DUT Differential Equations Seminar

In this talk, we study unbounded solutions of the 2D incompressible Euler equations. One of the motivating factors for this is that the usual functional framework for the Euler equations (e.g. based on finite energy conditions, such as L^2) does not respect some of the symmetries of the problem, such as Galileo invariance.
The first result, global existence and uniqueness of solutions for initial data with square-root growth O(|x|^{1/2 - }) and bounded vorticity, is based on two key ingredients. Firstly an integral decomposition of the pressure, and secondly examining local energy balance leading to solution estimates in weighted $L^2$ spaces. We also prove continuity of the initial data to solution map by a substantial adaptation of Yudovich's uniqueness argument.
The second result is global existence and uniqueness of solutions for initial data with sublinear growth O(|x|^{1-}). Here the Galileo symmetry plays an essential role, since the Leray projection is not available anymore.
This is joint work with Dimitri Cobb. The first part recently appeared online in ARMA, the second part is work in progress.

We establish the incompressible low-Mach/high-Reynolds limit for the Boltzmann equation for a broad class of initial data, without recourse to any asymptotic expansion. Exploiting the local Maxwellian manifold and the macro-micro decomposition in a new quasi-linear analysis, we derive quantitative estimates for the purely microscopic fluctuation, as well as bounds for the kinetic vorticity and the entropic fluctuation in terms of the initial data. As a consequence, in two space dimensions, the rescaled velocity and temperature converge to a global solution of the incompressible Euler equations coupled to a transported temperature, within the frameworks of DiPerna-Lions-Majda and Delort.

We establish Fourier restriction estimates for a class of degenerate simple curves in high dimensions under a higher-order convexity condition. By exploiting the finite-order structure of the torsion, we obtain a finite decomposition of the parameter interval together with a geometric inequality for the associated Jacobian. Combining these estimates with square-function and multilinear techniques, we derive the optimal range of \((p,q)\)-exponents, with sharpness up to the endpoints. We further demonstrate that the finite-order derivative condition is essential by constructing examples showing that the restriction constants necessarily deteriorate as the derivative order tends to infinity.

We obtain sharp logarithmic orders for the measure of the delta-neighbourhood of curved Kakeya sets as surfaces of revolution generated by a family of 2D curves satisfying a certain cinematic curvature condition. In particular, we obtain sharp logarithmic order for the measure of delta-neighbourhood of spherical Kakeya sets. This is joint with Xianghong Chen (SYSU).

In this talk, I will introduce some results on the linear stability of relative equilibria in the N-body problem obtained using the Maslov-type index. The talk consists of three parts.
In the first part, I will briefly introduce the basic definition of a relative equilibrium and describe the linear stability problem.
In the second part, I will introduce some basic background on the Maslov-type index and explain the idea of using the Maslov-type index to study the linear stability problem.
In the third part, I will present the fundamental solution of the Kepler problem and then some qualitative and quantitative results on the linear stability of relative equilibria in the N-body problem obtained using the idea introduced in the second part.
This talk is based on joint work with Prof. Xijun Hu and Prof. Yuwei Ou.

In this talk, we focus on local pattern formation in coupled elliptic PDE systems governed by transparent boundary conditions. Such systems bridge diverse areas, including inverse boundary problems via a single passive/active boundary measurement, the spectral geometry of transmission eigenfunctions, and the geometric characterization of invisibility phenomena and inverse shape problems in wave scattering. We also uncover and rigorously characterize a novel type of local pattern formation, establishing a sharp quantitative relationship between the difference in the PDEs' lower-order terms and the geometric and regularity parameters within product-type end-structures characterized by high extrinsic curvature in a generic domain. This foundational result yields new findings with novel physical insights and practical implications across these fields.

We establish a weighted restriction estimate in $\mathbb{R}^2$. Our proof combines new two-ends Furstenberg estimates with refined decoupling. As an application, our result yields decay estimates for the circular $L^p$-means of the Fourier transforms of fractal measures in $\mathbb{R}^2$.

Mathematics itself has been turned upside down by a recent counterexample to the Navier–Stokes Existence and Smoothness Millennium Problem published by OpenAI. In this talk, the mathematical details of the construction will be discussed: First, Fefferman's formulation of the problem and in particular the crucial inclusion of a smooth force field. Second, an overview of the basic properties of OpenAI's construction from early in their paper. Third, a few scattered insights into the key steps to make it work from later in the paper. Additionally, other essential progress from various mathematicians will be discussed parallel.

