非线性偏微分方程系列报告日程安排

2022年10月21日上午

（腾讯会议：ID 631-990-282）­­­

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2022年10月21日下午

组织者：

蒲学科，15876515526

王光武，18810995968

报告摘要

Well-posedness for the initial boundary problem of the generalized derivative nonlinear Schr\"odinger equation on the half-line for any large initial data

霍朝辉副研究员（中国科学院数学与系统科学研究院）

摘要：We cansider the well-posedness of the initial boundary for the generalized derivative nonlinear Schr\"odinger equation on the half-line

$$u_{t}-iu_{xx}=F( u,\bar{u}, u_x,\bar{u}_x), \ \ x\geq0;$$

where $F : \C^{4} \rightarrow \C $ is a polynomial with no constant or linear terms and no quadratic terms. We can show that the initial boundary problem is locally well-posed in $ C([-T,T]:H_x^{s})\bigcap C_x((0,\infty): H_t^{(2s+1)/4})$ with $s>3/2$ for any large initial data.

Moreover, using the above method, we can show that the Cauchy problem of the derivative nonlinear Schr\"odinger equation on the real line

$$u_{t}-iu_{xx}=F( u,\bar{u}, u_x,\bar{u}_x), \ \ (x,t) \in \R \times \R;$$

is locally well-posed in $ C([-T,T]:H_x^{s}) $ with $s>3/2$ for any large initial data.

On the attractors of primitive equations of the large-scale atmosphere and ocean

黄代文研究员（北京应用物理与计算数学研究所）

摘要：In this talk, we give some results on the attractors of primitive equations

of the large-scale ocean. Firstly, we recall the global

well-posedness and long-time dynamics for the viscous primitive

equations describing the large-scale oceanic motion . Secondly, we introduce some results on the global attractros of

primitive equations, such as the enhanced pullback attractors of 3D Primitive Equations.

Global solution for equations governing the low-frequency ion motion in plasma

张景军教授（嘉兴学院）

摘要：We consider the equations describing the interactions between Langmuir waves and the low-frequency response of ions. Using the analysis of higher order energy estimate and lower order decay estimate, existence of global smooth solution is established for suitably small initial data.

The Effect of Signal-dependent Motility in a Keller--Segel System of Chemotaxis

江杰副研究员（中国科学院精密测量科学与技术创新研究院）

摘要：In this talk, we would like to report our recent work on a Keller—Segel system of chemotaxis involving signal-dependent motility. This model was originally proposed by Keller and Segel in their seminal work in 1971, and has been used to provide a new mechanism for pattern formation in some recent Bio-physics work published in Science and PRL.

From a mathematical point of view, the model features a non-increasing signal-dependent motility function, which may vanish as the concentration becomes unbounded, leading to a possible degenerate problem. We develop systematic new methods to study the well-posedness problem. The key idea lies in an introduction of an elliptic auxiliary problem which enables us to apply delicate comparison arguments to derive the upper bound of concentration. Moreover, new iteration as well as monotonicity techniques are developed to study the global existence of classical solutions and their boundedness in any dimension. It is shown that the dynamic of solutions is closely related to the decay rate of the motility function at infinity. In particular, a critical mass phenomenon as well as an infinite-time blowup was verified in the two-dimensional case if the motility is a negative exponential function.

The talk is based on my recent joint works with Kentarou Fujie (Tohoku University), Philippe Laurençot (University of Toulouse and CNRS), Yanyan Zhang (ECNU), and Yamin Xiao (IAPCM).

Finite volume method for distributed-order diffusion-advection equations

李景副教授（长沙理工大学）

摘要：In this talk, we investigate the finite volume method (FVM) for a distributed-order spacefractional advection–diffusion (AD) equation. The mid-point quadrature rule is used to approximate the distributed-order equation by a multi-term fractional model. Next, the transformed multi-term fractional equation is solved by discretizing in space by the finite volume method and in time using the Crank–Nicolson scheme. We use a novel technique to deal with the convection term, by which the Riesz fractional derivative of order 0 < γ < 1 is transformed into a fractional integral form. An important contribution of our work is the use of nodal basis function to derive the discrete form of our model. The unique solvability of the scheme is also discussed and we prove that the Crank–Nicolson scheme is unconditionally stable and convergent with second-order accuracy.

The robust inverse scattering method for focusing Ablowitz–Ladik equation on the non-vanishing background

凌黎明教授（华南理工大学）

报告摘要： In this talk, we consider the robust inverse scattering method for the Ablowitz–Ladik (AL) equation on the non-vanishing background, which can be used to deal with arbitrary-order poles on the branch points and spectral singularities in a unified way. The Darboux matrix is constructed with the aid of loop group method and considered within the framework of robust inverse scattering transform. Various soliton solutions are constructed without using the limit technique. These solutions include general soliton, breathers, as well as high order rogue wave solutions. (joint with Y. Chen and B.-F. Feng)

Some new results of the d-D Euler equations

吴兴龙教授（武汉理工大学）

摘要：In this talk, we will recall some new progress in theory for the d-D Euler equations (such as global existence and blow-up), and present the relation between regularity of solution and energy conservation for the inhomogeneous incompressible and compressible Euler equations.

On the Boussinesq-MHD system

边东芬副教授（北京理工大学）

报告摘要: In this talk, we will show the stability for the Boussinesq-MHD system with partial dissipation. This is based on joint works with Xintong Ji, Jingjing Mao and Xueke Pu.

A Nonhomogeneous Initial Boundary-Value Problem for the Hirota Equation Posed on the Half Line.

郭春晓教授（中国矿业大学（北京））

报告摘要: We study a system described by a class of initial and boundary value problem (IBVP) of the Hirota equation posed on a half line with nonhomogeneous boundary conditions. In particular, using an explicit solution formula and contraction mapping ethod, we prove the local well-posedness of the IBVP in the Sobolev space $H^s (R^+)$ for any $s\geq 0$, and then we obtain the global well-posedness by the energy estimates of solution. The main difficulties of this model are caused by that the characteristic equation corresponding to Hirota equation is complicated and needs to be solved by construction, beyond that the Kato smoothness of the nonlinear terms $i\gamma(|u|^2u)_x$ and $|u|^2u$ are taken into consideration.

Optimal decay of compressible Navier-Stokes equations with

or without potential force

高金城副教授（中山大学）

摘要：In this talk, we investigate the optimal decay rate for the higher orderspatial derivative of globalsolution to the compressible Navier-Stokes equations with or without potential force in three-dimensional whole space. We will show that the higher order derivative of global solution will decay faster than the lower one.