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Zhaoyang Qiu

Publications and source records attributed to Zhaoyang Qiu.

16 recordsLinked to original sources
arXiv2026

Stability of invariant measures of the stochastic Landau-Lifshitz-Bloch equation with vanishing noise

In this paper, we investigate the limiting dynamics of invariant measures of the stochastic Landau-Lifshitz-Bloch equation driven by the Stratonovich noise defined on the entire space $\R^2$. We first prove the set of all invariant measures of the stochastic equation for small noise is tight in $H^1(\R^2)$, and then prove every limit of a sequence of invariant measures of the stochastic equation must be an invariant measure of the limiting system as the noise intensity approaches zero. The main difficulty of the paper is to establish the tightness of solutions which is caused by the low regularity of solutions and the non-compactness of Sobolev embeddings on unbounded domains. To solve the problem, we first consider a family of higher-order perturbed viscous systems and then use the regularity as well as the uniform tail-ends estimates of the perturbed solutions to establish the tightness of solutions of the original equation by a limiting process.

math.PR↗
arXiv2026

Homogenization and corrector results for the stochastic non-homogeneous incompressible Navier-Stokes equations with rapid oscillation

In this paper we are concerned with the homogenization property of stochastic non-homogeneous incompressible Navier-Stokes equations with rapid oscillation in a smooth bounded domain of $\mathbb{R}^d$, $d=2,3$, and driven by multiplicative infinite-dimensional Wiener noise. Using two-scale convergence, stochastic compactness and the martingale representation theory, we first show the solutions of original equations converge to the solution of a stochastic non-homogeneous incompressible homogenized system. Also, the energy equation of the homogenized system is established. Furthermore, a corrector result is proved which strengthens the two-scale convergence from weak to strong in the regularity space $H^1(\mathcal{O})$. Since the continuity equation which is of transport type cannot confer any regularization effect, there are some issues for proving the two results, including the difficulties for establishing the stochastic compactness and passing to the limit. We develop new regularity estimates, a stochastic version of lower semicontinuity as well as energy equation to overcome these difficulties.

math.PR↗
arXiv2024

Homogenization of the distribution-dependent stochastic abstract fluid models

In this paper, we study the homogenization of the distribution-dependent stochastic abstract fluid models by combining the $two\!-\!scale$ convergence and martingale representative approach. A general framework of the homogenization research is established for stochastic abstract fluid models which are the type of nonlinear partial differential equations including the (distribution-dependent) stochastic Navier-Stokes equations, stochastic magneto-hydrodynamic equations, stochastic Boussinesq equations, stochastic micropolar equations, stochastic Allen-Cahn equations.

math.AP↗
arXiv2024

Invariant measures and their limiting behavior of the Landau-Lifshitz-Bloch equation in unbounded domains

This paper deals with the existence and limiting behavior of invariant measures of the stochastic Landau-Lifshitz-Bloch equation driven by linear multiplicative noise and additive noise defined in the entire space $\mathbb{R}^d$ for $d=1,2$, which describes the phase spins in ferromagnetic materials around the Curie temperature. We first establish the existence and uniqueness of solutions by a domain expansion method. We then prove the existence of invariant measures by the weak Feller argument. In the case $d=1$, we show the uniform tightness of the set of all invariant measures of the stochastic equation, and prove any limit of a sequence of invariant measures of the perturbed equation must be an invariant measure of the limiting system. The cut-off arguments, stopping time techniques and uniform tail-ends estimates of solutions are developed to overcome the difficulty caused by the high-order nonlinearity and the non-compactness of Sobolev embeddings in unbounded domains.

math.AP↗
arXiv2023

Large deviations of invariant measure for the 3D stochastic hyperdissipative Navier-Stokes equations

In this paper, we consider the large deviations of invariant measure for the 3D stochastic hyperdissipative Navier-Stokes equations driven by additive noise. The unique ergodicity of invariant measure as a preliminary result is proved using a deterministic argument by the exponential moment and exponential stability estimates. Then, the uniform large deviations is established by the uniform contraction principle. Finally, using the unique ergodicity and the uniform large deviations results, we prove the large deviations of invariant measure by verifying the Freidlin-Wentzell large deviations upper and lower bounds.