The Intermediate Long Wave equation (ILW) models the internal wave propagation of the interface in a stratified fluid of finite depth, establishing a natural connection between the deep-water regime (the Benjamin-Ono (BO) regime) and the shallow-water regime (the Korteweg-de Vries (KdV) regime). In this talk, I will address convergence issues for ILW from a statistical mechanics viewpoint at different energy levels. By exploiting both the Hamiltonian and completely integrable structure of ILW, BO, and KdV, I will delve into the convergence of Hamiltonian and higher-order conservation laws for ILW, along with their associated measures and dynamics. Two interesting phenomena arise: (i) the modes of convergence of the measures in the deep-water and shallowwater limits differ. (ii) KdV, appearing in the shallow-water limit, possesses half as many conservation laws as ILW and BO, leading to a 2-to-1 collapse phenomenon.

This talk is based on joint works with Andreia Chapouto (Monash University), Ruoyuan Liu (University of Bonn), Tadahiro Oh (University of Edinburgh/BIT).

In this talk, we study the two-dimensional quadratic nonlinear wave equation (NLW) with Gaussian initial data in the regime where one of the stochastic objects arising in the construction of solutions has infinite variance. This regime occurs before the critical regularity predicted by the probabilistic scaling heuristics of Deng–Nahmod–Yue (2023). Consequently, standard probabilistic methods for constructing solutions fail.
Our main result shows that, for suitably renormalized, frequency-truncated Gaussian initial data, the corresponding solutions to the NLW converge locally in time to solutions of a stochastic NLW with zero initial data.
In particular, an interesting phenomenon emerges: the randomness propagates from the initial condition to the equation itself, giving rise to an effective time-correlated space-time Gaussian noise whose covariance exhibits a Bessel-type structure. Our proof combines paracontrolled calculus, the fourth moment method, and multilinear dispersive smoothing.
This talk is based on joint work with Guopeng Li (BIT), Jiawei Li (University of Edinburgh), Tadahiro Oh (University of Edinburgh / BIT), and Nikolay Tzvetkov (ENS Lyon).

The attractor is an important concept in dynamical systems, which can be used to characterize the long time asymptotic behavior of systems. In this talk, we will discuss random attractors for McKean-Vlasov stochastic differential equations (in short, MVSDEs). A difficulty arises from the distribution dependence in MVSDEs, which breaks the flow property of solutions. To address this, we consider the random attractor on product space $H\times \mathcal{P}(H)$, where $H$ is a separable Hilbert space and $\mathcal{P}(H)$ denotes the space of probability measures on $H$. This is based on a joint work with Xianjin Cheng and Zhenxin Liu.

This talk primarily presents singularity criteria for solutions to the compressible Navier-Stokes equations. It further extends a previously proven conjecture of Nash. Another part establishes the global existence of solutions for the MHD equations where the viscosity coefficients depend on the density. In this case, the initial density can be arbitrarily large.

As is well-known, the solution of the Patlak-Keller-Segel system in 3D will blow up in finite time regardless of any initial cell mass. In this talk, we are interested in the suppression of blow-up for the 3D Patlak-Keller-Segel-Navier-Stokes system via the stabilizing effect of the moving fluid. We prove that if the Couette flow is sufficiently strong, then the solutions for the system are global in time. This is a joint work with Shikun Cui, Wendong Wang and Juncheng Wei.

In this talk, we study the spectral stability of the standing periodic waves in the massive Thirring model. The Massive Thirring Model is complete integrable, the spectral stability of the standing periodic waves can be studied by using their Lax spectrum. We show analytically that each family of standing periodic waves is distinguished by the location of eight eigenvalues which coincide with the end points of the spectral bands of the Lax spectrum. The standing periodic waves are proven to be spectrally stable if the eight eigenvalues are located either on the imaginary axis or along the diagonals of the complex plane. By computing the Lax spectrum numerically, we show that this stability criterion is satisfied for some standing periodic waves. This is joint work with Prof. Dmitry Pelinovsky.

