河南大学

偏微分方程和数学物理国际会议

河南大学数学与统计学院

河南，开封

2018年6月22-24日

         

                                              会议日程（Program）

2018年6月23日上午 (The Morning of June 23, 2018)

        会议地点：河南大学数学与统计学院二楼南阶梯教室

7:50	参会人员在中州国际饭店大堂集合，统一坐车前往会场
时 间 (Time)	报告人(Speaker)	报告题目 (Title   of Talk)	主持人(Chairperson)
8:10-8:40	开幕式、照相 (Opening Ceremony, Photography)
8:50-9:30	丁时进	Strong   solutions to incompressible Navier-Stokes equations with Navier boundary conditions	郭柏灵
9:30-10:10	刘跃	Asymptotic model   equations arising in shallow water theory
10:10-10:30 Tea break
10:30-11:10	袁光伟	三维多面体网格上扩散格式	郭宗明
11:10-11:50	赵会江	Global spherical symmetric flows for a viscous radiative   and reactive gas in an exterior domain with large initial data
12:00-13:30 午餐（地点：中州金明酒店） (河南大学金明校区内)

2018年6月23日下午 (The Afternoon of June 23, 2018)

会议地点：河南大学数学与统计学院二楼南阶梯教室

时 间(Time)	报告人(Speaker)	报告题目(Title   of Talk)	主持人(Chairperson)
14:00-14:40	高洪俊	Stochastic strong solutions for   stochastic transport equations	王保祥
14:40-15:20	李再东	Solitons, breathers, and   rogue waves in ferromagnet
15:20-16:00	韩小森	Some nonlinear elliptic   PDEs arising in the Chern-Simons-Higgs theory
16:00-16:20   Tea break
16:20-16:50	赖柏顺	Some new developments on the forward self-similar solutions of the   incompressible Navier-Stokes equations	朱佩成
16:50-17:20	江杰	Convergence to equilibria of global solutions to quasilinear Keller--Segel systems
17:20-17:50	吕广迎	The   impact of noise on ODEs and PDEs	原保全
17:50-18:20	何道垠	半线性广义Tricomi方程的长时间行为
集体乘车回开封中州国际饭店 18:30-21:00   晚餐（开封中州国际饭店）

2018年6月24日上午 (The Morning of June 24, 2018)

自由讨论

8：30 开封市中州国际饭店大厅集合乘车

11：20 返回中州国际饭店

12：00 午餐

2018年6月24日下午 (The Afternoon of June 24, 2018)

离会

会议报告摘要（Abstract of Conference Talks）

Strong Solutions to Incompressible Navier-Stokes Equations with Navier Boundary Conditions

Shijing Ding

South China Normal University

Abstract: In this talk,we discuss the existence and uniqueness of strong solution to the Navier boundary value problem for the incompressible Navier-Stokes equations in a two dimensional slab domain. Since the solution is defined in an unbounded slab domain,we have to establish the Poincare inequalities and Gargniardo-Nirenberg inequalities for the functions with Navier boundary conditions and divergence free conditions. These inequalities themselves are useful in other applications. This is a joint work with Quanrong Li.

Stochastic strong solutions for stochastic transport equations

Hongjun Gao

Nanjing Normal University
Abstract: We investigate a stochastic transport equation driven by a multiplicative noise. For drift coefficients in $L^q(0,T;{ \mathcal C}^\alpha_b({ \mathbb R}^d))$ ($\alpha>2/q$) and initial data in $W^{1,r}({ \mathbb R}^d)$, we show the existence and uniqueness of stochastic strong solutions. Opposite to the deterministic case where the same assumptions on drift coefficients and initial data may induce
nonexistence of strong solutions, we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. However, for $\alpha+1<2/q$ with spatial dimension higher than one, we can choose proper initial data and drift coefficients so that the stochastic strong solutions do not exist. Moreover, if the drift coefficients belong to $L^q(0,T;W^{1,p}({ \mathbb R}^d))$, we also derive the global integrability for stochastic strong solutions. This responds
to the question raised by Fedrizzi and Flandoli in the case of drift coefficients in $L^q(0,T;L^p({ \mathbb R}^d))$, and we thus partially extend their earlier results.

 

Some Nonlinear Elliptic PDEs Arising in the Chern-Simons-Higgs Theory

Xiaosen Han

Henan University

Abstract: In this talk we review some results about the nonlinear elliptic pdes arising from Chern-Simons-Higgs(CSH) model. First we will recall some elliptic pdes with related existence results arising from the classical Abelian Higgs and CSH models. Then we review the pdes appear in the non-Abelian CSH model and present some existence results. In particular, we report some theorems concerning the existence of doubly periodic solutions, with a sketch of proof. At last, we mention some related open questions.

