Ken Abe
Title: A variational family of traveling vortex dipoles: transition, uniqueness, and stability
Abstract: I will discuss the existence of traveling vortex dipoles and their stability in the two-dimensional incompressible Euler equations.
This talk is based on a joint work with In-Jee Jeong (KIAS), Guolin Qin (AMSS), and Weicheng Zhan (Xiamen U).
William Feldman
Title: Perron extremal solutions of the Bernoulli one-phase problem
Abstract: I will discuss new results on the Perron extremal solutions of the Bernoulli one-phase problem. We show that the Perron solutions are also variational solutions. This allows us to prove new results on the fine structure of the free boundary for Perron solutions in $d=2$. Don't worry, PDEs in motion will play an important role! This is a joint work with Kerrek Stinson (Utah) and Farhan Abedin (Lafayette College).
Thierry Gallay
Title: Dynamics of a point vortex in a half-plane
Abstract: As a model for vortex-wall interactions, we consider the two-dimensional incompressible Navier-Stokes equations in a half-plane with no-slip boundary condition, and we focus on the paradigmatic case where the initial condition is a single point vortex in an otherwise stagnant fluid. We prove that this system has a unique global solution for all values of the Reynolds number, which is defined here as the ratio of the circulation of the vortex to the kinematic viscosity of the fluid. The solution we construct has finite energy for all positive times, and converges to zero in energy norm as time goes to infinity. A challenging question is to describe the motion of the vortex center in the vanishing viscosity limit. A recent result by C. Wang, J. Yue, and Z. Zhang shows that this is possible for short times in a suitable functional-analytic setting. Strong instabilities are expected to occur at later times due to vortex-induced boundary layer separation, a phenomenon that is well-documented in the physical literature. This talk is based on a joint work with Anne-Laure Dalibard (Paris).
Nicolás García Trillos
Title: A collective dynamics perspective on transformer dynamics beyond gradient flows.
Abstract: In this talk, I will discuss a collective dynamics perspective on transformers, the architecture at the heart of modern large language models. In particular, we will discuss how dimensionality reduction techniques akin to those used in the study of the Kuramoto model can be employed to explore the rich structure that the evolution of the distributions of tokens (particles) can have when selecting different values for the key, query, and value matrices parameterizing a transformer model with multiple self-attention layers . This perspective will allow us to explore the structure of token dynamics beyond the gradient flow setting obtained by very specific choices of model parameters. In particular, we will discuss how certain parameter choices induce cyclical behavior, consensus formation without stability, and Hamiltonian dynamics. While our theoretical discussion will focus exclusively on 2-dimensional token embeddings, I will also discuss numerical experiments that suggest that our theoretical findings can be extrapolated to general multi-dimensional settings.
This is joint work with Sixu Li (UW-Madison), Jan Peszek (Warsaw), Trevor Teolis (Rice), Konstantin Riedl (Oxford), Jake Maranzatto (Maryland), Semih Akkoc (Maryland), and Sennur Ulukus (Maryland).
De Huang
Title: Existence of Sadovskii’s vortex patch via a fixed-point approach
Abstract: The Sadovskii vortex patch——a steady contiguous anti-symmetric vortex-patch dipole solution of the 2D incompressible Euler equation——was numerically discovered over 50 years ago, whose shape was also observed as an accurate approximation of the large-time asymptotic profile in the head-on collision of two anti-symmetric vortex rings. In this talk, we present the first proof of existence of the Sadovskii vortex patch with 90-degree touching angles via a fixed-point approach. In particular, we show that the upper boundary of the Sadovskii vortex patch is given by a smooth even function that is monotonic on one side. This is based on a joint work with Jiajun Tong.
Tsukasa Iwabuchi
Title: Incompressible Euler equations in 3D bounded domains in a critical space
Abstract: We discuss the local-in-time existence of solutions to the incompressible Euler equations in a bounded domain. The solutions are constructed via the inviscid limit of the Navier–Stokes equations. The function space in which the existence of solutions is proved is a Besov space, which serves as a critical space in the sense of well-posedness.
Matt Jacobs
Title: On the singular limit of Brinkman's law to Darcy's law
Abstract: In this talk, I will discuss the singular limit of Brinkman's law to Darcy's law in the context of congestion driven motion models. These models arise in various scenarios, in particular, in the continuum description of cellular and tissue growth. Our result shows that different versions of these models are related and can be obtained from one another. The main ingredient of our analysis is a family of energy evolution equations and their dissipation structures, which are novel and of independent interest. This strategy allows us to establish our result for a much larger family of pressure laws than was previously possible in the literature. Furthermore, our analysis can handle the joint limit starting from a compressible Brinkman's model and converging to an incompressible Darcy's law model, where the latter is a Hele-Shaw type free boundary problem.