This report is about the traveling wave solutions to the pseudo-parabolic equation, a kind of non-classical diffusion equation characterized by the mixed third-order derivative term. We demonstrate that the ratio of the mixed third-order derivative coefficient to the diffusion coefficient $\frac{\tau}{D}$ can serve as a bifurcation parameter. In detail, when $\frac{\tau}{D}\leq1$, the equation possesses monotone traveling waves; when $\frac{\tau}{D}>1$, traveling waves are not monotonic and oscillate around the steady state $u=1$. The precise form of the minimal wave speed $c^*(\tau,D)$ is also derived, exhibiting a monotonic increase with respect to $\tau$ and converging to $2\sqrt{D}$ as $\tau$ approaches 0. Numerical simulations confirm and support our theoretical results. They further show that the larger the value of $\tau$ is, the more non-monotonic the traveling waves become. Our findings regarding oscillating traveling waves predict saturation overshoot—a behavior that contradicts classical diffusion-like behavior yet is widely observed in unsaturated porous media. Mathematically, the threshold value of $\frac{\tau}{D}$ reveals the essential role of the dynamic capillary effect in the fundamental overshoot mechanism.

We propose a first-order, time-discrete stochastic consensus model for global optimization. The model draws on interaction-based mechanisms to incorporate objective-function information and handles non-convex, non-differentiable, and even discontinuous functions. It is motivated by the Consensus-Based Optimization (CBO) paradigm, which promotes consensus among agents toward a global optimum through simple stochastic dynamics amenable to rigorous mathematical analysis. Despite these promises, the actual behavior of agents in its time-discrete implementation remains largely unknown. We address this issue by the novel observation that the consensus point governs the entire ensemble. We further demonstrate competitive performance across various problems.

Mean motion resonance (MMR), a phenomenon occurring when two celestial bodies have orbital periods in a commensurable ratio, plays a pivotal role in both stabilizing and destabilizing orbital motions within the Solar System. For highly eccentric orbits, the risk of close encounters introduces significant complexity. When such eccentric orbits are trapped in resonance, perturbations can induce chaotic motions, leading to rapid changes in orbital elements and transitions of different dynamical states. In this talk, we will numerically demonstrate the limits of application of canonical perturbation theory, specifically the validity of the normal form approximations, for the study of MMRs of the circular restricted three-body problem in the domain of orbits which cross, or are close to cross, the orbit of the secondary body. The external 1:2 and 5:6 MMRs for values of the mass ratios representing the Sun-Jupiter and Sun-Neptune cases will be used as model examples.

Yang-Mills theory is fundamental for both mathematics and theoretical physics. It underlies electrodynamics with the first unitary group U(1) as gauge group or quantum chromodynamics with the third special unitary group SU(3) as gauge group. It has also lead to a deeper understanding of four-dimensional manifolds, for example the Fields Medal winning proof of Donaldson’s theorem. All applications are based on the study of the Yang-Mills equations, which are partial differential equations of second order as well as linear for an abelian gauge group and non-linear for a non-abelian gauge group. Since the formulation of the Yang-Mills equations connects differential forms with Lie algebras, a differential and algebraic perspective are inseparable in Yang-Mills theory.

In this talk, I will introduce some uniqueness of inverse source scattering and inverse obstacle scattering problems with multi-frequency data. I will also present some direct sampling methods for inverse source scattering problems. With sparse data, these numerical methods can reconstruct the support of source. The stability estimation based on dense data are also provided. Additionally, I will propose some direct sampling methods for the simultaneous reconstruction of both the obstacle shape and the impedance value in the context of inverse acoustic scattering problems, backscattering data are used in this problem.

