Special Session 81: Analytic and numerical progress in complex fluids and related PDE models

Global current-vortex sheets in the two-dimensional ideal incompressible MHD

Yuan Cai Fudan University Peoples Rep of China Co-Author(s):    Zhen Lei

Abstract: The magnetohydrodynamic current-vortex sheet is a free boundary problem involving a moving free surface separating two plasma regions. We show the global nonlinear stability of current-vortex sheet in the two dimensional ideal incompressible magnetohydrodynamics under the strong horizontal background magnetic field. This appears to be the first result on the global solutions of the free boundary problems for the ideal (inviscid and non-resistive) incompressible rotational fluids. The strong magnetic field plays a crucial role in the global in time stabilization effect. The proof relies on the understanding of the interplay between the dynamics of the fluids inside the domain and on the free interface, a design of multiple-level energy estimates with different weights, and the inherent structures of the problem. This is based on the joint work with Professor Zhen Lei.

Flux Globalization Based Well-Balanced Path-Conservative Central-Upwind Scheme for Shallow Water Equations

Yangyang CAO Shenzhen MSU-BIT University Peoples Rep of China Co-Author(s):

Abstract: In this work, we propose flux globalization based well-balanced path-conservative central-upwind schemes for several shallow water models. We focus on development of the well-balanced schemes capable of exactly preserving quite complicated steady-state solutions the studied systems admit when the bottom topography is discontinuous. In such cases, nonconservative product terms naturally appear and they require a special treatment. To this end, we incorporate the nonconservative product terms into the global fluxes using the path-conservative technique implemented within a framework of simple---yet highly accurate and robust---Riemann-problem-solver-free central-upwind schemes. This results in new flux globalization based central-upwind schemes, which are more accurate than their existing counterparts. The advantages of the proposed schemes are demonstrated on a number of numerical examples.

The local regularity theory for the Stokes and Navier-Stokes equations near the curved boundary

Hui Chen Zhejiang University of Science and Technology Peoples Rep of China Co-Author(s):    Hui Chen, Su Liang, Tai-Peng Tsai

Abstract: In this talk, we study local regularity of the solutions to the Stokes equations near the curved boundary under no-slip and slip boundary conditions. As an application, we propose a new definition of boundary regular points for the incompressible Navier-Stokes equations that guarantees higher spatial regularity.

Parameter-limit preserving low-regularity method in fluid models: analytic and numerical results

Xinyu Cheng Fudan University Peoples Rep of China Co-Author(s):

Abstract: Among the study of PDEs arising from fluid dynamics and material sciences, parameter stability including vanishing viscosity/damping limits is a very interesting research topic. Indeed, it is deeply related to boundary layers, perturbation theory, and many other areas. However, parameter-limit structure is not always preserved in numerical simulations. On the other hand, solutions with low regularity play an important role in the study of fluid PDEs and are deeply connected to the well-known Onsager conjecture. Therefore, computing such low-regularity solutions while preserving parameter stability is of great significance. In this talk, we will discuss some recent progress in the structure-preserving methods for low-regularity problems arising from fluid models, supported by both analytical and numerical results.

Scattering and energy cascade for the 2D Klein-Gordon-Zakharov

Shijie Dong Southern University of Science and Technology Peoples Rep of China Co-Author(s):    Zihua Guo and Kuijie Li

Abstract: We study the Klein-Gordon--Zakharov system in two spatial dimensions, an important model in plasma physics. For small, smooth, and spatially localized initial data, we establish the global existence of solutions and characterize their sharp long-time behavior, including sharp time decay, scattering, and growth of Sobolev norms. This is joint with Prof. Zihua Guo and Prof. Kuijie Li.

Global well-posedness and stability for the non-resistive magnetohydrodynamical systems

Bin Han Donghua University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we consider the global existence results on the compressible and incompressible magnetohydrodynamics without magnetic diffusivity. The absence of dissipation in the magnetic field and density leads to significant analytical difficulties. To overcome this problem, we introduce the deformation gradient, which reveals an effective dissipative structure relating the magnetic field to the inverse deformation tensor. By developing time-weighted energy estimates, we establish the global well-posedness of classical solutions with small initial data around equilibrium

On the differentiability of local times of ($1+\beta$)-stable super-Brownian motion

Jieliang Hong Southern University of Science and Technology Peoples Rep of China Co-Author(s):    Ziyi Chen

Abstract: We consider the local times of $(1 + \beta)$-stable super-Brownian motion with $0 < \beta < 1$. It is known from Sugitani (J. Math. Soc. Japan, 41(3), 437--462, 1989) that for $\beta = 1$, the local time is differentiable in $d=1$. For $0 < \beta < 1$, Mytnik and Perkins (Ann. Probab., 31(3), 1413--1440, 2003) proved that the local time, denoted by $L(t, x)$, is jointly continuous in $d = 1$ while it is locally unbounded in $x$ in $d \ge 2$ where it exists. This paper strengthens the results of Mytnik and Perkins for $d=1$ by showing that the local time $L(t, x)$ is continuously differentiable in the spatial parameter $x$. Moreover, we give a representation of the spatial derivative, denoted by $\frac{\partial}{\partial x}L(t, x)$, and further prove that the derivative is locally $\gamma$-H\older continuous in $x$ with any index $\gamma \in (0, \frac{\beta}{1+\beta})$.