Joint work with Noemi David and Inwon Kim.
Kyungkeun Kang
Title: Separation point of the Stokes system in the half space with localized boundary data
Abstract: We consider the Stokes system in the half-space with localized boundary data. We show that there exist boundary influxes for which the resulting flow exhibits flow reversal and boundary-layer separation. In the case of boundary-layer separation, we further investigate the dynamics and asymptotic behavior of the separation point. Moreover, through a perturbative argument, we construct solutions to the Navier–Stokes equations in the half-space that exhibit the same qualitative behavior as in the Stokes case. This is a joint work with T.-K. Chang and C.-H. Min.
Moon-Jin Kang
Title: Hydrodynamic Limit of the Boltzmann Equation toward Riemann Solutions with Shocks
Abstract: I will talk about the hydrodynamic limit of the one-dimensional Boltzmann equation with hard-sphere collision toward Riemann solutions of the compressible Euler system containing shock waves.
The Riemann solutions include generic superpositions of elementary waves, in particular shock-contact-shock and rarefaction-contact-shock configurations.
For suitably well-prepared initial data and sufficiently small wave strengths, we obtain global-in-time Boltzmann solutions and convergence to the corresponding Riemann solution in a global space-time energy norm, without removing the initial layer or the shock layer.
This is a joint work with Mingi Choe(KAIST) and Chanwoo Kim (UW Madison).
Jeongho Kim
Title: Spherically symmetric stationary solutions to the Navier-Stokes-Korteweg equations
Abstract: In this talk, I will present the existence and asymptotic stability of the spherically symmetric stationary solution of the Navier-Stokes-Korteweg equations. First, we show that the Navier-Stokes-Korteweg equations admit a unique stationary solution, under several assumptions on the boundary and far-field data. In particular, when the impermeable wall boundary condition is implemented, the stationary solution exponentially decays. Then, we show that this stationary solution with impermeable wall problem is nonlinearly stable, showing that if the boundary condition and the initial perturbation is small, the solution to the Navier-Stokes-Korteweg system converges to the stationary solution.
Junha Kim
Title: Lagrangian uniqueness for the $\alpha$-SQG equations
Abstract: We consider the $\alpha$-SQG equations in $\mathbb{R}^2$ including the 2D Euler equations ($\alpha = 0$) and the critical SQG equation ($\alpha = 1$). We establish uniqueness of solutions in two families of function spaces: spaces defined by a modulus of continuity, and $varphi$-mean oscillation spaces, in each case under the assumption that the induced velocity field satisfies the Osgood condition. The proof is purely Lagrangian: we close a Grönwall-type inequality directly for the difference of the associated flow maps.
Tim Laux
Title: Energy convergence for continuous-in-time median filters
Abstract: Median filters are popular nonlinear – but computationally efficient – image processing techniques to remove noise while preserving sharp edges. Effectively, iterative application of a median filter is in fact equivalent to evolving each contour line by mean curvature flow. In this talk, I will introduce a continuous-time counterpart of the filter that gradually denoises a given image. In the limit of vanishing stencil size, our results show that the evolution converges to level set mean curvature flow. Surprisingly, and for the first time for any median-type filter scheme, we can show the convergence of energies in this limit. Such strong convergence results are of major interest as they often have to be assumed in the literature to prove convergence of related schemes. Moreover, this result carries crucial geometric information, namely the unit multiplicity of interfaces. The proof relies on a compensated compactness argument inspired by the work of Evans and Spruck.
This is based on joint work with Fabius Krämer (Heidelberg University).
Quoc Hung Nguyen
Title: Flexibility through moving vortices: 3D Navier–Stokes and SQG
Abstract: We discuss a common convex-integration mechanism for producing flexible weak solutions of the three-dimensional incompressible Navier–Stokes equations and the inviscid surface quasi-geostrophic (SQG) equation. The main building blocks are localized coherent vortices traveling along carefully chosen trajectories. Their motion cancels the dominant transport and self-interaction errors before antidivergence, while long-orbit averaging reconstructs the Reynolds stress.
For Navier–Stokes, the perturbations are rescaled copies of Hill’s spherical vortex. This yields weak solutions in (C_tL_x^2), with gradients in (C_tL_x^{6/5+\varepsilon}) for an explicit (\varepsilon>0), that approximately connect arbitrary finite-energy divergence-free states. Time localization then gives exact nonuniqueness for a dense set of initial data. For SQG, localized traveling profiles yield analogous endpoint flexibility in (C_tL_x^{4/3+\varepsilon}), again for an explicit (\varepsilon>0), together with nonuniqueness for a dense set of initial data.
