Special Session 5: Recent developments in Partial Differential Equations from Physics

The effective medium generated by a cluster of highly contrasting nanoparticles with periodic and nonperiodic distribution

Xinlin Cao The Hong Kong Polytechnic University Peoples Rep of China Co-Author(s):    Ahcene Ghandriche and Mourad Sini

Abstract: We derive the electromagnetic medium equivalent to a collection of all-dielectric nano-particles (enjoying high refractive indices) distributed periodically in a smooth domain. Then we extend and generalize the study to the medium with Van-Der-Waals Heterostructure possessing locally non-periodic distribution. For the periodic distribution case, we figure out that the effective medium is an alteration of the permeability that keeps the permittivity unchanged, which provides regimes under which the effective permeability can be positive or negative valued. For the heterostructure case, we build up a local distribution tensor, which models the local strong interaction of the nano-particles. To our best knowledge, such tensors are new in both the mathematical and engineering oriented literature.

Well-posedness for local and nonlocal quasilinear evolution equations in fluids and geometry

Ke Chen The Hong Kong Polytechnic University Hong Kong Co-Author(s):    Ruilin Hu, Quoc-Hung Nguyen

Abstract: In this talk, I will present some recent results on local and global well-posedness results of some nonlocal and nonlinear evolution equations. These results are based on a Schauder type estimate for general nonlocal parabolic equations.

Incompressible limit of compressible viscoelastic system with vanishing shear viscosity

Xiufang Cui Lanzhou University Peoples Rep of China Co-Author(s):

Abstract: We study the vanishing shear viscosity limit and the incompressible limit of compressible viscoelastic system near the equilibrium. The large value of volume viscosity forces the limit system to be incompressible and the vanishing shear viscosity indicates that the limit system is inviscid. The generalized energy estimate and the ghost weight method are used to guarantee the global convergence from compressible viscoelastic system to the incompressible elastodynamics.

Steady compressible Navier-Stokes-Fourier system with slip boundary conditions arising from kinetic theory

Renjun Duan The Chinese University of Hong Kong Hong Kong Co-Author(s):    Junhao Zhang

Abstract: This talk concerns the boundary value problem on the steady compressible Navier-Stokes-Fourier system in a channel domain $(0,1)\times\mathbb{T}^2$ with a class of generalized slip boundary conditions that were systematically derived from the Boltzmann equation by Coron [JSP, 1989] and later by Aoki et al. [JSP, 2017]. We establish the existence and uniqueness of strong solutions in $(L_{0}^{2}\cap H^{2}(\Omega))\times V^{3}(\Omega)\times H^{3}(\Omega)$ provided that the wall temperature is near a positive constant. The proof relies on the construction of a new variational formulation for the corresponding linearized problem and employs a fixed point argument. The main difficulty arises from the interplay of velocity and temperature derivatives together with the effect of density dependence on the boundary.

Extrinsic Derivative Formula for Distribution Dependent SDEs

Panpan P Ren City University of Hong Kong Hong Kong Co-Author(s):

Abstract: To characterize the regularity of nonlinear Fokker-Planck equations with respect to weighted variational distances, we establish for the first time a Bismut type formula for the extrinsic derivative of distribution dependent SDEs (DDSDEs). As an application, the Lipschitz continuity in the weighted variational distance is derived for the associated nonlinear Fokker-Planck equation, which can be regarded as the counterpart of the classical contraction property in the linear setting. The main results are illustrated by non-degenerate DDSDEs with space-time singular drift, as well as degenerate DDSDEs with weakly monotone coefficients.

Ill/well-posedness of non-diffusive active scalar equations with physical applications

Anthony Suen The Education University of Hong Kong Hong Kong Co-Author(s):    Susan Friedlander and Fei Wang

Abstract: We consider a general class of non-diffusive active scalar equations with constitutive laws obtained via an operator $\mathcal{T}$ that is singular of order $r_0\in[0, 2]$. We obtain ill/well-posedness results for various values for $r_0$. We then apply the results to several physical problems including the magnetogeostrophic equation, the incompressible porous media equation and the singular incompressible porous media equation. This is a joint work with Susan Friedlander and Fei Wang.

Regularity structure and asymptotic behavior of energy conservative solutions to the Hunter-Saxton equation

Tak Kwong Wong The University of Hong Kong Hong Kong Co-Author(s):    Yu Gao and Hao Liu

Abstract: The Hunter-Saxton equation is an integrable equation, and can be used to study the nonlinear instability in the director field of a nematic liquid. In this talk, we will first introduce a new generalized framework for energy conservative solutions of the Hunter-Saxton equation, and then discuss how to apply this new framework to investigate the regularity structure and long-time behavior of these solutions. In particular, some new observations have been found and rigorously shown: (i) singularities for the energy measure may only appear at at most countably many times, and are completely determined by the absolutely continuous part of initial energy measure; (ii) the temporal and spatial locations of singularities are explicitly determined by the initial data; and (iii) the long-time behavior of energy conservative solution is given by a kink-wave that is determined by the total energy of the system only.

