Conference Title:

Conference on Dispersive Partial Differential Equations

Introduction:

The nonlinear dispersive partial differential equations (PDEs) describe the dispersion phenomena that widely exist in nature, and their applications include quantum physics, optics, condensed matter physics, surface waves, etc. In the past three decades, many breakthroughs have been made in the study of nonlinear dispersive PDEs, while many important open questions still need to be answered. We wish to bring together some experts in these areas to expound the new ideas and to introduce the recent developments to graduate students, postdoctoral researchers and junior faculty members.

The conference will be held online (Tencent Meeting).

Time:

2022.11.12-2022.11.13

Place:

Online (Tencent ID: 70178335658)

Organizing Committee:

Chenjie Fan (AMSS, CAS)

Qingtang Su (MCM, CAS)

Haitian Yue (ShanghaiTech University)

Sponsors:

Institute of Mathematics, Academy of Mathematics and Systems Science, CAS

Morningside Center of Mathematics, CAS

Invited Speakers:

Xing Cheng (Hohai University)

Jiaxi Huang (Beijing Institute of Technology)           

Yang Lan (Tsinghua University)

Baoping Liu (Peking University)

Ruipeng Shen (Tianjin University)

Chengbo Wang (Zhejiang University)

Yifei Wu (Tianjin University)

Guixiang Xu (Beijing Normal University)         

Jianwei Yang (Beijing Institute of Technology)

Kai Yang (Southeast University)

Huali Zhang (Changsha University of Science & Technology)

Zehua Zhao (Beijing Institute of Technology)

Jiqiang Zheng (Institute of Applied Physics and Computational Mathematics)

Schedule: 

Titles and Abstracts:

Speaker: Xing Cheng (Hohai University)

Title: The long time behavior of the nonlinear Klein-Gordon equations (online)

Abstracts: In this talk, we will review some results on the long time behavior of the mass-critical nonlinear Klein-Gordon equations, and then I will talk about our latest progress on the complex-valued mass-critical nonlinear Klein-Gordon equations. 

Speaker: Yifei Wu (Tianjin University)

Title: Low regularity Fourier integrators for some nonlinear dispersive equations

Abstracts: In recent years, more and more attention has been paid to the low regularity numerical study based on the practical needs. In this talk, some Fourier integrators are proposed for solving the KdV equation and the nonlinear Schrodinger equation. The designation of the scheme is based on the exponential-type integration and the Phase-Space analysis of the nonlinear dynamics. By the rigorous analysis, the new schemes provide the first-order or second-order accuracy in Sobolev spaces for rough data, and reduce the regularity requirement of existing methods so far for optimal convergence.

Speaker: Chengbo Wang (Zhejiang University)

Title: Strauss conjecture的一个注记

Abstracts: 我们将讨论经典Strauss conjecture的一个遗留问题以及我们的一个补充注记（与邵科润合作）。时间允许的情况下，我们还将讨论一般时空流形中的相关结果与展望。

Speaker: Huali Zhang (Changsha University of Science & Technology)

Title: Improvement on the rough solutions of 3D compressible Euler equations

Abstracts: For the Cauchy problem of irrotational Euler equations and incompressible Euler equations, the sharp regularity has been obtained for controlling the local existence of solutions. However, the sharp regularity problem for compressible Euler equations with non-trvial vorticity and entropy remains open. The best known 3D result is proved by Qian Wang(2022, Ann. of Math., 195(2), pages 509-654 ). In this talk, we will present an improvent result by using Smith-Tataru's method on quasilinear wave equations. This is a joint work with Lars Andersson.

Speaker: Ruipeng Shen (Tianjin University)

Title: Radiation fields and channel of energy method

Abstracts: Radiation fields and channel of energy method have become powerful tools in the study of global and asymptotic behaviours of solutions to nonlinear wave equations in the recent years. In this talk we deduce the explicit formula of free waves in term of its radiation profile and the explicit formula between radiation profiles in two time directions. As an application we figure out all radial weakly non-radiative free waves and prove the exterior energy estimate of free waves.

Speaker: Kai Yang (Southeast University)

Title: Dynamics of threshold solutions for energy critical NLW with inverse square potential

Abstracts: We consider the focusing energy critical NLW with inverse square potential in dimensions d = 3, 4, 5. Solutions on the energy surface of the ground state are characterized. We prove that solutions with kinetic energy less than that of the ground state must scatter to zero or belong to the stable/unstable manifold of the ground state. In the latter case, they converge to the ground state exponentially in the energy space either as t →∞ or t →−∞. When the kinetic energy is greater than that of the ground state, we show that all solutions with finite mass blow up in finite time in both time directions in d = 3, 4. In d = 5, a finite mass solution

can either have finite lifespan or lie on the stable/unstable manifolds of the ground state. The proof relies on the detailed spectral analysis of the linearized operator, local invariant manifold theory, and global Virial analysis. This is joint work with Prof. Xiaoyi Zhang.

