ICM Satellite Meeting: New Methods in Evolution Partial Differential Equations

Monday, July 20

09:00-09:15: Morning coffee

09:15-10:15: Natasa Pavlovic

10:15-11:15: Minh-Binh Tran

11:15-12:15: Mikaela Iacobelli

Lunch break

14:30-15:30: Hao Jia

15:30-16:00: Afternoon coffee break

16:00-17:00: Bjoern Bringmann

Tuesday, July 21

09:00-09:15: Morning coffee

09:15-10:15: Sijue Wu

10:15-11:15: Matthieu Cadiot

11:15-12:15: Delia Ionescu-Kruse

Lunch break

14:30-15:30: Fabio Pusateri

15:30-16:00: Afternoon coffee break

16:00-17:00: Jiajie Chen

Wednesday, July 22

09:00-09:15: Morning coffee

09:15-10:15: Sylvia Serfaty

10:15-11:15: Benoit Pausader

11:15-12:15: Patrick Gerard

Bjoern Bringmann: Construction of the 2D Yang-Mills-Higgs measure.

Abstract: We construct the 2D Yang-Mills-Higgs (YMH) measure via stochastic quantization. To this end, we show global well-posedness and uniform-in-time bounds for the associated Langevin dynamics, which is given by the 2D stochastic YMH equations. A key component of our approach is the further development of techniques in stochastic geometric analysis, combining ideas from geometric analysis and stochastic analysis. These methods yield a manifestly gauge-covariant local existence theory, refined estimates for covariant stochastic objects, and a decay mechanism driven by unstable Yang-Mills connections.

Matthieu Cadiot: Constructive existence proofs of branches of overhanging waves

Abstract: In this talk, we present a computer-assisted method for the constructive existence proof of 2D periodic steady water waves of finite depth with constant vorticity. The approach is based on a Fourier series representation of the solution, which transforms the water-wave problem into an equivalent system for the Fourier coefficients. We then apply a Newton–Kantorovich framework: starting from a numerically computed approximate solution u, we rigorously prove the existence of a true solution in a neighborhood of u.

This methodology enables us, in particular, to establish the existence of overhanging waves. Moreover, the approach can be extended to validate global branches of solutions parameterized by the relative mass flux. Specifically, we prove the existence of a global branch connecting the flat solution to an overhanging wave with self-intersection on the trough line, thereby confirming the conjectured global bifurcation scenario of Constantin, Strauss, Varvaruca (2016).

This is a joint work with Susanna Haziot (Princeton).

Jiajie Chen: Asymptotically self-similar blowup for 3D incompressible Euler with $C^{1, 1/3-}$ velocity

Abstract: Whether the incompressible Euler equations in $R^3$ can develop a finite-time singularity from smooth initial data is a long-standing open problem in mathematical fluid mechanics. In the axisymmetric setting without swirl, global regularity is known for $C_c^\alpha$ initial vorticity for all $\alpha \geq 1/3$. Below this regularity threshold, for any $\alpha \in (0,1/3)$, we construct exact $C^\alpha$ self-similar blowup profiles for the vorticity of the 3D axisymmetric Euler without swirl, and build on them to prove asymptotically self-similar blowup from $C_c^\alpha$ initial vorticity and $C^{1,\alpha} \cap L^2$ initial velocity. Moreover, we provide a complete characterization of the limiting behavior of these $C^\alpha$ vorticity profiles and the associated blowup solutions as $\alpha$ tends to $1/3$. Our construction is based on lifting $C^\infty$ blowup profiles for a 1D nonlocal model and exploits the anisotropy of the 3D self-similar flow. To the best of our knowledge, our results provide the first example in which a singularity from a 1D nonlocal fluid model is lifted to construct blowup for incompressible fluid equations in $R^2$ or $R^3$.

Patrick Gerard: The half-wave maps flow on the energy space

Abstract: The half-wave maps equation is a quasilinear hyperbolic system in one space dimension, with target the two-dimensional sphere. It is related to the theory of half-harmonic maps and can be seen as a continuum limit of Calogero-Moser spin systems. On the one dimensional torus, it admits a flow on the critical energy space, with almost periodic trajectories. I will discuss the main ingredients of the proof of this fact, which relies on an explicit representation formula of the solutions derived from a Lax pair. This talk is based on a joint work with Enno Lenzmann.

