Special Session 123: New trends in elliptic and parabolic PDEs

Green functions for stationary Stokes systems in two dimensions

Jongkeun Choi Pusan National University Korea Co-Author(s):

Abstract: This talk presents Green functions for the stationary Stokes system with measurable coefficients in two dimensions, along with a unified approach to constructing Green functions for both elliptic and Stokes systems with various boundary conditions in two-or higher-dimensional domains.

The Stokes System on Convex Domains

Jun Geng Lanzhou University Peoples Rep of China Co-Author(s):    Zhongwei Shen

Abstract: We investigate the Neumann problem for Stokes system on convex domain $\Omega$. The $L^p$ solvability and $W^{1,p}$ solvability are obtained for certain ranges of $p$. The ranges of $p$ are sharp for $d=2$ and these intervals are larger than the known interval on Lipschitz domain.

Conformal metrics of constant scalar curvature with unbounded volumes

Liuwei Gong The Chinese University of Hong Kong Hong Kong Co-Author(s):    Yanyan Li

Abstract: When $n>24$, Brendle and Marques constructed a smooth metric on $S^n$ such that there exists a sequence of conformal metrics with the same positive constant scalar curvature but with unbounded Ricci curvatures. We find a ``worse`` blowup phenomenon when $n>24$: a smooth metric on $S^n$ such that there exists a sequence of conformal metrics with the same positive constant scalar curvature but with unbounded volumes (and, in particular, unbounded Ricci curvatures). This is a joint work with Yanyan Li.

Concentration of weak solutions in compressible flows

Xianpeng Hu The Hong Kong Polytechnic University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we will discuss some recent progress in the mathematical analysis of weak solutions for compressible flows. Global existences of weak solutions will be the main subject. The oscillation and concentration of approximating solutions are two main obstacles.

Blow up phenomena of mean field type equations

Yeyao Hu Central South University Peoples Rep of China Co-Author(s):    Bartolucci, Cheng, Gu, Gui, Jevnikar, Li, Xie and Yang

Abstract: We constructed blow-up solutions of mean field type equations on degenerate surfaces, including flat torus and standard sphere in dimension two. More recently, we built the blow-up solutions of mean field equation on four dimensional standard sphere, basing on point configurations inherited from the dimension two. New progress towards the planar Neumann problem will also be mentioned, particularly the construction of boundary and interior bubble assemblies.

Parabolic equations with a half-order time derivative and their application to boundary value problems

Doyoon Kim Korea University Korea Co-Author(s):    Pilgyu Jung, Hongjie Dong

Abstract: We present parabolic equations in divergence form that include a half-order time derivative term on the right-hand side. We discuss the motivation for considering such equations, particularly their usefulness (or necessity) when dealing with parabolic equations in divergence form with highly irregular coefficients or domains. As an application, we demonstrate the $L_p$ solvability of parabolic equations with the conormal derivative boundary condition in very irregular domains, assuming the coefficients are only measurable in time. A key challenge to solvability, when obtaining necessary estimates, is the presence of the time derivative of the solution on the right-hand side, which may lack sufficient regularity to belong to $L_p$ spaces. To address this, we reformulate the term as the half-order time derivative of a function in $L_p$ spaces and employ the aforementioned results.

Harnack inequality for parabolic equations in double-divergence form with singular lower order coefficients

Seick Kim Yonsei University Korea Co-Author(s):    Istvan Gyongy

Abstract: This paper investigates the Harnack inequality for nonnegative solutions to second-order parabolic equations in double divergence form. We impose conditions where the principal coefficients satisfy the Dini mean oscillation condition in , while the drift and zeroth-order coefficients belong to specific Morrey classes. Our analysis contributes to advancing the theoretical foundations of parabolic equations in double divergence form, including Fokker-Planck-Kolmogorov equations for probability densities.

Some Liouville type theorems about Q-curvature

Mingxiang Li Chinese University of Hong Kong Hong Kong Co-Author(s):    Juncheng Wei

Abstract: In this talk, I will introdue a Case-Gursky-V\`etois formula on compact Einstein manifolds and make use of such identity to estbalish some Liouville type theorems on compact Einstein manifolds related to Paneitz operator. This is a joint work with Prof. Juncheng Wei.

Sobolev estimates for degenerate linear equations on the upper half space

Junhee Ryu Korea University Korea Co-Author(s):    Hongjie Dong

Abstract: In this talk, we present both divergence and nondivergence degenerate equations on the upper half space. The coefficient matrices of the equations are the product of $x_d^2$ and bounded uniformly elliptic matrices. Under a partially weighted mean oscillation assumption on the coefficients, we obtain the wellposedness and regularity of solutions in weighted Sobolev spaces. This talk is based on a joint work with Hongjie Dong.

The scaling limit of the continuous solid-on-solid model

Wei Wu NYU Shanghai Peoples Rep of China Co-Author(s):    Scott Armstrong

Abstract: We prove that the scaling limit of the continuous solid-on-solid model in Zd is a multiple of the Gaussian free field, based on methods from degenerate stochastic homogenization.

Resonant modes of two hard inclusions within a soft elastic material and their stress estimates

Longjuan Xu Capital Normal University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we will discuss subwavelength resonant modes of two hard inclusions embedding in soft elastic materials to realize negative materials in elasticity. We will show that the resonant modes are categorized into dipolar, quadrupolar, and hybrid groups, facilitating the effective realization of negative mass density, negative shear modulus and double-negative properties in elastic metamaterials. Moreover, we will analyze the stress distribution between two hard inclusions when they are closely touching.

On the fast growth of some active scalar equations

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

Abstract: In this talk, we discuss the small scale creation and the optimal growth in PDEs of fluid mechanics, especially the Euler equations and the related models. We first give a clear and understandable picture of so-called hyperbolic flow restricted in 1D. Then, we will look into the hyperbolic flow in 2D, based on the study of Green functions of elliptic equations.

Scale separation in multiscale elliptic homogenization

Jinping Zhuge Academy of Mathematics and Systems Science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):    Weisheng Niu

Abstract: Multiscale homogenization is a mathematical method used to analyze partial differential equations (PDEs) in heterogeneous media with coefficients that vary on multiple oscillating scales. The classical homogenization theory, which addresses PDEs with a single oscillating scale, has been well-established so far. The multiscale homogenization is more complicated due to the interactions between different oscillating scales, particularly when these scales are not separated. In this talk, I will discuss a new scale separation idea in multiscale elliptic homogenization by using the simultaneous Diophantine approximation from number theory.