Speaker: Jiaxi Huang (Beijing Institute of Technology) 

Title: Global regularity of skew mean curvature flow for small data

Abstracts: In this talk, I will introduce the skew mean curvature flow, which is an evolution equation for a codimension two immersed submanifold, and moves along the binormal direction with a speed proportional to its mean curvature. I will provide the proof of small data global regularity in low-regularity Sobolev spaces for the skew mean curvature flow in dimensions $d\geq 4$. This extends the local wellposedness result in our previous paper.

Speaker: Baoping Liu (Peking University)

Title: Large time asymptotics for nonlinear Schr\"{o}dinger equation

Abstracts: We consider the Schr\"{o}dinger equation with a general interaction term, which is localized in space. Under the assumption of radial symmetry, and uniformly boundedness of the solution in $H^1(\mathbb{R}^3)$ norm, we prove it is asymptotic to a free wave and a weakly localized part. We derive further properties of the localized part such as smoothness and boundedness of the dilation operator. This is joint work with A. Soffer.

Speaker: Guixiang Xu (Beijing Normal University) 

Title: Minimal mass blow-up solutions for the $L^2$-critical NLS with the Delta potential in one dimension

Abstracts: Local well-posedness theory together with sharp Gagliardo-Nirenberg inequality and the conservation laws of mass and energy implies that the solution with mass less than $\|Q\|_{2}$ is global existence in $H^1(\R)$, where $Q$ is the ground state of the $L^2$-critical NLS without the delta potential.

We are interested in the dynamics of the solution with threshold mass $\|u_0\|_{2}=\|Q\|_{2}$ in $H^1(\R)$. First, for the case $\mu=0$, such blow-up solution exists due to the pseudo-conformal symmetry of the equation,  and is unique up to the symmetries of the equation  in $H^1(\R)$ by F. Merle and recently in $L^2(\R)$ by B. Dodson. Second, for the case $\mu<0$, simple variational argument with the conservation laws of mass and energy implies that radial solutions with threshold mass exist globally in $H^1(\R)$. Last, for the case $\mu>0$, we show the existence of radial threshold solutions with blow-up speed determined by the sign (i.e. $\mu>0$) of the delta potential perturbation since the refined blow-up profile to the rescaled equation is stable in a precise sense.  The key ingredients here including the Energy-Morawetz argument and compactness method as well as the modulation analysis, which are close to Raphael-Szeftel’s argument for inhomogeneous case. This is a joint work with Xingdong Tang (UIST, Nanjing).

Speaker: Zehua Zhao (Beijing Institute of Technology)

Title: On decaying property of NLS

Abstracts: In this talk, we discuss the decaying properties for solutions of NLS. This method can be applied for some other dispersive models. It is based on the recent joint works with C. Fan (AMSS) and G. Staffilani (MIT).

Speaker: Jiqiang Zheng (Institute of Applied Physics and Computational Mathematics)

Title: Dispersive and Strichartz estimate for dispersive equations with scaling-critical electromagnetic potential

Abstracts: In this talk, we study the dispersive equation with Aharonov-Bohm magnetic potential. We prove sharp time-decay estimates in the purely magnetic case, and Strichartz estimates for the complete model, involving a critical electromagnetic field. The novel ingredients are the Schwartz kernels of the spectral measure and heat propagator of the Schrödinger operator in Aharonov-Bohm magnetic fields. In particular, we explicitly construct the representation of the spectral measure and resolvent of the Schrödinger operator with Aharonov-Bohm potentials, and show that the heat kernel in critical electromagnetic fields satisfies Gaussian boundedness. This talk is based on a series of joint works with Xiaofen Gao, Luca Fanelli, Zhiqing Yin and Junyong Zhang.

Speaker: Yang Lan (Tsinghua University)

Title: Strongly interacting multi-soliton for generalized Benjamin-Ono equations

Abstracts: We consider the generalized Benjamin-Ono equation: $\partial_t u-(|D|u-|u|^{p-1}u)_x=0$ with L^2-supercritical power p>3 or L^2-subcritical power 2<p<3. We will construct strongly interacting multi-solitary wave of the form: $\sum_{i=1}^n Q(\cdot-t-x_i (t))$, where the parameters x_i(t) satisfying $x_i(t)-x_{i+1}(t)\sim \sqrt{t}$ as t goes to infinity. We will also prove the uniqueness of such solutions in the case of n=2 and p>3.

Speaker: Jianwei Yang (Beijing Institute of Technology)

Title: Semilinear wave equations with energy supercritical growth

Subtitle: Some recent trends in nonlinear dispersive equations, new equilibria, stability and dynamics

Abstracts: The longtime dynamics of the nonlinear wave equations with energy supercritical growth has a new landscape to explore. Unlike the energy-critical case, very few results are known. In this talk, we shall summarize some recent works on this model. We first consider the classification on the scattering/blow-up dichotomy via the channel of energy method. Then, we study the Cauchy-Dirichlet problem outside an obstacle and show how the boundary drastically influences the picture of the dynamics, by generating a family of stationary solutions. We shall focus on  the known results and some doable open problems around this object. Some of the results are still in progress. joint with Thomas Duyckaerts.