Assume $n\geq3$ and $u\in \dot{H}^1(\mathbb{R}^n)$. Recently, Piccione, Yang and Zhao [JDE, 2025] established a nonlocal version of Struwe's decomposition, i.e., if $\Gamma(u):=\left\|\Delta u+D_{n,\alpha}\int_{\mathbb{R}^{n}}\frac{|u|^{p_{\alpha}}(y) }{|x-y|^{\alpha}}\mathrm{d}y |u|^{p_{\alpha}-2}u\right\|_{H^{-1}}\rightarrow0$ and $u\geq 0$, then $dist(u,\mathcal{T})\to 0$, where $dist(u,\mathcal{T})$ denotes the $\dot{H}^1(\mathbb{R}^n)$-distance of $u$ from the manifold of sums of Talenti bubbles. In this paper, we establish the nonlocal version of the quantitative estimates of Struwe's decomposition in Ciraolo, Figalli and Maggi [IMRN, 2018] for one bubble and $n\geq3$, Figalli and Glaudo [Arma, 2021] for $3\leq n\leq5$ and Deng, Sun and Wei [Duke, 2025] for $n\geq6$ and two or more bubbles. We prove that for $n\geq 3$, $\alpha\le n$ and $0\le \alpha\leq 4$, $$ dist (u,\mathcal{T})\leq C\begin{cases} \Gamma(u)\left|\log \Gamma(u)\right|^{\frac{1}{2}}\quad&\text{if } \,\, n=6 \,\, \text{and} \,\, \alpha=4,\\ \Gamma(u) \quad&\text{for any other cases.}\end{cases} $$ Furthermore, we show that this inequality is sharp for $n=6$ and $\alpha=4$.

In this talk, we focus on the life span estimates of solutions to semilinear heat equations with inner and initial sources. We first prove that solutions blow up in finite time when only a single inner source non-rarefied at infinity, and derive the optimal upper and lower bounds on the life span. In recent work, we find only a single source dominates the asymptotic life span in this model with non-trivial non-negataive scaled sources, and establish sharp asymptotic estimates of life span on blow-up solutions.

Seiberg-Witten theory is fundamental for both mathematics and theoretical physics. It was derived from N = 2 supersymmetric Yang-Mills theory and then effectively applied to the study of four-dimensional manifolds. This was a remarkable instance of mathematics gaining from theoretical physics instead of the usual other way around. Seiberg-Witten invariants, based on the Seiberg-Witten equations with the abelian gauge group U(1), also both extended and simplified calculations of Donaldson invariants, based on the Yang-Mills equations with the non-abelian gauge group SU(2), which also concerns the study of four-dimensional manifolds. This was another remarkable aspect since an abelian theory is in many aspects simpler than a non-abelian theory. Seiberg-Witten theory is in particular able to distinguish smooth structures on four-dimensional manifolds, based on the fact that they are encoded in partial differential equations on it like the Seiberg-Witten equations. But the smooth Poincaré conjecture about smooth structures on the four-dimensional sphere is still open, so research continues to find other partial differential equations to encode the necessary information, which could come from physics again.

In 2004, Dombrowski et al. demonstrated that suspensions of aerobic bacteria often generate fluid flows through the interplay of chemotaxis and buoyancy, described by the chemotaxis-Navier-Stokes model, and observed self-concentration manifesting as turbulence characterized by transient, self-reconstituting, high-speed jets that entrain nearby fluid to form paired, oppositely signed vortices. Investigating the properties of these vortices (singular points) is of significant interest, and one approach is to follow the partial regularity theory of Caffarelli-Kohn-Nirenberg in studying the singularity properties of suitable weak solutions. In this paper, we first establish the existence of suitable weak solutions for the three-dimensional chemotaxis-Navier-Stokes equations, where the primary difficulty lies in deriving an appropriate local energy inequality of weak type. Moreover, we derive a new a priori estimate for the linear chemotaxis-Stokes model and present several $\varepsilon$-regularity criteria.

Scalar balance laws are advection-reaction equations, that appear either in biology or physic when one mesures the variation of a quantity over time. In this presentation, we will focus on specific solutions of these equations, namely propagated waves that connect two distinct constant states. A large variety of such waves can be constructed, and we will discuss the stability of some of them. It is a joint work with L. M. Rodrigues.

The Klein-Gordon equation is a wave equation with an additional mass damping term. In this presentation, I will review some literature about the dynamic of such equation when settled on an unbounded one-dimensional spatial domain. I will further present some new results regarding the global existence and long time behaviour of solutions that are initialy close to constant or periodic equilibria. Most notably, I will talk about a viscous approximation of this equation, as well as describe how uniformly local orbital stability can be obtained from polar decomposition. This is joint work with Björn de Rijk and Emile Bukieda.

We introduce the NLS hierarchy and discuss its well-posedness with a nonzero boundary condition. Here the main difficulty is local well-posedness, since good conserved energies have already been constructed by H. Koch and X. Liao. We adopt a perturbative formulation and determine its structure sufficiently in order to apply techniques for general dispersive nonlinear PDE.

