arXiv subjects

Shihu Li

Publications and source records attributed to Shihu Li.

14 recordsLinked to original sources

arXiv2025

Stochastic Forced 3D Navier-Stokes Equations in $\mathbb{H}^{1/2}$-Space

In the classical work [FK], Fujita and Kato established the local existence of solutions to the 3D Navier-Stokes equations in the critical $\mathbb{H}^{1/2}$-space. In this paper, we are concerned with the global well-posedness of the stochastic forced 3D Navier-Stokes equations in the $\mathbb{H}^{1/2}$-space under general initial conditions, where the stochastic forcing comprises a transport forcing and a nonlocal turbulent forcing. In this setting, the random noise is shown to provide a regularization effect on the energy estimates, which we obtain by constructing suitable Lyapunov functions. However, its nonlocality also brings analytical challenges. We develop a bootstrap type estimate based on the kinematic viscosity together with a delicate stopping time argument to prove the global existence and uniqueness of solutions, as well as continuous dependence on the initial value. Furthermore, we also investigated the long-time behavior of the stochastic forced 3D Navier-Stokes equations.

math.PR↗

arXiv2025

Mean Field Stochastic Partial Differential Equations with Nonlinear Kernels

This work focuses on the mean field stochastic partial differential equations with nonlinear kernels. We first prove the existence and uniqueness of strong and weak solutions for mean field stochastic partial differential equations in the variational framework, then establish the convergence (in certain Wasserstein metric) of the empirical laws of interacting systems to the law of solutions of mean field equations, as the number of particles tends to infinity. The main challenge lies in addressing the inherent interplay between the high nonlinearity of operators and the non-local effect of coefficients that depend on the measure. In particular, we do not need to assume any exponential moment control condition of solutions, which extends the range of the applicability of our results. As applications, we first study a class of finite-dimensional interacting particle systems with polynomial kernels, which are commonly encountered in fields such as the data science and the machine learning. Subsequently, we present several illustrative examples of infinite-dimensional interacting systems with nonlinear kernels, such as the stochastic climate models, the stochastic Allen-Cahn equations, and the stochastic Burgers type equations.

math.PR↗

arXiv2025

McKean-Vlasov SPDEs driven by Poisson random measure: Well-posedness and large deviation principle

In this work, we investigate the McKean-Vlasov stochastic partial differential equations driven by Poisson random measure. By adapting the variational framework, we prove the well-posedness and large deviation principle for a class of McKean-Vlasov stochastic partial differential equations with monotone coefficients. The main results can be applied to quasi-linear McKean-Vlasov equations such as distribution dependent stochastic porous media equation and stochastic p-Laplace equation. Our proof is based on the weak convergence approach introduced by Budhiraja et al. for Poisson random measures, the time discretization procedure and relative entropy estimates. In particular, we succeed in dropping the compactness assumption of embedding in the Gelfand triple in order to deal with the case of bounded and unbounded domains in applications.

math.PR↗

arXiv2024

Stochastic PDEs with Generalized Coercivity: Global Well-Posedness and Finite Time Extinction

This work investigates the global existence, uniqueness, and Feller property for stochastic partial differential equations under generalized coercivity conditions, particularly in cases where the corresponding deterministic equations possess only local solutions. Furthermore, we reveal a novel phenomenon: for a potentially explosive deterministic system, the introduction of appropriate multiplicative noise not only prevents blow-up but also leads to the finite-time extinction of the stochastic dynamics. Our main results are applicable to a broad range of models, including stochastic 3D Navier-Stokes equations, stochastic surface growth models, and stochastic $p$-Laplace equations with heat sources.

math.PR↗

arXiv2023

McKean-Vlasov Stochastic Partial Differential Equations: Existence, Uniqueness and Propagation of Chaos

In this paper, we provide a general framework for investigating McKean-Vlasov stochastic partial differential equations. We first show the existence of weak solutions by combining the localizing approximation, Faedo-Galerkin technique, compactness method and the Jakubowski version of the Skorokhod representation theorem. Then under certain locally monotone condition we further investigate the existence and uniqueness of (probabilistically) strong solutions. The applications of the main results include a large class of McKean-Vlasov stochastic partial differential equations such as stochastic 2D/3D Navier-Stokes equations, stochastic Cahn-Hilliard equations and stochastic Kuramoto-Sivashinsky equations. Finally, we show a propagation of chaos result in Wasserstein distance for weakly interacting stochastic 2D Navier-Stokes systems.

math.PR↗

arXiv2023

Multi-Scale McKean-Vlasov SDEs: Moderate Deviation Principle in Different Regimes

The main aim of this paper is to study the moderate deviation principle for McKean-Vlasov stochastic differential equations with multiple scales. Specifically, we are interested in the asymptotic estimates of the deviation processes $\frac{X^{\delta}-\bar{X}}{\lambda(\delta)}$ as $\delta\to 0$ in different regimes (i.e. $\varepsilon=o(\delta)$ and $\varepsilon=O(\delta)$), where $\delta$ stands for the intensity of the noise and $\varepsilon:=\varepsilon(\delta)$ stands for the time scale separation. The rate functions in two regimes are different, in particular, we show that it is strongly affected by the noise of the fast component in latter regime, which is essentially different from the former one and the case of large deviations (cf. \cite{HLLS}). As a by-product, the explicit representation formulas of the rate functions in all of regimes are also given. The main techniques are based on the weak convergence approach and the functional occupation measure approach.

math.PR↗

arXiv2022

Diffusion Approximation for Multi-Scale McKean-Vlasov SDEs Through Different Methods

