology [18], electrical engineering [26], chemical reaction theory [8], and so on.…”

Section: Introductionmentioning

confidence: 99%

“…Very recently, on invariant measures, Zhao, Xue and Lukaszewicz [40] established the existence of invariant Borel probability measures of nonautonomous discrete Klein-Gordon-Schrödinger equations. Later, based on the work mentioned above, Wu and Huang [33] further investigated the statistical solution for the discrete Klein-Gordon-Schrödinger type equations.…”

Section: Introductionmentioning

confidence: 99%

“…Remark 3. In this article we mainly develop the method in [33,37,40] to construct invariant measures and statistical solutions for the nonautonomous discrete Schrödinger-Boussinesq equations. We mention that the method in this paper is relatively simpler than that used in [37] for continuous Klein-Gordon-Schrödinger equations, which phenomenon maybe an important difference between continuous PDEs and LDSs, and is caused by the LDSs' intrinsic characterization defined on a space of infinite sequences (see also [43]).…”

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confidence: 99%

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Statistical solution and Liouville type theorem for coupled Schrödinger-Boussinesq equations on infinite lattices

Wang

2022

DCDS-B

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<p style='text-indent:20px;'>In this article, we are concerned with statistical solutions for the nonautonomous coupled Schrödinger-Boussinesq equations on infinite lattices. Firstly, we verify the existence of a pullback-<inline-formula><tex-math id="M1">\begin{document}$ {\mathcal{D}} $\end{document}</tex-math></inline-formula> attractor and establish the existence of a unique family of invariant Borel probability measures carried by the pullback-<inline-formula><tex-math id="M2">\begin{document}$ {\mathcal{D}} $\end{document}</tex-math></inline-formula> attractor for this lattice system. Then, it will be shown that the family of invariant Borel probability measures is a statistical solution and satisfies a Liouville type theorem. Finally, we illustrate that the invariant property of the statistical solution is indeed a particular case of the Liouville type theorem.</p>

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“…Let t ∈ R and ε > 0 be arbitrary given numbers. By (33), we have e −αt t −∞ e αs f (s) 2 ds < +∞, from which it can be seen that there exists…”

Section: Preliminariesmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Statistical solution and Liouville type theorem for coupled Schrödinger-Boussinesq equations on infinite lattices

Wang

2022

DCDS-B

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Dynamics of a globally modified Navier–Stokes model with double delay

Yang

Chen

Caraballo

2022

Z. Angew. Math. Phys.

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This paper investigates the dynamics of a class of three-dimensional globally modified Navier-Stokes equations with double delay in the forcing and convective terms. We first prove the well-posedness of solutions of such system, which enables us to establish suitable non-autonomous dynamical systems. We then show the existence and uniqueness of pullback attractors for the associated dynamical systems. Finally, by using the generalized Banach limit, we construct a family of invariant Borel probability measures, which are supported on the pullback attractors.

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Statistical solution and Kolmogorov entropy for the impulsive discrete Klein-Gordon-Schrödinger-type equations

Lin

Zhao

et al. 2023

DCDS-B

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<p style='text-indent:20px;'>This paper studies the impulsive discrete Klein-Gordon-Schrödinger-type equations. We first prove that the problem of the discrete Klein-Gordon-Schrödinger-type equations with initial and impulsive conditions is global well-posedness. Then we establish that the solution operators form a continuous process and that this process possesses a pullback attractor and a family of invariant Borel probability measures. Further, we prove that this family of Borel probability measures satisfies the Liouville type theorem piecewise and is a statistical solution of the impulsive discrete Klein-Gordon-Schrödinger-type equations. Finally, we formulate the concept of Kolmogorov entropy for the statistical solution and estimate its upper bound.</p>

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Invariant Measure and Statistical Solutions for Non-Autonomous Discrete Klein-Gordon-Schrödinger-Type Equations

Cited by 4 publications

References 21 publications

Statistical solution and Liouville type theorem for coupled Schrödinger-Boussinesq equations on infinite lattices

Statistical solution and Liouville type theorem for coupled Schrödinger-Boussinesq equations on infinite lattices

Dynamics of a globally modified Navier–Stokes model with double delay

Statistical solution and Kolmogorov entropy for the impulsive discrete Klein-Gordon-Schrödinger-type equations

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