We consider long-wave solutions to the NLS hierarchy with nonzero boundary condition. Rewriting the conserved energies and momenta in hydrodynamic variables, we find that they are approximated by the conserved energies of the KdV hierarchy. Combining this with a similar approximation of the symplectic structures yields a formal approximation result for tthe hierarchies. We quantify this using energy estimates and previously established well-posedness results.

We shall discuss the global nonlinear stability of vortex sheets for the Navier-Stokes equations. When the Mach number is small, we allow both the amplitude and the initial vorticity of the vortex sheets to be large. We introduce an auxiliary flow and reformulate the problem as a new vortex sheet with small vorticity but subjected to a large perturbation. Based on the decomposition of frequency, the largeness of the perturbation is encoded in the zero modes of the tangential velocity. We find an essential cancellation property that there are no nonlinear interactions among these large zero modes in the zero-mode perturbed system. This cancellation is owing to the shear structure inherent in the vortex sheets. These observations enable us to derive the global estimates for strong solutions that are uniform with respect to the Mach number. As a byproduct, we can justify the incompressible limit.

Einstein's Field Equations describe gravity as the curvature of spacetime. Following the discovery of the equivalence of energy and mass, an expansion of Newton's theory of gravity became necessary. Since it doesn't treat the motion or rotation of celestial bodies as a source of gravity, although these carry kinetic and rotational energy, a contradiction appears. Fixing it requires an expansion to combine all these information. In particular the new prospect that then even gravity becomes its own source, since gravitational waves also carry energy, leads to a non-linear formulation and the precise partial differential equations were in development for an entire decade. By now, their unique structure has been studied for over a century. Countless strange solutions, for example allowing time travel, have already been found.

We consider quasilinear partially dissipative hyperbolic systems with time-dependent damping in the whole space $R^d$, with $d \ge 1$. Using an approach similar to that developed by Crin-Barat and Danchin, we establish the global existence of small-amplitude solutions for systems endowed with a damping term of the form $-\frac{K z}{(1+t)^{\alpha}}, \quad 0 < \alpha \le 1.$ We assume that the linearized system satisfies the Shizuta--Kawashima (SK) condition, which ensures that the dissipation acts on all characteristic components through coupling. The key idea is to construct a Lyapunov-type functional that compensates for the lack of full dissipation. Such a functional was first introduced by Beauchard and Zuazua in the framework of control theory.

In this talk, I will present recent work on numerical approximations of the stochastic Allen–Cahn equation driven by multiplicative trace-class noise. We consider a fully discrete scheme combining a drift-implicit Euler method in time with a spectral Galerkin approximation in space.
The main focus is on weak convergence analysis. I will explain how the spatial weak convergence rate improves upon the corresponding strong rate—by nearly one order in dimensions $d =1,2$ and by nearly one-half order in dimension $d=3$. For the temporal discretization, we obtain weak convergence rates close to order one in $d=1,2$ and close to $3/4$in $d=3$.
A key ingredient of the analysis is the derivation of suitable a priori estimates for the Kolmogorov equations associated with the spectral Galerkin semi-discretization. In addition, I will introduce techniques for handling operator traces involving stochastic integrals in the temporal weak error analysis.

The Birkhoff normal form provides a systematic way to simplify Hamiltonian dynamics near an equilibrium or relative equilibrium and remains one of the basic tools in local dynamical analysis. For Hamiltonian systems with symmetry, however, the relevant dynamics is the reduced dynamics, and the reduced phase space may be singular. This creates a serious obstacle for local normal form methods, which are usually most effective in smooth canonical coordinates.
In this talk I will present a local approach that avoids performing the normal form construction directly on the reduced space. The idea is to work instead on a smooth momentum level set and to model the reduced dynamics locally by means of a transverse system of second-class constraints. The associated Dirac bracket then provides an effective Poisson structure on the slice. I will explain a condition on the quadratic part of the Hamiltonian ensuring that the dynamics on the momentum level, the Dirac-bracket dynamics on the slice, and the reduced dynamics on the corresponding symplectic stratum agree locally. This gives a practical framework for constructing Birkhoff normal forms in symmetric Hamiltonian systems without having to normalize directly on a possibly singular quotient.