math.AP↗
arXiv2022

Stochastic Cahn-Hilliard-Navier-Stokes equations with the dynamic boundary: Martingale weak solution, Markov selection

The existence of global martingale weak solution for the 2D and 3D stochastic Cahn-Hilliard-Navier-Stokes equations driven by multiplicative noise in a smooth bounded domain is established. In particular, the system is supplied with the dynamic boundary condition which accounts for the interaction between the fluid components and the rigid walls. The proof is completed by a three-level approximate scheme combining a fixed point argument and the stochastic compactness argument, overcoming challenges from strong nonlinearity, dynamic boundary and random effect. Then, we prove the existence of an almost surely Markov selection to the associated martingale problem following the abstract framework established by F. Flandoli and M. Romito.

math.PR↗
arXiv2021

Well-posedness for the stochastic electrokinetic flow

We consider the stochastic electrokinetic flow in a smooth bounded domain $\mathcal{D}$, modelled by a Nernst-Planck-Navier-Stokes system with a blocking boundary conditions for ionic species concentrations, perturbed by multiplicative noise. Several results are established in this paper. In both $2d$ and $3d$ cases, we establish the global existence of weak martingale solution which is weak in both PDEs and probability sense, and also the existence and uniqueness of the maximal strong pathwise solution which is strong in PDEs and probability sense. Particularly, we show that the maximal pathwise solution is global one in $2d$ case without the restriction of smallness of initial data.

math.AP↗
arXiv2020

Strong solution for compressible liquid crystal system with random force

We study the three-dimensional compressible Navier-Stokes equations coupled with the $Q$-tensor equation perturbed by a multiplicative stochastic force, which describes the motion of nematic liquid crystal flows. The local existence and uniqueness of strong pathwise solution up to a positive stopping time is established where ``strong" is in both PDE and probability sense. The proof relies on the Galerkin approximation scheme, stochastic compactness, identification of the limit, uniqueness and a cutting-off argument. In the stochastic setting, we develop an extra layer approximation to overcome the difficulty arising from the stochastic integral while constructing the approximate solution. Due to the complex structure of the coupled system, the estimates of the high-order items are also the challenging part in the article.

math.AP↗
arXiv2020

Martingale Solution for Stochastic Active Liquid Crystal System

The global weak martingale solution is built through a four-level approximation scheme to stochastic compressible active liquid crystal system driven by multiplicative noise in a smooth bounded domain in $\mathbb{R}^{3}$ with large initial data. The coupled structure makes the analysis challenging, and more delicate arguments are required in stochastic case compared to the deterministic one \cite{11}.

math.AP↗
arXiv2020

Strong pathwise solution and large deviation principle for the stochastic Boussinesq equations with partial diffusion term

We establish the existence and uniqueness of local strong pathwise solutions to the stochastic Boussinesq equations with partial diffusion term forced by multiplicative noise on the torus in $\mathbb{R}^{d},d=2,3$. The solution is strong in both PDE and probabilistic sense.In the two dimensional case, we prove the global existence of strong solutions to the Boussinesq equations forced by additive noise using a suitable stochastic analogue of a logarithmic Gronwall's lemma. After the global existence and uniqueness of strong solutions are established, the large deviation principle (LDP) is proved by the weak convergence method. The weak convergence is shown by a tightness argument in the appropriate functional space.

math.AP↗
arXiv2020

Invariant measure for 2D stochastic Cahn-Hilliard-Navier-Stokes equations

Using the Maslowski and Seidler method, the existence of invariant measure for 2-dimensional stochastic Cahn-Hilliard-Navier-Stokes equations with multiplicative noise is proved in state space $L_x^2\times H^1$, working with the weak topology. Also, the existence of global pathwise solution is investigated using the stochastic compactness argument.

math.AP↗
arXiv2020

Well-posedness for the Cahn-Hilliard-Navier-Stokes equation with random initial data