Long-time behavior to the 3D isentropic compressible Navier-Stokes equations

Guochun Wu Huaqiao University Peoples Rep of China Co-Author(s):    Guochun Wu and Xin Zhong

Abstract: We are concerned with the long-time behavior of classical solutions to the isentropic compressible Navier-Stokes equations in $\mathbb R^3$. Under the assumption that the density is uniformly bounded, we establish the global stability of classical solutions to the isentropic compressible Navier-Stokes equations. The main ingredient of the proof relies on the techniques involving blow-up criterion, a key time-independent positive upper and lower bounds of the density, and a regularity interpolation trick.

Transonic Shock with Large Swirl Velocity in a Finite Cylinder

Wei XIANG City University of Hong Kong Hong Kong Co-Author(s):

Abstract: In this talk, we introduce our recent work on the existence and unique location of the three-dimensional transonic shock with large swirl velocity for axisymmetric full Euler equations in a cylinder, with appropriate boundary conditions at the entrance of the nozzle and the receiver pressure at the exit of the nozzle. It is the first mathematical result on the three-dimensional transonic shock with large swirl velocity. We investigate the important role of the non-zero swirl in uniquely determining the shock location.

Long time instability of compressible planar Poiseuille flows

Andrew Yang City University of Hong Kong Hong Kong Co-Author(s):    Zhu Zhang

Abstract: It is well-known that at high Reynolds numbers, the linearized Navier-Stokes equations around the inviscid stable shear profile admit growing mode solutions due to the destabilizing effect of small viscosities. This phenomenon, which is related to Tollmien-Schlichting instability, has been rigoriously justified by Grenier-Guo-Nguyen [Adv. Math. 292 (2016); Duke J. Math. 165 (2016)] on incompressible Navier-Stokes equations. In this work, we aim to construct the Tollmien-Schlichting waves for the compressible Navier-Stokes equations over symmetric shear flows in a channel. We will also discuss the effect of temperature fields on the stability of these shear flows.

The 3D kinetic Couette flow via the Boltzmann equation in the diffusive limit

Anita Yang The Chinese University of Hong Kong Hong Kong Co-Author(s):    Renjun Duan, Shuangqian Liu, Robert M. Strain

Abstract: In this talk, we will study the Boltzmann equation in the diffusive limit in a channel domain $\T^2\times (-1,1)$ for the 3D kinetic Couette flow. Our results demonstrate that the first-order approximation of the solutions is governed by the perturbed incompressible Navier-Stokes-Fourier system around the fluid Couette flow. Moreover, in the absence of external forces, the 3D kinetic Couette flow asymptotically converges over time to the 1D steady planar kinetic Couette flow. This is a joint work with Prof. Renjun Duan, Prof. Shuangqian Liu and Prof. Robert M. Strain.

Pattern formation in Landau-de Gennes theory

Yong Yu The Chinese University of Hong Kong Hong Kong Co-Author(s):    Ho Man Tai

Abstract: In this talk, I will introduce the spherical droplet problem in the Landau-de Gennes theory. With a novel bifurcation diagram, we find solutions with ring and split-core disclinations, respectively. This work theoretically confirms the numerical results of Gartland and Mkaddem in 2000.

Some global existence and uniqueness of the strong solution for the multi-dimensional viscoelastic flows

Ting Zhang Zhejiang University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we mainly focus on the multi-dimensional viscoelastic flows of Oldroyd-B type. Considering a system of equations related to the compressible viscoelastic fluids of Oldroyd-B type with the general pressure law, $P^\prime(\bar{\rho})+\alpha>0$, with $\alpha >0$ being the elasticity coefficient of the fluid, we prove the global existence and uniqueness of the strong solution in the critical Besov spaces when the initial data $u_0$ and the low frequency part of $\rho_0$, $\tau_0$ are small enough compared to the viscosity coefficients. The proof we display here does not need any compatible conditions. In addition, we also obtain the optimal decay rates of the solution in the Besov spaces. At Last, considering the multi-dimensional compressible Oldroyd-B model, which is derived by Barrett, Lu, and Suli (Comm. Math. Sci. 2017) through the micro-macro analysis of the compressible Navier-Stokes-Fokker-Planck system in the case of Hookean bead-spring chains. We would provide a unified method to study the system with the background polymer number density $\eta_\infty\geq0$, including the vanishing case and the nonvanishing case, and establish the global-in-time existence of the strong solution for the associated Cauchy problem when the initial data are small in the critical Besov spaces.

Nonlinear stability of entropy waves for the Euler equations

Wenbin Zhao Renmin University of China Peoples Rep of China Co-Author(s):

Abstract: In this talk, we consider a special class of the contact discontinuity in the full compressible Euler equations, namely the entropy wave, where the velocity is continuous across the interface while the density and the entropy can have jumps. By deriving the evolution equation of the interface in the Eulerian coordinates, we relate the Taylor sign condition to the hyperbolicity of this evolution equation, which yields a stability condition for the entropy waves. With the optimal regularity estimates of the interface, we can derive the a priori estimates without loss of regularity.