Both the Schrödinger and the Klein–Gordon equation from quantum mechanics are derived from energy-momentum relations, non-relativistic for the former and relativistic for the latter. Both are also mathematically interesting as they are closely related to the heat and wave equation, respectively. But going deeper into quantum field theory requires more structure, mainly to properly describe the spin of particles. A suitable mathematical method is provided by Dirac operators, which when applied twice yield a generalized Laplace operator. Dirac operators then not only lead to the Dirac equation, which predicted antimatter, but also to rich mathematical applications, for example in combination with the famous Atiyah–Singer index theorem.

Developed in the 1980s, paradifferential calculus has become a robust tool in the analysis of nonlinear partial differential equations. This mini-course offers a beginner-friendly introduction to the construction and fundamental properties of paradifferential operators, alongside their applications to some problems in nonlinear PDEs. The first part of the course reviews the algebraic structure of classical differential operators and motivates the transition to pseudodifferential operators. We will then introduce the paraproduct and paradifferential operators, and establish their basic calculus. In the second part, we explore some advanced properties and end with applications in elliptic estimates and the symmetrization of hyperbolic systems.

The instability of the water jet system under long-wave perturbation—the Rayleigh-Plateau instability—has been observed and studied in experimental and theoretical physics since the 19th century. This talk provides a rigorous mathematical justification for this phenomenon. We consider the water jet system, modeled by the incompressible irrotational Euler equation with surface tension, and prove that it possesses a local hyperbolic structure around the trivial steady state. The core of our method is the construction of “paradifferential propagator" corresponding to linear paradifferential hyperbolic systems, effectively balancing the loss of regularity due to the quasilinear nature of this system. This enables the use of Lyapunov-Perron type arguments to construct the stable/unstable manifolds and a center invariant set, with or without spectral gap. We expect that such method could be extended to other quasilinear models. This is a joint work with Chengyang Shao.

This talk is about a physical motivation for the damping term u_xxt. The first part presents the explicit capillary pressure relation \(p^n - p^w = p^c - \tau \dot{S}^w\) in multiphase porous media flow, as formulated by Hassanizadeh and Gray, highlighting its derivation from conservation laws and the second law of thermodynamics; the second part presents microforce balance framework, which yields the generalized Ginzburg-Landau and Cahn-Hilliard equations by introducing constitutive relations and the local dissipation inequality; the final part is connected with the theory of Rivlin-Ericksen fluids and for an incompressible second-order fluid undergoing nonsteady simple shearing flow between parallel plates the equation reduces to the linear third-order PDE \(u_t = u_{xx} - u_{xxt}\) on a strip. These examples illustrate how fundamental physical laws guide the development of mathematical models across multiphase flow, phase-field theory, and non-Newtonian fluid mechanics.

Maxwell's equations are a coupled system of four partial differential equations, which describe the electromagnetic field: Coulomb's law describes electrostatics, Gauß's law describes magnetostatics, Faraday's law describes the evolution of the electric field and Ampére's law describes the evolution of the magnetic field. Maxwell's equations are both interesting from a mathematical and physical perspective: A suitable combination results in wave equations for the electromagnetic field with their speed matching exactly that of light, which historically first showed that light is in fact an electromagnetic wave. Maxwell's equations therefore serve as a prime example of the connection between elegant mathematics and physical phenomena. Formulating them shows the key applications of vector analysis, their generlization in differential forms and a path to decouple partial differential equations to then transform differential relations into algebraic relations, which can be solved easier.

In this talk, we first introduce the concept of first integrals of ordinary differential equations and their connection with first-order linear partial differential equations. We then present the method of characteristics for solving first-order linear PDE, which reduces PDE problems to a system of ODE along characteristic curves. Furthermore, we extend this method to the initial value problem of the one-dimensional linear wave equation and derive the d'Alembert formula. We illustrate the geometric meaning of characteristics. This talk shows how the method of characteristics serves as a unified tool in analyzing several basic types of partial differential equations.

The Kadomtsev-Petviashvili (KP) equation is a nonlinear PDE that describes nonlinear wave propagation in two spatial dimensions. It has infinitely many conserved quantities, and is a member of an infinite family of mutually compatible PDEs called the KP hierarchy. I will give an overview of a structural result that recovers the equations of the KP hierarchy from a minimal amount of information, and discuss the connection to its "tau-structure". If time permits, I will also mention reductions of the KP hierarchy, in particular the BKP hierarchy.

