The Schedule of the Conference on Infinite Dynamical Systems

无穷维动力系统研讨会日程表

Mixed well-posedness and applications to the statistical solutions for the semi-dissipative Boussinesq syste

赵才地 温州大学

Abstract：

This article investigates the semi-dissipative Boussinesq system involving a dissipative parabolic equation for the velocity field and a conservative hyperbolic equation for the temperature. The authors employ the strong topology in space $\mathbb{H}^1$ for the velocity field and employ the weak topology in space $L^2$ for the temperature. They first exploit the mixed well-posedness of the addressed system and prove that the generated evolution process of the solution mappings possesses a sigma compact pullback attractor in space $\mathbb{H}^1\times L^2$ under the mixed topology. Afterwards, they establish that the evolution process possesses a family of invariant Borel probability measures with supports contained in the obtained pullback attractor. Finally, they prove that this family of probability measures satisfies Liouville's equation and is a statistical solution for the addressed Boussinesq system.

Almost sure spatial decay and almost sure nonlinear smoothing for two classes of stochastic dispersive equations

闫威 河南师范大学

Abstract:

In this paper, we consider the almost sure nonlinear smoothing, the almost sure spatial decay and the almost sure uniform convergence for the stochastic mKdV equation and the stochastic cubic KdV-Benjamin-Ono equation.

Firstly, when $\Phi_{1}\in L_{2}^{0,s}$ and $f\in H^{s}(\R)(s\geq\frac{1}{4})$, we establish the local well-posedness for the stochastic mKdV equation. Secondly, when $\Phi_{2}\in L_{2}^{0,s}$ and $g\in H^{s}(\R)(s\geq\frac{1}{4})$, we prove the local well-posedness for the stochastic cubic KdV-Benjamin-Ono equation. Thirdly, we establish the almost sure nonlinear smoothing for the stochastic mKdV equation and the stochastic cubic KdV-Benjamin-Ono equation. Finally, by using the almost sure nonlinear smoothing, we obtain the almost sure spatial decay and the almost sure uniform convergence of the integral term in the pathwise solutions of the stochastic mKdV equation and the stochastic cubic KdV-Benjamin-Ono equation. More precisely, we have the following results: for the stochastic mKdV equation, let $s>\frac{1}{3}$, $\Phi_{1}\in L_{2}^{0,s}$, and $f\in H^{s}(\R)$. Then, the local pathwise solution $u$ satisfies

\begin{eqnarray*}

&&\mathbb{P}\Big(\Big\{\omega: \lim_{t\rightarrow0}\Big\|u-U(t)f-\int_{0}^{t}U(t-s)\Phi_{1}dW(s)\Big\|_{L_{x}^{\infty}}=0\Big\}\Big)=1,\\

&&\mathbb{P}\Big(\Big\{\omega: \forall t\in[0,T_{\omega}], \lim_{|x|\rightarrow\infty}\Big(u-U(t)f-\int_{0}^{t}U(t-s)\Phi_{1}dW(s)\Big)=0\Big\}\Big)=1.

\end{eqnarray*}

For the stochastic cubic KdV-Benjamin-Ono equation, let $s>\frac{1}{3}$, $\Phi_{2}\in L_{2}^{0,s}$, and $g\in H^{s}(\R)$. Then the local pathwise solution $v$ satisfies

\begin{eqnarray*}

&&\mathbb{P}\Big(\Big\{\omega: \lim_{t\rightarrow0}\Big\|v-V(t)g-\int_{0}^{t}V(t-s)\Phi_{2}dW(s)\Big\|_{L_{x}^{\infty}}=0\Big\}\Big)=1,\\

&&\mathbb{P}\Big(\Big\{\omega: \forall t\in[0,T_{\omega}],\lim_{|x|\rightarrow\infty}\Big(v-V(t)g-\int_{0}^{t}V(t-s)\Phi_{2}dW(s)\Big)=0\Big\}\Big)=1.

\end{eqnarray*}

Inertial manifolds for modified Navier-Stokes equations

李新华 兰州大学

Abstract:

An inertial manifold (IM) is a finite-dimensional invariant manifold that contains the global attractor and attracts all orbits at an exponential rate; it is also the graph of a Lipschitz continuous function. If a partial differential equation admits an inertial manifold, its dynamics can be completely described by a system of ordinary differential equations. The classical theory of inertial manifolds requires a spectral gap condition for their construction. In this talk, we introduce a spatial averaging method that allows the construction of inertial manifolds without the spectral gap condition, and discuss its application to the two-dimensional modified Navier–Stokes equations. The original motivation for the theory of inertial manifolds was to address the Navier–Stokes equations; however, this problem remains open to this day. This talk will review key results on inertial manifolds for modified Navier–Stokes equations and present our recent contributions to this topic.

Approximation of the Heaviside function by sigmoidal functions in reaction-diffusion equations

孙文龙 湖北民族大学

Abstract:

A reaction-diffusion system with a Heaviside reaction term on a smooth bounded domain in R d is studied by using sigmoidal functions to approx imate the Heaviside term. First, the global well-posedness of solutions to the sigmoidal reaction-diffusion system and the existence of solutions to the Heaviside reaction-diffusion system are proved. Then the lattice systems resulting from Galerkin expansions of the solutions to the sigmoidal system and the Heaviside system are studied. In particular, solutions of the sigmoidal lattice system are shown to converge to the solution of the Heaviside lattice system as the steepness parameter \epsilon goes to 0, through an inflated lattice

system. Moreover, dynamics of the lattice systems are utilized to show that solutions of the sigmoidal reaction-diffusion system tend to the solution of the Heaviside reaction-diffusion system as \epsilon goes to 0. Finally, relations be tween the attractors for the sigmoidal system and the Heaviside system are established.

On stochastic Landau-Lifshitz-Bloch equation in unbounded domains

黄代文 北京应用物理与计算数学研究所

Abstract:

In this talk, we give some results on stochastic Landau-Lifshitz-Bloch equation in unbounded domains. Firstly, we recall the random dynamics of the stochastic Landau-Lifshitz-Bloch equation with colored noise. Secondly, we introduce some results on invariant measures and their limiting behavior of the stochastic Landau-Lifshitz-Bloch equation in unbounded domains.

Kolmogorov ε-entropy for uniform attractors of dissipative PDEs

孙春友 东华大学

Abstract:

In this talk, we focus on the Kolmogorov $\varepsilon$-entropy estimations for the uniform attractors of dissipative PDEs, especially for the case that the symbol space is not compact. This talk based on our recent results that joint with Yangmin Xiong, A.Kostianko, and S.Zelik.

一维周期色散方程的最佳能观测常数估计

王明 中南大学

Abstract:

In this talk, we establish a quantitative observability inequality for dispersive equations of the form $\partial_t u = iP(D)u$ on the one-dimensional torus $\mathbb{T}$. The observation set is of the form $[0, T] \times E$, where $E \subset \mathbb{T}$ is an arbitrary measurable set with positive measure. We provide a unified approach that yields the sharp dependence of the control cost on both the measure of the set $E$ and the time horizon $T$ in the limits $|E| \to 0$ and $T \to 0^+$.