The two constructions share the same geometric blueprint but face different analytic obstacles: viscosity and the critical Hill-vortex scaling for Navier–Stokes, and derivative loss from the nonlocal constitutive law for SQG, overcome through a bilinear null-form estimate.
This talk is based on joint work with Zexi Wang and on joint work with Elia Bruè and Rui Jin.
Mitake Hiroyoshi
Title: Optimal semiconcavity with fractional modulus for Hamilton-Jacobi equations with Neumann boundary conditions
Abstract: We study the semiconcavity property of viscosity solutions to Hamilton--Jacobi equations with Neumann boundary conditions. Unlike the state-constraint case, minimizing trajectories associated with the Neumann problem may fail to be $C^1$, so the classical approach based on the regularity of minimizers is no longer available. To overcome this difficulty, we introduce a comparison argument between the constrained action associated with the Skorokhod problem and the unconstrained action, avoiding any use of higher regularity of minimizing trajectories. Under a structural decomposition assumption on the Hamiltonian at the boundary, we establish the global semiconcavity with a fractional modulus and show the optimality of the fractional exponent. This is a joint work with Panrui Ni (Waseda Univ.).
Tatsuya Miura
Title: Calibration energy and mean curvature flow
Abstract: We introduce the calibration energy for oriented immersions into Euclidean space, quantifying the deviation from calibrated geometry. A key property is that this energy may remain finite for infinite-volume immersions, while a null-Lagrangian structure ensures that it has the same first variation as the volume functional. We establish an exact dissipation identity for the calibration energy along oriented, proper mean curvature flows in arbitrary dimensions and codimensions, under a mild local-volume bound. This provides a new, finite variational framework for mean curvature flow beyond the finite-volume setting. Our result yields several applications, including rigidity for solitons and convergence for two-dimensional immortal solutions. This talk is based on joint work with Fabian Rupp (University of Vienna).
Jaemin Park
Title: Global existence of weak solutions to the inviscid SQG equation
Abstract: In this talk, I will discuss global existence of weak solutions to the 2D inviscid SQG equation. We construct weak solutions as a limit of Leray-type weak solutions to the viscous SQG equation. We adapt concentration compactness principle to show that the sequence of viscous solutions actually converge strongly in a suitable function space, leading to the conservation of the Hamiltonian for the weak solutions. This is a joint work with Luigi De Rosa (GSSI) and Mickael Latocca (Univ. Evry).
Jiewon Park
Title: Gradient flow methods for quantitative stability in a free-boundary variational problem
Abstract: We present a gradient-flow approach to quantitative stability for a free-boundary variational problem. The model case is the relative isoperimetric problem outside a convex obstacle, whose constrained gradient flow is a free-boundary area-preserving curve shortening flow. This gives rise to a nonlocal parabolic evolution with an orthogonality boundary condition and a finite-dimensional family of equilibria. We prove sharp Łojasiewicz estimates for the constrained energy near this equilibrium family, with explicit constants and optimal exponents. These estimates imply exponential convergence of the flow to a unique equilibrium and yield quantitative stability for minimizers through energy dissipation along the flow. This is joint work with Elena Mäder-Baumdicker, Robin Neumayer, and Melanie Rupflin.
Keisuke Takasao
Title: New phase field model for the multi-phase mean curvature flow
Abstract: We consider the phase-field method for the multi-phase mean curvature flow. It is well known that the solution to the system of Allen-Cahn equation converges to the multi-phase mean curvature flow under suitable assumptions, and this has been extensively studied through numerical computations. In this talk, we introduce a new system whose singular limit is Brakke's mean curvature flow and present its properties. This talk is based on joint work with Tim Laux (Heidelberg University).
Jiajun Tong
Title: Uniformly rotating vortex patches with 90-degree corners for the 2-D incompressible Euler equation
Abstract: We will discuss recent progress on constructing uniformly rotating vortex patches with 90-degree corners for the 2-D incompressible Euler equation. This is based on a joint work with De Huang and Xiaopeng Zheng.
Xiaoqian Xu
Title: AI Proofs in PDEs
Abstract: In this talk, I will discuss recent results on fluid equations done by AI. I will first outline the relationship between the mixing properties of incompressible flows and the diffusion process, including dissipation enhancement. I will then focus on the so-called unmixing phenomenon and on the existence of Batchelor scales for certain flows, both of which are results recently proved using AI.
Yao Yao
Title: Stability and growth for the incompressible Euler equations
Abstract: The incompressible Euler equations describe how the velocity field of an incompressible, inviscid fluid evolves over time. Even in the regimes where the global well-posedness of solutions is known, there are many open questions regarding the long-time dynamics and infinite-in-time growth of solutions. In this talk, I will discuss some recent results on the nonlinear stability of traveling wave solutions and infinite-in-time growth results.