In this paper, we aim to study the diffusion approximation for multi-scale McKean-Vlasov stochastic differential equations. More precisely, we prove the weak convergence of slow process $X^\varepsilon$ in $C([0,T];\mathbb{R}^n)$ towards the limiting process $X$ that is the solution of a distribution dependent stochastic differential equation in which some new drift and diffusion terms compared to the original equation appear. The main contribution is to use two different methods to explicitly characterize the limiting equations respectively. The obtained diffusion coefficients in the limiting equations have different form through these two methods, however it will be asserted that they are essential the same by a comparison.

math.PR↗

arXiv2022

Strong Convergence Rates in Averaging Principle for Slow-Fast McKean-Vlasov SPDEs

In this paper, we aim to study the asymptotic behaviour for a class of McKean-Vlasov stochastic partial differential equations with slow and fast time-scales. Using the variational approach and classical Khasminskii time discretization, we show that the slow component strongly converges to the solution of the associated averaged equation. In particular, the corresponding convergence rates are also obtained. The main results can be applied to demonstrate the averaging principle for various McKean-Vlasov nonlinear SPDEs such as stochastic porous media type equation, stochastic $p$-Laplace type equation and also some McKean-Vlasov stochastic differential equations.

math.PR↗

arXiv2021

Central Limit Type Theorem and Large Deviation Principle for Multi-Scale McKean-Vlasov SDEs

In this paper, we aim to study the asymptotic behavior for multi-scale McKean-Vlasov stochastic dynamical systems. Firstly, we obtain a central limit type theorem, i.e, the deviation between the slow component $X^{\varepsilon}$ and the solution $\bar{X}$ of the averaged equation converges weakly to a limiting process. More precisely, $\frac{X^{\varepsilon}-\bar{X}}{\sqrt{\varepsilon}}$ converges weakly in $C([0,T],\RR^n)$ to the solution of certain distribution dependent stochastic differential equation, which involves an extra explicit stochastic integral term. Secondly, in order to estimate the probability of deviations away from the limiting process, we further investigate the Freidlin-Wentzell's large deviation principle for multi-scale McKean-Vlasov stochastic system. The main techniques are based on the Poisson equation for central limit type theorem and the weak convergence approach for large deviation principle.

math.PR↗

arXiv2021

Freidlin-Wentzell Type Large Deviation Principle for Multi-Scale Locally Monotone SPDEs

This work is concerned with Freidlin-Wentzell type large deviation principle for a family of multi-scale quasilinear and semilinear stochastic partial differential equations. Employing the weak convergence method and Khasminskii's time discretization approach, the Laplace principle (equivalently, large deviation principle) for a general class of multi-scale SPDEs is derived. In particular, we succeed in dropping the compactness assumption of embedding in the Gelfand triple in order to deal with the case of bounded and unbounded domains in applications. Our main results are applicable to various multi-scale SPDE models such as stochastic porous media equations, stochastic p-Laplace equations, stochastic fast-diffusion equations, stochastic 2D hydrodynamical type models, stochastic power law fluid equations and stochastic Ladyzhenskaya models.

math.PR↗

arXiv2021

Large Deviation Principle for McKean-Vlasov Quasilinear Stochastic Evolution Equations

This paper is devoted to investigating the Freidlin-Wentzell's large deviation principle for a class of McKean-Vlasov quasilinear SPDEs perturbed by small multiplicative noise. We adopt the variational framework and the modified weak convergence criteria to prove the Laplace principle for McKean-Vlasov type SPDEs, which is equivalent to the large deviation principle. Moreover, we do not assume any compactness condition of embedding in the Gelfand triple to handle both the cases of bounded and unbounded domains in applications. The main results can be applied to various McKean-Vlasov type SPDEs such as distribution dependent stochastic porous media type equations and stochastic p-Laplace type equations.

math.PR↗

arXiv2019

Small Time Asymptotics for SPDEs with Locally Monotone Coefficients

This work aims to prove the small time large deviation principle (LDP) for a class of stochastic partial differential equations (SPDEs) with locally monotone coefficients in generalized variational framework. The main result could be applied to demonstrate the small time LDP for various quasilinear and semilinear SPDEs such as stochastic porous media equations, stochastic $p$-Laplace equations, stochastic Burgers type equation, stochastic 2D Navier-Stokes equation, stochastic power law fluid equation and stochastic Ladyzhenskaya model. In particular, our small time LDP result seems to be new in the case of general quasilinear SPDEs with multiplicative noise.

math.PR↗

arXiv2018

Averaging principle for two dimensional stochastic Navier-Stokes equations

The averaging principle is established for the slow component and the fast component being two dimensional stochastic Navier-Stokes equations and stochastic reaction-diffusion equations, respectively. The classical Khasminskii approach based on time discretization is used for the proof of the slow component strong convergence to the solution of the corresponding averaged equation under some suitable conditions. Meanwhile, some powerful techniques are used to overcome the difficulties caused by the nonlinear term and to release the regularity of the initial value.

math.PR↗

arXiv2018

Stochastic 3D Leray-$α$ Model with Fractional Dissipation

In this paper, we establish the global well-posedness of stochastic 3D Leray-$α$ model with general fractional dissipation driven by multiplicative noise. This model is the stochastic 3D Navier-Stokes equation regularized through a smoothing kernel of order $θ_1$ in the nonlinear term and a $θ_2$-fractional Laplacian. In the case of $θ_1 \ge 0, θ_2 > 0$ and $θ_1+θ_2 \geq\frac{5}{4}$, we prove the global existence and uniqueness of strong solutions. The main results cover many existing works in the deterministic cases, and also generalize some known results of stochastic models as our special cases such as stochastic hyperviscous Navier-Stokes equation and classical stochastic 3D Leray-$α$ model.

math.AP↗