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Guoli Zhou

Publications and source records attributed to Guoli Zhou.

12 recordsLinked to original sources
arXiv2026

Global Existence and Stability of 3D Stochastic NSEs in Bounded and Unbounded Domains Driven by a Special Multiplicative Wiener Process

The aim of this work is to extend the results from a recent paper by Hong, Li and Liu, from bounded domains to both bounded and unbounded domains. We show the global existence and uniqueness of 3D stochastic Navier-Stokes equations with nonlinear multiplicative noise for every initial data from the Sobolev space $H^1$. We do not use any tightness argument. Instead, we firstly show the existence of a local maximal solution and then we use Lyapunov function to prove the solution is global. Our approach is motivated by a recent paper of the first named author with Ferrario, Maurelli and Zanella about a similar result for stochastic nonlinear Schr\"odinger Equations. The main idea is that once the local existence of strong solutions is established, a very strong noise pushing toward the origin, will make blow-up impossible.

math.PR↗
arXiv2023

Global well-posedness and regularity of 3D Burgers equation with multiplicative noise

In this paper, we develop low regularity theory for 3D Burgers equation perturbed by a linear multiplicative stochastic force. This method is new and essentially different from the deterministic partial differential equations(PDEs). Our results and method can be widely applied to other stochastic hydrodynamic equations and the deterministic PDEs. As a further study, we establish a random version of maximum principle for random 3D Burgers equations, which will be an important tool for the study of 3D stochastic Burgers equations. As we know establishing moment estimates for highly nonlinear stochastic hydrodynamic equations is difficult. But moment estimates are very important for us to study the probabilistic properties and long-time behavior for the stochastic systems. Here, the random maximum principle helps us to achieve some important moment estimates for 3D stochastic Burgers equations and lays a solid foundation for the further study of 3D stochastic Burgers equations.

math.PR↗
arXiv2020

Exponential stability of 3D stochastic primitive equations driven by fractional noise

In this article, we study the stability of solutions to 3D stochastic primitive equations driven by fractional noise. Since the fractional Brownian motion is essentially different from Brownian motion, lots of stochastic analysis tools are not available to study the exponential stability for the stochastic systems. Therefore, apart from the standard method for the case of Brownian motion, we develop a new method to show that 3D stochastic primitive equations driven by fractional noise converge almost surely exponentially to the stationary solutions. This method may be applied to other stochastic hydrodynamic equations and other noises including Brownian motion and Lévy noise.

math.PR↗
arXiv2019

Global well-posedness of stochastic nematic liquid crystals with random initial and random boundary conditions driven by multiplicative noise

The flow of nematic liquid crystals can be described by a highly nonlinear stochastic hydrodynamical model, thus is often influenced by random fluctuations, such as uncertainty in specifying initial conditions and boundary conditions. In this article, we consider the $2$-D stochastic nematic liquid crystals with the velocity field perturbed by affine-linear multiplicative white noise, with random initial data and random boundary conditions. Our main objective is to establish the global well-posedness of the stochastic equations under certain sufficient Malliavin regularity of the initial conditions and the boundary conditions. The Malliavin calculus techniques play important roles in proving the global existence of the solutions to the stochastic nematic liquid crystal models with random initial and random boundary conditions. It should be pointed out that the global well-posedness is also true when the stochastic system is perturbed by the noise on the boundary.

math.PR↗
arXiv2018

Random attractor for the 2D stochastic nematic liquid crystals flows with multiplicative noise

Under non-periodic boundary conditions, we consider the long-time behavior for stochastic 2D nematic liquid crystals flows with velocity and orientations perturbed by additive noise and multiplicative noise respectively. It is the first result for the long-time behavior of stochastic nematic liquid crystals under Dirichlet boundary condition for velocity field and Neumann boundary condition for orientation field.

