Special Session 10: Recent Developments in Regularity Theory for PDEs

Entire solutions and asymptotic behavior to a class of parabolic k-Hessian equations

Ning Cao Southeast University Peoples Rep of China Co-Author(s):    Cao Ning and Jiang Feida

Abstract: In this talk, we make a systematic investigation of the existence, uniqueness and nonexistence of entire separable variable radial solutions to the following class of parabolic k-Hessian equations with the parameter: $$ -u_t [S_k(D^2u)]^{\alpha}=1 $$ We also study the asymptotic behavior and its refined version of the solution at infinity. We particularly note that our results fully generalize the result of An, Bao, et al. in Nonlinear Anal. 239:113441 (2024) for the parabolic Monge-Amp{\`e}re equation in the form of $-u_t \det D^2u=1$. The main tools employed in this work comprise Euler`s broken line method, local boundedness estimate, Keller-Osserman type criteria, the generalized L`Hospital`s rule, and asymptotic stability analysis.

Stationary Stokes systems in non-divergence and double divergence form

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

Abstract: We study stationary Stokes systems in non-divergence form with DMO coefficients and data. We prove that if $(u, p)$ is a strong solution of the system, then $(D^2u, \nabla p)$ is continuous. The corresponding boundary regularity result on a $C^{2, \rm Dini}$ domain is also established. To this end, we introduce the adjoint system associated with the non-divergence form Stokes system, which is formulated in double divergence form.

The hydrodynamic limit of the discrete-velocity BGK Boltzmann equation

Zhongyang Gu Shenzhen University Peoples Rep of China Co-Author(s):    Xin Hu, Tsuyoshi Yoneda

Abstract: In this talk, we will present the construction of a global weak solution to the $3$-dimensional incompressible Navier-Stokes equations through the study of the hydrodynamic limit of the discrete-velocity BGK Boltzmann equation, provided the smallness of the initial data. In particular, the derivation of the diffusion term from the microstructure will be explained. This talk is based on a joint work with Xin Hu (Wuhan University) and Tsuyoshi Yoneda (Hitotsubashi University).

The free boundary for a superlinear system

Seongmin Jeon Hanyang University Korea Co-Author(s):    Daniela De Silva, Henrik Shahgholian

Abstract: In this talk, we discuss superlinear systems that give rise to free boundaries. Such systems appear for example from the minimization of the energy functional $$ \int_{\Omega}\left(|\nabla\mathbf{u}|^2+\frac2p|\mathbf{u}|^p\right),\quad 0

Dirichlet problem and regular boundary points for elliptic equations in non-divergence and double divergence form

Dong-ha Kim Research Institute of Mathematics, Seoul National University. Korea Co-Author(s):    Hongjie Dong, Seick Kim

Abstract: We consider the Dirichlet problem for second-order elliptic equations in non-divergence form $L$ and double divergence form $L^*$. We introduce a potential theory framework for these operators, including Perron`s method, capacity theory, and Wiener`s criterion. We establish the equivalence between regular boundary points for these operators $L$ and $L^*$ with those for the Laplace operator, assuming that the principal coefficients satisfy the Dini mean oscillation condition.

Improving Physics-Informed Neural Networks via Sobolev Trace Regularization

Doyoon Kim Korea Unversity Korea Co-Author(s):    Junbin Song

Abstract: We study the formulation of Physics-Informed Neural Networks (PINNs) for elliptic boundary value problems from the perspective of Sobolev space theory. Standard PINN approaches enforce boundary conditions using discrete $L_2(\partial\Omega)$ penalties, which are not consistent with the trace space of $H^1(\Omega)$. To address this mismatch, we introduce Trace Regularity Physics-Informed Neural Networks (TRPINNs), in which boundary data are enforced in the Sobolev-Slobodeckij space $H^{1/2}(\partial\Omega)$. We develop a computationally efficient approximation of the corresponding semi-norm that retains its essential structure while avoiding numerical instabilities. We show that this formulation yields convergence in the $H^1(\Omega)$ norm and provide numerical evidence demonstrating improved performance, particularly for problems with highly oscillatory boundary data.

On the Aleksandrov-Bakelman-Pucci estimate for 1-Laplace-type equations

Shuhei Kitano Waseda University Japan Co-Author(s):

Abstract: In this talk, I will present a new $L^\infty$ bound for solutions to the Poisson problem associated with the weighted $1$-Laplacian. This estimate is analogous to the classical result for uniformly elliptic equations established independently by Aleksandrov, Bakelman, and Pucci in the 1960s, and it is based on an analysis of the contact set of the solutions and their quasiconcave envelopes.

Junhee Ryu KIAS Korea Co-Author(s):

Abstract:

The Isometric Study of Wasserstein Spaces- Bounded Intervals

Chuanlong Sun Southeast University Peoples Rep of China Co-Author(s):    Tiren Huang

Abstract: The isometric rigidity of Wasserstein spaces and PDEs are inherently interconnected through the optimal transport theory, with the Monge-Amp\`{e}re equation serving as the core link. In this talk, based on the work of Geher, Titkos, and Virosztek, we extend their rigidity results on [0,1] \to [0,1] to different bounded intervals. Particularly, we give the characterization of isometries of p=1 and p>1, and we find the exotic isometries for p=1.

Some studies on the regularity of solutions to degenerate or singular equations with gradient terms

Jiangwen Wang School of Mathematics, Southeast University Peoples Rep of China Co-Author(s):    Jiangwen Wang, Feida Jiang

Abstract: In this talk, we consider $ C^{1,\alpha} $ regularity of solutions to degenerate normalized $ p $-Laplacian equations and degenerate or singular fully nonlinear integro-differential equations. If time allows, we will also introduce related applications, such as dead-core problems. This talk is based on joint works with Prof. Feida Jiang.

Mixing flow and advection-diffusion equations

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.