In classical mechanics, observables such as position, momentum, energy, and angular momentum appear to be measurable independently and simultaneously. The state of a particle is described by a point $(q,p)\in\mathbb{R}^3\times\mathbb{R}^3$ in phase space, evolving according to Hamilton's equations.
In quantum mechanics, this picture changes. In this talk, I will first introduce the basic concepts and axioms of quantum mechanics. Then, starting from the Schrödinger equation and the momentum operator in the position representation, I will derive them formally, as an analogy with classical mechanics. Using properties of the Fourier transform, I will also show that the momentum operator becomes a multiplication operator in momentum representation, which is an example of the spectral theorem.

In this talk, we will present recent progress on the suppression of blow-up for the three-dimensional Patlak-Keller-Segel-Navier-Stokes system via the Couette flow. As is well known, for the two-dimensional PKS system, solutions blow up in finite time when the initial cell mass exceeds 8π; for the three-dimensional PKS system, blow-up may occur in finite time for any positive initial cell mass. We prove that when the Couette flow are sufficiently strong, the solutions of the coupled system do not blow up in finite time. It is a joint work with Lili Wang and Wendong Wang.

The Lamb-Chaplygin dipole is a traveling wave solution to the 2D incompressible Euler equation, whose orbital stability was established in [Abe-Choi, 2022] and [Abe-Choi-Jeong, 2025] assuming the odd symmetry in x_2 (O) and non-negativity in upper half-plane (N). This talk will further study its stability in the following two aspects. Firstly, we will show the spectral stability of the linearized operator around the Lamb-Chaplygin dipole without conditions (O) or (N), based on the index theory established in [Lin-Zeng, 2022]. Secondly, assuming (O) and (N), we refine the orbital stability results in [Abe-Choi, 2022] and [Abe-Choi-Jeong, 2025] quantitatively. If time permits, I will introduce how to use the orbital stability of Lamb-Chaplygin dipole to obtain the superlinear gradient growth result for smooth and compactly supported vorticity in R^2 for 2D Euler equation.
This talk is based on the joint papers with In-Jee Jeong (SNU), Zexing Li (CY Cergy), Peicong Song (Caltech) and Yao Yao (NUS).

The mathematical description of two-phase flows of immiscible materials is a fundamental topic in materials science and fluid dynamics, with numerous applications in biology, chemistry, and engineering. Over the past decades, various diffuse-interface models have been proposed to describe such systems, among them the class of Navier-Stokes-Cahn-Hilliard models. Standard formulations are typically complemented by no-slip boundary conditions for the velocity field and homogeneous Neumann boundary conditions for the phase field and the chemical potential. However, these conditions are often insufficient when a precise description of the dynamics close to the boundary is required. In this mini-course, I will present a recently developed diffuse-interface model for incompressible viscous fluid mixtures with bulk-surface interaction. The system couples a bulk Navier-Stokes-Cahn-Hilliard model with a surface Navier-Stokes-Cahn-Hilliard system posed on the boundary, allowing for the description of phase separation and fluid motion both in the bulk and along the surface. The first lecture will focus on the convective bulk-surface Cahn-Hilliard subsystem. We will discuss the mathematical structure of the model and address analytical questions such as the existence and uniqueness of weak solutions, propagation of regularity, and, time permitting, the long-time behavior. In the second lecture, we will turn to the full bulk-surface Navier-Stokes-Cahn-Hilliard system. We will outline the construction of weak solutions via a semi-Galerkin approximation scheme, where the velocity fields are approximated using eigenfunctions of a novel bulk-surface Stokes operator. In the two-dimensional setting, we even obtain a (unique) global strong solution by the same approach. Finally, I will briefly discuss ongoing and future research directions related to these models.

The Navier–Stokes equations describe the motion of incompressible fluids and give rise to many fundamental and challenging mathematical problems. In this talk, I will present new results for the two-dimensional free boundary problem for two immiscible fluids. We prove global-in-time well-posedness in the nonnegative density regime and establish the persistence of the natural $C^{1+\gamma}$ regularity of the interface under sharp assumptions on the initial velocity. We also consider the case of density-dependent viscosity and obtain global regularity provided the viscosity contrast is sufficiently small.