半线性广义Tricomi方程的长时间行为

何道垠

复旦大学

Abstract: Tricomi方程是一类退化双曲方程，它不但是数学中非常有意义的课题，同时也在物理研究中起到重要的作用。从数学家的观点出发，Tricomi方程可以看作是波方程的推广，具有弯曲的特征锥以及在时间 处的退化。从物理学家的角度来看，Tricomi方程与跨音速气流的研究之间有着密切的关系。

我们考虑以下的Cauchy问题：

我们研究此问题局部解的爆破或整体存在性与指标值的依赖关系。假设，， 其中，。我们确定了临界指标，当时，我们用试验函数方法以及一些对于修正型Bessel函数的技巧导出爆破结果。对于的情况， 我们对于广义Tricomi算子建立了加权及非加权的Strichartz估计。基于这些估计的不等式和压缩映像原理，我们得到了解的全局存在性的证明。

Convergence to Equilibria of Global Solutions to Quasilinear Keller--Segel Systems

Jie Jiang

Wuhan Institute of Physics and Mathematics, CAS

Abstract: In this talk, I will present some recent results on the longtime behavior of global solutions to initial-boundary value problems of the quasilinear Keller--Segel system. We consider this system with non-degenerate as well as degenerate diffusions. For the former case, with the help of a non-smooth version of Lojasiewicz-Simon inequality, we prove that any globally bounded classical solution will converge to an equilibrium as time goes to infinity. If diffusion is degenerate, we consider the typical porous medium case. We establish global existence of weak solutions. In addition, asymptotic behavior was studied via Lojasiewicz-Simon approach.

Some new developments on the forward self-similar solutions of the incompressible Navier-Stokes Equations

Baishun Lai

Henan University

Abstract: In this talk, we first recall the research history on self-similar solutions of the incompressible Navier-Stokes Equations. Secondly, I will introduce our new result about this field. More precisely, by establishing the precisely regularity of solution for the corresponding elliptic system, we prove that the solution constructed by Korobkov-Tsai [Anal. PDE 9 (2016), 1811-1827] satisfies the decay estimate which implies this solution enjoys the same property with that solution was constructed in [Jia and \v{S}ver\'{a}k, Invent. Math. 196 (2014), 233-265]. This is a joint work with Miao Changxing and Zheng xiaoxing

Solitons, Breathers, and rogue waves in ferromagnet

Zaidong Li

Hebei University of Technology

In terms of Darboux transformationwe investigate the dynamic process of spin wave passing through amagnetic soliton. It causes nonlinear excitations, such as Akhmediev breathers solution and Kuznetsov-Ma soliton. The former case demonstrates a spatial periodic process of a magnetic soliton forming the petal with four pieces. The spatial separation of adjacent magnetic petals increases rapidly, while one valley splits into two and the amplitude of valley increases gradually with the increasing amplitude of spin wave. The other case shows a localized process of the spin-wave background. In the limit case, we get rogue waves and clarify its formation mechanism.

We also theoretically investigate the effect of Dzyaloshinskii-Moriya interaction on magnetic soliton in anisotropic ferromagnetic nanowires driven by the adiabatic spin-transfer torque. The Dzyaloshinskii-Moriya interaction changes the formation region of soliton solution and the corresponding phase diagram is given for different soliton solution type.

Asymptotic Model Equations Arising in Shallow Water Theory

Yue Liu

University of Texas at Arlington, USA

Abstract: The study of water waves has a long history starting from Euler in 1752, and continues to be a very active area to the present day. Mathematically, the water wave equations describe the motion of water bounded above by a free surface. This free surface is subject to a constant (atmospheric) pressure, while gravity acts as an external force.
In this talk, I will start by demonstrating the underlying complexity of the physical system,  and then I will  discuss possible simplifications in the "shallow water" regime along with the relevant physical phenomena. In particular, I will focus on the singularity formation of the Cauchy problem for the simplified nonlocal shallow-water models, such as Camassa-Holm-type equations in 1D and 2D cases.

The impact of Noise on ODEs and PDEs

Guangying Lv

Henan University

Abstract: In this talk, we begin with the reason why we consider the SDEs and SPDEs. We first consider the impact of noise on equilibrium point, then we compare the role of nonlinear terms and noise term, lastly we consider the impact of noise on the existence of positive solutions for parabolic equations. Furthermore, we will give the index of solution when the strength of noise goes to infinity.

三维多面体网格上扩散格式

袁光伟

北京应用物理与计算数学研究所

Abstract: 本报告简要介绍三维一般多面体网格上扩散方程单元中心型离散格式的研究进展。

Spherical Symmetric Flows for a Viscous Radiative and Reactive Gas in an Exterior Domain with Large Initial Data

Huijiang Zhao

Wuhan University

Abstract: In this talk, we study the global existence, uniqueness and large-time behavior of spherically symmetric solution of a viscous radiative and reactive gas in an unbounded domain exterior to the unit sphere in for . The key point in the analysis is to deduce certain uniform estimates on the solutions, especially on the uniform positive lower and upper bounds on the specific volume and the temperature.