河南省第十二届非线性偏微分方程 学术研讨会

河南省第十二届非线性偏微分方程

学术研讨会日程安排

10月23日

10月24日下午学术报告

地点：B502会议室

10月25日上午学术报告

地点：B502会议室

会议报告摘要

Existence and multiplicity of solutions to semilinear Dirichlet problem for subelliptic operator with a free perturbation

陈化

武汉大学

Abstract

In this talk, we shall report some results on existence and multiplicity for Dirichlet problem of semilinear subelliptic equation with free perturbation term. By using the degenerate Rellich-Kondrachov compact embedding theorem, precise lower bound estimates of Dirichlet eigenvalues for the finitely degenerate elliptic operator and minimax method, we can obtain the existence and multiplicity of weak solutions for the problem.

数理医学中的偏微分方程问题

孔德兴

浙江大学

Properties of positive solutions to nonlinear fractional p-Laplacian equation

李凤泉

大连理工大学

Abstract

In this talk, I will discuss the qualitative properties of solutions to Hardy type problems and Hénon type problems involving the fractional p-Laplacian. First, the symmetry and monotonicity results are proved by the method of moving planes. Finally, as p=2, by a comparison with the first eigenfunction associated with the fractional Laplacian, we obtain a nonexistence result for a Hénon type problem on unbounded domain. This work is joined with Cai Miaomiao.

Global Regularity of the 2D Incompressible Generalized Magneto-micropolar Equations

原保全

河南理工大学

Abstract

In this talk I will present the global regularities of the generalized 2D magneto-micropolar equations with partial magnetic diffusion and fractional dissipation. For the first case the velocity field is ideal, the micro-rotational velocity is with Laplacian dissipation and the magnetic field has fractional partial diffusion $(-partial^{eta}_{22}b_1, -partial^{eta}_{11}b_2)$ with $eta>1$. In the second case, the velocity has a fractional Laplacian dissipation $(-Delta)^{alpha}u$ with any $alpha>0$, the micro-rotational velocity is with Laplacian dissipation and the magnetic field has partial diffusion $(-partial_{22}b_1,-partial_{11}b_2)$. I also present the results for a Cauchy problem of the 2D Leray-$alpha$ regularized incompressible magneto-micropolar equations. The global smooth solution of the Cauchy problem for the equations with zero angular viscosity and zero magnetic diffusion or with only angular viscosity and magnetic diffusion are established.

The cancel property of Green tensor of Stokes system in R^n(or R_+^n), and its application

赖柏顺

河南大学

Abstract

I will introduce two recent results in this talk on some fine properties of Green tensor of Stokes system. First, I will give an alternative proof of cancel property of Green tensor (usually called as Oseen kernel) of Stokes system in R^n, which is more simple and direct. As an important application, we obtain the optimal decay estimate of forward self-similar solutions the 3D incompressible Navier-Stokes Equations, constructed by Korobkov-Tsai. This work is the subsequence to our recent work in [Advance in math 352 (2019), 981-1043]. Secondly, I sketch our another recent result about the pointwise estimates of the Green tensor for the Stokessystem with non-zero external force and zero initial-boundary condition in the half-space. In contrast to the Solonnikov's work, the external force needs not be divergence free. The first work is joint with Changxing Miao and Xiaoxin Zheng, and the second is joint with Kyungkeun Kang, Chen-Chih Lai and Tai-Peng Tsai.

Some problems on the compressible Euler equations of Chaplygin gases and related models

王玉柱

华北水利水电大学

Abstract

In this topic, I shall review and report some resluts on global calssical solutions tothe compressible Euler equations of Chaplygin gases and related models and some open problems are proposed. This talk is based on joint works with Prof. De-Xing Kong, Dr. Changhua Wei.

Liouville-type theorems for generalized Hénon-Lane-Emden Schrödinger systems in R2 and R3

李奎

郑州大学

Abstract

In this talk, we study the Liouville-type theorems for generalized Hénon-Lane-Emden elliptic systems in RN. By the methods of spherical averages, Rellich-Pohozaev type identities, Sobolev inequalities on SN-1, feedback and measure arguments, and scale invariance of the solutions, we show that if the pair of exponents is subcritical, then this system has no positive solutions for N=2 and no bounded positive solutions for N=3.