On self-similar finite-time blowups of the incompressible Euler equations and related models

De Huang Peking University Peoples Rep of China Co-Author(s):

Abstract: It remains an open problem whether the 3D incompressible Euler equations can develop finite-time singularity from smooth initial data in the whole space. In this talk, I will review some most recent results on finite-time blowups mostly with self-similar features, divided into two parts. In the first part, I will talk about traditional self-similar finite-time blowup and the dynamic rescaling method for establishing asymptotically self-similar finite-time blowup. In the second part, I will talk about recent findings on potential self-similar finite-time blowups of the 3D Euler equations with multi-scale features, which are closely related to traveling wave solutions and provide a new approach towards Euler singularity.

Morawetz type estimates for wave/Schr\{o}dinger equations and its applications

Ning-An Lai Zhejiang Normal University Peoples Rep of China Co-Author(s):

Abstract: In this talk, I will talk about some Morawetz type estimates for the wave/Schr\{o}dinger equations. And some related results will be discussed.

Asymptotic behavior for the Maxwell-Dirac system in Lorenz and Coulomb gauges

Kiyeon Lee KAIST Korea Co-Author(s):    Yonggeun Cho

Abstract: In this talk, I will present the nonlinear scattering results for the Maxwell-Dirac system in the Lorenz and Coulomb gauges. In the Lorenz gauge, we obtained the nonlinear scattering results in Cho-Lee (\cite{1}). Using this approach, we also prove nonlinear scattering in the Coulomb gauge. Although the Coulomb gauge lacks Lorentz invariance, it allows the elliptic nature of the scalar potential to be exploited. We introduce a spinorial null structure to overcome the lack of Lorentz invariance and to obtain the desired asymptotic behavior. Moreover, the vector field method combined with various resonance analyses adapted to the Coulomb gauge will be discussed.

On the steady Navier-Stokes equations in a system of channels: existence, uniqueness and rigidity

Xiao Ren Center for Applied Mathematics, Fudan University Peoples Rep of China Co-Author(s):    Giovanni P. Galdi, Filippo Gazzola, Mikhail Korobkov, Gianmarco Sperone

Abstract: The steady motion of a viscous incompressible fluid in a junction of unbounded channels is modeled through the Navier-Stokes equations under inhomogeneous Dirichlet boundary conditions. In contrast to many previous works, the domain is not assumed to be simply-connected and the fluxes are not assumed to be small. In this very general setting, we prove the existence of a solution with a uniformly bounded Dirichlet integral in every compact subset. This is a generalization of the classical Ladyzhenskaya-Solonnikov result obtained under the additional assumption of zero boundary conditions. For small data of the problem we prove the unique solvability and attainability of Couette-Poiseuille flows at infinity. Further, by extending and refining an approach initially introduced by J.B. McLeod, we prove the rigidity of a generic class of Couette-Poiseuille flows, without any restriction on the size of the flux or the periodicity of perturbation. Based on joint works with Giovanni P. Galdi, Filippo Gazzola, Mikhail Korobkov and Gianmarco Sperone.

Neural network methods for non-smooth PDE-constrained optimization

Yongcun Song Nanyang Technological University Singapore Co-Author(s):

Abstract: We present neural network methods for solving non-smooth PDE-constrained optimization problems. Our investigation focuses on three challenging categories: (1) optimization with non-smooth regularization, (2) optimal control of PDEs involving interfaces, and (3) optimal control of elliptic variational inequalities. For each category, we develop tailored neural network algorithms that exploit the specific mathematical structure of the underlying problem. The principal advantages of our methods are that they are mesh-free, thus avoiding grid generation challenges; computationally scalable to high dimensions and complex domains; and straightforward to implement. Extensive numerical experiments demonstrate their computational efficiency and accuracy on benchmark problems

Mixing flow and its applications

Xiaoqian Xu Duke Kunshan University Peoples Rep of China Co-Author(s):

Abstract: In the study of incompressible fluid, one fundamental phenomenon that arises in a wide variety of applications is dissipation enhancement by so-called mixing flow. In this talk, I will give a brief introduction to the idea of mixing flow and examples of such flows. In addition, I will also discuss the un-mixing property of the diffusion process.

Decoupling for degenerate hypersurfaces

Tongou Yang Southern University of Science and Technology Peoples Rep of China Co-Author(s):    Jianhui Li

Abstract: The Fourier decoupling theory developed by Bourgain and Demeter in 2015 has found its applications in harmonic analysis as well as dispersive PDEs. Their proof relies heavily on the nonzero curvature property of the hypersurfaces. We seek to generalise decoupling inequalities to smooth hypersurfaces, by using different parttioning rectangles in the Fourier space.

Boundary Regularity and Global Classical Solution of Dynamic Prandtl Equation

Cheng Yuan School of Mathematical Sciences, Fudan University Peoples Rep of China Co-Author(s):    Hao Jia, Zhen Lei

Abstract: In this talk, I will present the boundary regularity theory and the global well-posedness of regular classical solutions on the dynamic Prandtl equations. To begin, we first establish the up-to-boundary regularity theory for the dynamic Prandtl system. The main obstacle lies in the lack of an explicit expression for the fundamental solution of a certain ultra-parabolic operator in the half-space. Our key strategies in overcoming this regularity issue include identifying the collaboration mechanism between diffusion and transport for a simplified operator, and then combining Fourier analysis, enhanced dissipation theory, and iterative methods to establish a series of hypoelliptic estimates for linear and quasilinear equations. By combining the established boundary regularity with the presented local theory, we also prove the global--in--$t,x$ well-posedness of regular classical solutions for the Prandtl system. This is joint work with Prof. Hao Jia (University of Minnesota) and Prof. Zhen Lei (Fudan University).