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Young Scholars' Lectures in Analysis and PDE (online seminar)

TIME:June-4 ,2020
REGISTRATION DEADLINE:No Need to Register
LOCATION:LOCATION:Zoom online conference
    Description
    Speaker: Guilong Gui (Northwest University)    
    Time: 14:00-15:00   June 4 ,2020
    Title: Semiclassical limit of Gross-Pitaevskii equation with Dirichlet boundary condition

    Abstract: In this talk, we justify the semiclassical limit of Gross-Pitaevskii equation with Dirichlet boundary condition on the 3-D upper space under the assumption that the leading order terms to both initial amplitude and initial phase function are sufficiently small in some high enough Sobolev norms. We remark that the main difficulty of the proof lies in the fact that the boundary layer appears in the leading order terms of the amplitude functions and the gradient of the phase functions to the WKB expansions of the solutions. In particular, we partially solved the open question proposed in [Chiron and Rousset2009, Pham, Nore and Brachet2005] concerning the semiclassical limit of Gross-Pitaevskii equation with Dirichlet boundary condition. This is a joint work with Prof. Ping Zhang.

     

    Speaker: Weixi Li (Wuhan University)

    Time:15:15- 16:15   June 4 ,2020
    Title:Well-posedness in Gevrey function space for 3D Prandtl equations without Structural Assumption

    Abstract:We establish the well-posedness in Gevrey function space with optimal class of regularity 2 for the three dimensional Prandtl system without any structural assumption. The proof combines in a novel way a new cancellation in the system with some of the old ideas to overcome the difficulty of the loss of derivatives in the system. This shows that the three dimensional instabilities in the system leading to ill-posedness are not worse than the two dimensional ones



    To join ZOOM Conference:

    https://zoom.com.cn/j/61862377513?pwd=YThOeWJwZkczN0crRHYzaS9RUFE4UT09

    Conference ID:618 6237 7513

    Password: PKU_MATH



    Speaker:Yong Lv(Nanjing University)

    Time:14:00-15:00   June 11,2020

    Title:Space-time resonances and high-frequency Raman instabilities in the two-fluid Euler-Maxwell system

    Abstract:We apply the symbolic flow method to the two-fluid Euler-Maxwell system, and show that space-time resonances induce high-frequency Raman and Langmuir instabilities. A consequence is that the Zakharov WKB approximation to Euler-Maxwell is unstable for non-zero group velocities. A key step in the proof is the reformulation of the set of resonant frequencies as the locus of weak hyperbolicity for linearized equations around the WKB solution. Due to large transverse variations in the WKB profile, the equation satisfied by the symbolic flow around resonant frequencies is a linear  partial differential equation. At space-time resonances corresponding to the Raman or Langmuir instability, we observe a fast growth of the symbolic flow, which translates into an instability result for the original system. 

    Speaker:Liming Ling(South China University of Technology)

    Time:15:15- 16:15   June 11 ,2020

    Title: Infinite order rogue waves of the focusing nonlinear Schrödinger equations
    Abstract:In this talk, we would like to introduce the fundamental rogue wave solutions of the focusing nonlinear Schrödinger equation in the limit of large order. Using a recently proposed Riemann–Hilbert representation
    of the rogue wave solution of arbitrary order k, we establish the existence of a limiting profile of the rogue wave in the large-k limit when the solution is viewed in appropriate rescaled variables capturing the near-field region where the solution has the largest amplitude. The limiting profile is a new particular solution of the focusing nonlinear Schrödinger equation in the rescaled variables—the rogue wave
    of infinite order—which also satisfies ordinary differential equations with respect to space and time. The spatial differential equations are identified with certain members of the Painlevé-III hierarchy. We compute the far-field asymptotic behavior of the near-field limit solution and compare the asymptotic formulas with the exact solution
    using numerical methods for solving Riemann–Hilbert problems.  In a certain transitional region for the asymptotics, the near-field limit function is described by a specific globally defined tritronquée solution of the Painlevé-II equation. These properties lead us to regard the rogue wave of infinite order as a new special function.(joint with Deniz Bilman and Peter D. Miller)

    Organizer

     Zhifei Zhang  Chao Wang, Di Wu, Lan Zeng, Jianfeng Zhou


    To join ZOOM Conference:

    https://zoom.com.cn/j/6858797481?pwd=V3g2cUduUTJ5alNYeVRIZ1djVUxDUT09

    Conference ID:685 879 7481


    Password: PKU_MATH



    Speaker: Huanyao Wen (South China University of Technology)
    Time:14:00-15:00   June 18 ,2020
    Title:Global existence of weak solution to a two-fluid model with large initial data