In this paper, we consider the almost sure well-posedness of the Cauchy problem to the Cahn-Hilliard-Navier-Stokes equation with a randomization initial data on a torus $\mathbb{T}^3$. First, we prove the local existence and uniqueness of solution. Furthermore, we prove the global existence and uniqueness of solution and give the relative probability estimate under the condition of small initial data.

math.AP↗
arXiv2020

Asymptotic behavior for the 1-D stochastic Landau Lifshitz Bloch equation

The stochastic Landau-Lifshitz-Bloch equation describes the phase spins in a ferromagnetic material and has significant role in simulating heat-assisted magnetic recording. In this paper, we consider the deviation of the solution to the 1-D stochastic Landau-Lifshitz-Bloch equation, that is, we give the asymptotic behavior of the trajectory $\frac{u_\varepsilon-u_0}{\sqrt{\varepsilon}λ(\varepsilon)}$ as $\varepsilon\rightarrow 0+$, for $λ(\varepsilon)=\frac{1}{\sqrt{\varepsilon}}$ and $1$ respectively. In other words, the large deviation principle and the central limit theorem are established respectively.

math.AP↗
arXiv2018

BCI-Based Strategies on Stroke Rehabilitation with Avatar and FES Feedback

Stroke is the leading cause of serious and long-term disability worldwide. Some studies have shown that motor imagery (MI) based BCI has a positive effect in poststroke rehabilitation. It could help patients promote the reorganization processes in the damaged brain regions. However, offline motor imagery and conventional online motor imagery with feedback (such as rewarding sounds and movements of an avatar) could not reflect the true intention of the patients. In this study, both virtual limbs and functional electrical stimulation (FES) were used as feedback to provide patients a closed-loop sensorimotor integration for motor rehabilitation. The FES system would activate if the user was imagining hand movement of instructed side. Ten stroke patients (7 male, aged 22-70 years, mean 49.5+-15.1) were involved in this study. All of them participated in BCI-FES rehabilitation training for 4 weeks.The average motor imagery accuracies of the ten patients in the last week were 71.3%, which has improved 3% than that in the first week. Five patients' Fugl-Meyer Assessment (FMA) scores have been raised. Patient 6, who has have suffered from stroke over two years, achieved the greatest improvement after rehabilitation training (pre FMA: 20, post FMA: 35). In the aspect of brain patterns, the active patterns of the five patients gradually became centralized and shifted to sensorimotor areas (channel C3 and C4) and premotor area (channel FC3 and FC4).In this study, motor imagery based BCI and FES system were combined to provided stoke patients with a closed-loop sensorimotor integration for motor rehabilitation. Result showed evidences that the BCI-FES system is effective in restoring upper extremities motor function in stroke. In future work, more cases are needed to demonstrate its superiority over conventional therapy and explore the potential role of MI in poststroke rehabilitation.

cs.HC↗
arXiv2016

Optimized motor imagery paradigm based on imagining Chinese characters writing movement

Motor imagery (MI) is a mental representation of motor behavior that has been widely used as a control method for a brain-computer interface (BCI), allowing communication for the physically impaired. The performance of MI based BCI mainly depends on the subject's ability to self-modulate EEG signals. Proper training can help naive subjects learn to modulate brain activity proficiently. However, training subjects typically involves abstract motor tasks and is time-consuming. To improve the performance of naive subjects during motor imagery, a novel paradigm was presented that would guide naive subjects to modulate brain activity effectively. In this new paradigm, pictures of the left or right hand were used as cues for subjects to finish the motor imagery task. Fourteen healthy subjects (11 male, aged 22-25 years, mean 23.6+/-1.16) participated in this study. The task was to imagine writing a Chinese character. Specifically, subjects could imagine hand movements following the sequence of writing strokes in the Chinese character. This paradigm was meant to find an effective and familiar action for most Chinese people, to provide them with a specific, extensively practiced task and help them modulate brain activity. Results showed that the writing task paradigm yielded significantly better performance than the traditional arrow paradigm (p<0.001). Questionnaire replies indicated that most subjects thought the new paradigm was easier and more comfortable. The proposed new motor imagery paradigm could guide subjects to help them modulate brain activity effectively. Results showed that there were significant improvements using new paradigm, both in classification accuracy and usability.

cs.HC↗