math.AP↗
arXiv2018

The asymptotic behavior of primitive equations with multiplicative noise

This paper is concerned with the existence of invariant measure for 3D stochastic primitive equations driven by linear multiplicative noise under non-periodic boundary conditions. The common method is to apply Sobolev imbedding theorem to proving the tightness of the distribution of the solution. However, this method fails because of the non-linearity and non-periodic boundary conditions of the stochastic primitive equations. To overcome the difficulties, we show the existence of random attractor by proving the compact property and the regularity of the solution operator. Then we show the existence of invariant measure.

math.PR↗
arXiv2017

Random attractor of the 3D viscous primitive equations driven by fractional noises

We develop a new and general method to prove the the existence of the random attractor (strong attractor) for the primitive equations (PEs) of large-scale ocean and atmosphere dynamics under $non$-$periodic$ boundary conditions and driven by infinite-dimensional additive fractional Wiener processes. In contrast to our new method, the common method, compact Sobolev embedding theorem, is to obtain the uniform $a$ $priori$ estimates in some Sobolev space whose regularity is high enough. But this is very complicated for the 3D stochastic PEs with the $non$-$periodic$ boundary conditions. Therefore, the existence of universal attractor ( weak attractor) was established in previous work. The main idea of our method is that we first derive that $\mathbb{P}$-almost surely the solution operator of stochastic PEs is compact. Then we construct a compact absorbing set by virtue of the compact property of the the solution operator and the existence of a absorbing set. We should point out that our method has some advantages over the common method of using compact Sobolev embedding theorem, i.e., if the random attractor in some Sobolev space do exist in view of the common method, our method would then further implies the existence of random attractor in this space. The present work provides a general way for proving the existence of random attractor for common classes of dissipative stochastic partial differential equations and improves the existing results concerning random attractor of stochastic PEs. In a forth coming paper, we use this new method to prove the existence of strong attractor for the stochastic moist primitive equations, improving the results, the existence of weak (universal) attractor of the deterministic model.

math.PR↗
arXiv2016

Finite dimensionality of global attractor for the solutions to 3D viscous primitive equations of large-scale moist atmosphere

Under general boundary conditions we consider the finiteness of the Hausdorff and fractal dimensions of the global attractor for the strong solution of the 3D moist primitive equations with viscosity. Firstly, we obtain time-uniform estimates of the first-order time derivative of the strong solutions in $L^2(\mho)$. Then, to prove the finiteness of the Hausdorff and fractal dimensions of the global attractor, the common method is to obtain the uniform boundedness of the strong solution in $H^2(\mho)$ to establish the squeezing property of the solution operator. But it is difficult to achieve due to the boundary conditions and complicated structure of the 3D moist primitive equations. To overcome the difficulties, we try to use the uniform boundedness of the derivative of the strong solutions with respect to time $t$ in $L^2(\mho)$ to prove the uniform continuity of the global attractor. Finally, using the uniform continuity of the global attractor we establish the squeezing property of the solution operator which implies the finiteness of the Hausdorff and fractal dimensions of the global attractor.

math.AP↗
arXiv2016

The global attractor for the 3-D viscous primitive equations of large-scale moist atmosphere

Absorbing ball in $H^{1}(\mho)$ is obtained for the strong solution to the three dimensional viscous moist primitive equations under the natural assumption $Q_{1},Q_{2}\in L^{2}(\mho)$ which is weaker than the assumption $Q_{1},Q_{2}\in H^{1}(\mho)$ in $\cite{GH2}$. In view of the structure of the manifold and the special geometry involved with vertical velocity, the continuity of the strong solution in $H^{1}(\mho)$ is established with respect to time and initial data. To obtain the existence of the global attractor for the moist primitive equations, the common method is to obtain the absorbing ball in $H^{2}(\mho)$ for the strong solution to the equations. But it is difficult due to the complex structure of the moist primitive equations. To overcome the difficulty, we try to use Aubin-Lions lemma and the continuous property of the strong solutions to the moist primitive equations to prove the the existence of the global attractor which improves the result, the existence of weak attractor, in $\cite{GH2}$.

math.AP↗