Steady configurations in plasma dynamics are often described by the magneto-hydrostatic equations, which couple the magnetic field, the current density, and the plasma pressure. A classical question is whether such equilibria can be reconstructed from partial boundary data prescribed on suitable inflow and outflow parts of the boundary. This leads to a delicate class of mixed elliptic-transport problems in which the geometry of magnetic field lines plays a central role.
In this mini-course, I will discuss recent analytical progress on this problem, moving from the two-dimensional theory to the genuinely three-dimensional setting. In two dimensions, I will explain how the boundary value problem can be reformulated as a nonlinear fixed-point problem combining transport of the current with a nonlocal reconstruction of the magnetic field. I will then discuss the three-dimensional extension, where the divergence-free character of the current and genuinely three-dimensional transport effects require new tools, including pseudo-differential operators of limited regularity. Time permitting I will also comment on some ongoing work for the  steady MHD equations.

In this talk, I will derive asymptotic interface models for an elastic Muskat free boundary problem describing Darcy flow beneath an elastic membrane. In a weakly nonlinear regime of small interface steepness, we obtain nonlocal evolution equations that capture the free-boundary dynamics up to quadratic order. In the long-wave thin-film regime, we rewrite the kinematic condition in flux form, flatten the moving domain, and derive a lubrication-type equation. Moreover, we establish well-posedness for these models in suitable Wiener spaces. This is joint work with R. Granero-Belinchón.

In this mini-course, we offer a welcoming introduction to the theory of Boltzmann equations. We begin with the spatially homogeneous Boltzmann equation in the classical Grad cutoff setting, presenting a fundamental existence result and the H-Theorem. This provides a basic understanding of the equation’s structure and the role of entropy. We then discuss an existence result in the non-cutoff setting for the spatially inhomogeneous Boltzmann equation.

In this talk, we formulate the three-wave and four-wave kinetic equations within the GENERIC framework. We derive the small-angle limit for a four-wave kinetic operator, which is analogous to the well-known grazing limit from the Boltzmann collision operator to the Landau collision operator. We also show the GENERIC structure of the limiting system.
This talk is based on joint work with Hong Duong (Birmingham).

In this talk, we consider deterministic homogenization for fast-slow systems, where the fast variables are generated by iterating a nonuniformly expanding map. We show that the slow variables, suitably scaled in time, converge to the solution of a stochastic differential equation. If time permits, we also give the rate of convergence in Wasserstein distance.

I will discuss a directed, non-exchangeable particle approximation of McKean-Vlasov diffusions, where particle \(i\) interacts only with its predecessors through their empirical measure. Although the system has the same McKean-Vlasov limit as the classical exchangeable model, its sequential structure leads to different quantitative and fluctuation behavior.
The first part concerns propagation of chaos via an incremental path-space relative entropy, giving sharp \((i-1)^{-1}\) one-step entropy estimates and \(N^{-1/2}\)-scale empirical convergence in negative Sobolev norms. A key ingredient is the construction of auxiliary random measures from conditional laws, which replace predecessor empirical measures by averaged conditional measures. The second part presents a central limit theorem: the limiting averaged fluctuation is Gaussian, but its equation is coupled to a hierarchy of logarithmically weighted fluctuation fields.

The hydroelastic wave problem describes the interaction between elastic structures and hydrodynamic excitation. It is a free boundary problem in mathematical fluid mechanics and can be written as a quasilinear dispersive system of order 5/2. In this talk, I will talk about the low regularity local well-posedness of the two-dimensional deep hydroelastic waves. By constructing a cubic modified energy that incorporates a paradifferential weight chosen carefully, we prove that the hydroelastic waves are locally well-posed in $H^{s+3/2}\times H^{s}$ for $s>3/4$, This is based on the joint work with Lizhe Wan.

We prove the existence of a class of orbitally stable bound state solutions to nonlinear Schrödinger equations with super-quadratic confinement in two and three spatial dimensions. These solutions are given by time-dependent rotations of a non-radially symmetric spatial profile which in itself is obtained via a doubly constrained energy minimization. One of the two constraints imposed is the total mass, while the other is given by the expectation value of the angular momentum around the z-axis. Our approach also allows for a new description of the set of minimizers subject to only a single mass constraint.