耦合LPD方程和耦合导数NLS方程的高阶半有理解和怪波对

徐涛

郑州轻工业大学

Abstract

基于达布变变换方法和极限技巧，研究了耦合LPD（Lakshmanan–Porsezian–Daniel）方程两种特殊类型的精确解：针对Lax对空间部分的特征函数存在二重根时，构造了高阶怪波和多亮、暗孤子以及多呼吸子的相互作用；针对特征函数存在三重根时，构造了新型的高阶怪波结构—高阶怪波对。研究了耦合DNLS（导数非线性薛定谔）方程的新型半有理解，分成了三种类型：退化成高阶怪波；一个分量是怪波和振幅变化的孤子，另一个分量是怪波和亮孤子；两个分量都是怪波和呼吸子。这种振幅变化的孤子能够对怪波的预测提供一定的理论指导。

Sensitivity analysis and incompressible Navier-Stokes-Poisson limit of Vlasov-Poisson-Boltzmann equations with uncertainties

张旭

郑州大学

Abstract

For the Vlasov-Poisson-Boltzmann equations with random uncertainties from the initial data or collision kernels, we proved the sensitivity analysis and energy estimates uniformly with respect to the Knudsen number in the diffusive scaling using hypocoercivity method. As a consequence, we also justified the incompressible Navier-Stokes-Poisson limit with random inputs. In particular, for the first time, we obtain the precise convergence rate { without} employing any results based on Hilbert expansion. We not only generalized the previous deterministic Navier-Stokes-Fourier-Poisson limits to random initial data case, also improve the previous uncertainty quantification results to the case where the initial data include both kinetic and fluid parts. This is the first uncertainty qualification (UQ) result for spatially high dimension kinetic equations in diffusive limits containing Navier-Stokes dynamics, and generalizes the previous UQ results which does not contain fluid equations. This is a joint work with Prof. Ning Jiang in Wuhan university.

Well-posedness and attractor on the 2D Kirchhoff–Boussinesq models

冯娜

中原工学院

Abstract

The report studies the well-posedness and the existence of attractors for a class of 2D Kirchhoff-Boussinesq models. We show that: (i) the IBVP of the equations is well-posed in natural energy space and strong solution space, respectively; (ii) the related solution semigroup has a global and an (generalized) exponential attractor provided that the damping parameter k is suitably large; (iii) in particular when γ = 0, the corresponding Boussinesq model has a subclass J of limit solutions and the subclass J has a weak global attractor in energy space without any upper bound restriction for the growth exponent of g(u).

Some recent results on global strong solutions to the Cauchy problem of 1-D compressible MHD equations

叶嵎林

河南大学

Some Regularity Results for the Naiver-Stokes Equations

周道国

河南理工大学

Abstract

First, we report some one scale regularity criteria for the three dimensional Navier-Stokes equations, which improve previous results due to Caffarelli, Kohn, Nirenberg, et al. Then we discuss the regularity of NSE in in the largest critical space.This talk is based on joint works with Prof.Cheng He, Prof.Quansen Jiu, Dr. Zhouyu Li, Prof. Gregory Seregin，Dr.Yanqing Wang, Dr. Gang Wu.

非线性波动方程的长时间动力学行为

丁鹏燕

河南工业大学

Abstract

本报告主要介绍具有分数阶耗散的非线性波动方程在超临界情形下的整体适定性和吸引子的存在性。

Convergence rates for homogenization of nonlinear equations

赵杰

中原工学院

Abstract

In this talk, we will study the convergence rates of solutions for homogenization of quasilinear elliptic equations with the mixed Dirichlet-Robin boundary conditions. Thanks to the smoothing operator as well as homogenization tools, we could handle the different boundary conditions in a uniform fashion. As a consequence, we establish the sharp rates of convergence in H^{1} and L^{2} , which may be regarded as an extension from the classical linear equations Dirichlet or Neumann problems to a nonlinear case with the mixed boundary settings.

Monge-Ampere方程解的无穷远渐近行为

贾小标

华北水利水电大学

Abstract

对于Monge-Ampere方程，全空间时，已有著名的Jogens-Calabi-Pogorelov定理（Loiuville型定理）以及Caffarelli-Yi给出的解无穷远处渐近性为。半空间时，Mooney以及Savin也已给出相应的Loiuville型定理。 本报告主要介绍半空间上Monge-Ampere方程解的无穷远处渐近性为。

Multiple positive solutions for quasilinear elliptic problems with combined critical Sobolev–Hardy terms

李园园

华北水利水电大学

Abstract

In this paper, we investigate the quasilinear elliptic equations involving multiple critical Sobolev–Hardy terms with Dirichlet boundary conditions on bounded smooth domains Ω ⊂ R^N (N ≥ 3), and prove the multiplicity of positive solutions by employing Ekeland’s variational principle and the maximum principle.

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