    Abstract:We shall talk about the global existence of weak solutions of Dirichlet problem for a one-velocity viscous two-fluid model. By relying on weak compactness tools we obtain existence results within the class of weak solutions in one dimension and in high dimensions respectively. An essential novel aspect of the analysis, compared with previous works, is that the solution space is large enough to allow for transition to single-phase flow without any constraints in one dimension. In  multi-dimensions, similar result can be obtained subject to some constraints for the adiabatic index. 


    Speaker: Chengjie Liu(shanghai Jiaotong University)
    Time:15:15-16:15   June 18 ,2020
    Title:Stabilization effect of magnetic field in the inviscid limit for MHD equations on the half space
     
    Abstract:In this talk, I will discuss the stabilization effect of magnetic field in the inviscid limit for MHD equations on the half space with non-slip boundary condition for the velocity.I will talk about two cases of the problem to show respectively that the tangential component and the normal component of magnetic field have stabilization effect in the inviscid limit process. It’s bases on the joint work with Prof. Feng Xie and Tong Yang.

    To join ZOOM Conference:
    https://zoom.com.cn/j/6858797481?pwd=V3g2cUduUTJ5alNYeVRIZ1djVUxDUT09
    ID:685 879 7481

    Password: PKU_MATH


    Speaker:  Jiqiang Zheng(Institute for Applied Physics and Computational Mathematics)
    Time:14:00-15:00   June 25 ,2020
    Title:Laplacian operator with Hardy potential and applications to nonlinear dispersive equations

    Abstract: In this talk, we first discuss the Sobolev space theory and harmonic analysis tools(such as Littlewood-Paley theory) for the Laplacian operator associated with Hardy potential. And then we consider the energy-critical nonlinear wave equation in the presence of an inverse-square potential in dimensions three and four.  In the defocusing case, we prove that arbitrary initial data in the energy space lead to global solutions that scatter. In the focusing case, we prove scattering below the ground state threshold. This talk is based on a series of joint works with Rowan Killip, Changxing Miao, Jason Murphy, Monica Visan and Junyong Zhang.

    Speaker:  Yong Wang (CAS)
    Time:15:15-16:15   June 25 ,2020
    Title:Global Solutions of the Compressible Euler Equations with Large Initial Data of Spherical Symmetry and Positive Far-Field Density
    Abstract:We are concerned with the global existence theory for spherically symmetric solutions of the multidimensional compressible Euler equations with large initial data of positive far-field density so that the total initial-energy is unbounded. The central feature of the solutions is the strengthening of waves as they move radially inward toward the origin. For the large initial data of positive far-field density, various examples have shown that the spherically symmetric solutions of the Euler equations blow up near the origin at certain time. A fundamental unsolved problem is whether the density of the global solution would form concentration to become a measure near the origin for the case when the total initial-energy is unbounded. Another longstanding problem is whether a rigorous proof could be provided for the inviscid limit of the multidimensional compressible Navier-Stokes to Euler equations with large initial data. In this paper, we establish a global existence theory for spherically symmetric solutions of the compressible Euler equations with large initial data of positive far-field density and relative finite-energy. This is achieved by developing a new approach via adapting a class of degenerate density-dependent viscosity terms, so that a rigorous proof of the vanishing viscosity limit of global weak solutions of the Navier-Stokes equations with the density-dependent viscosity terms to the corresponding global solution of the Euler equations with large initial data of spherical symmetry and positive far-field density can be obtained. One of our main observations is that the adapted class of degenerate density-dependent viscosity terms not only includes the viscosity terms for the Navier-Stokes equations for shallow water (Saint Venant) flows but also, more importantly, is suitable to achieve our key objective of this paper. These results indicate that concentration is not formed in the vanishing viscosity limit even when the total initial-energy is unbounded, though the density may blow up near the origin at certain time. The talk is based on a joint work with Gui-Qiang Chen.


    To join the Zoom conference:
    https://zoom.com.cn/j/6858797481?pwd=V3g2cUduUTJ5alNYeVRIZ1djVUxDUT09

    ID:685 879 7481
    Password:PKU_MATH

    Organizer

     Zhifei Zhang  Chao Wang, Di Wu, Lan Zeng, Jianfeng Zhou





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