Mikaela Iacobelli: Quasineutral limits in plasma models

Abstract: I will begin with a short introduction to kinetic models for collisionless plasmas, focusing on Vlasov-type systems (Vlasov-Poisson and the variant with thermalized/massless electrons). I will then turn to the quasineutral regime. For Vlasov-Poisson and for the model with massless electrons, I will present almost-optimal stability of the quasineutral limit obtained using the kinetic Wasserstein distance, together with refined control of the exponential Poisson coupling. In the magnetized setting, I will describe a new result establishing the quasineutral limit from the relativistic Vlasov-Maxwell system to electron-MHD within an analytic-regularity framework; the analysis yields uniform-in-ε estimates and strong (filtered) convergence to the limiting dynamics.

Delia Ionescu-Kruse: Nonlinear water waves with arbitrary vorticity: dynamics of an embedded point vortex

Abstract: We investigate the two-dimensional water-wave problem for an inviscid incompressible fluid with a flat bed, a free surface, and a general non-zero vorticity field. The nonlinear governing equations describing the intricate interaction between the rotational motion in the bulk of the fluid and the surface motion, are formulated through the Dirichlet–Neumann operator and the Green function associated with the Laplace operator in the free-boundary domain; new explicit representations for both objects are derived (see D. Ionescu-Kruse and R. Ivanov, J. Differential Equations 368, 2023). As an application, we study the dynamics of a point vortex embedded in the fluid and its interaction with the free surface. In the small-amplitude, long-wave Boussinesq and KdV regimes, by using appropriate approximations of the Green function in an infinite strip and asymptotic expansions of the Dirichlet–Neumann operator, we derive a coupled system governing the evolution of the surface variables and the vortex motion. Our analysis shows that surface solitary waves are not destroyed by the interaction with the vortex and remain practically unaffected over a significant range of vortex strengths. This observation leads to a further simplification of the model, in which the vortex motion beneath propagating solitons is described by a decoupled system of ordinary differential equations, capturing the qualitative features of the interaction. Analytical results and supporting numerical simulations are presented (see D. Ionescu-Kruse, R. Ivanov and M. Todorov, J. Nonlinear Sci 36, 2026).

Hao Jia: Global well-posedness of the dynamical Prandtl equation

Abstract: The Prandtl equation is a fundamental equation in the study of boundary layers for the Navier Stokes equations in the vanishing viscosity limit. Under important pressure and monotonicity conditions on the initial and in-flow data, Oleinik first proved the local existence of classical solutions to the dynamical Prandtl equation. Recently Xin-Zhang and Xin-Zhang-Zhao established a global theory of weak solutions and proved their interior regularity in the Crocco coordinates. However due to the nonlocal nature of the Crocco transform, it is difficult to obtain smoothness of the solutions in the physical coordinates from this, even in the interior. As a consequence, the existence of global classical solutions to the dynamical Prandtl equation remained open. In a joint work with Z. Lei and C. Yuan, we prove up to boundary smoothness of weak solutions to the dynamical Prandtl equation in both the Crocco and the physical coordinates. Using this result, we then establish the global existence and regularity of the dynamical Prandtl equation (in two dimensions). We also allow very general matching rates of initial and in-flow data towards the outflow, including the important Blasius profile which is the global attractor, in the downstream variable, of the Prandtl equation.

Benoit Pausader: Linear stability for small BGK waves

Abstract: We consider the Vlasov-Poisson system in 1+1 dimension in periodic setting. The simplest states are homogeneous equilibria, and they have been widely studied. When a suitable Penrose-type criterion is satisfied, these equilibria are (nonlinearly) stable under small smooth perturbations. When the Penrose criterion is violated, the picture is more complex and involves another type of steady states: the (inhomogeneous) BGK waves. This is a joint work with D. Bian, E. Grenier and W. Huang.

Natasa Pavlovic: What happens when bosons are mixed with fermions

Abstract: Investigating mixtures of bosons and fermions is an active area of research in experimental physics for constructing and understanding novel quantum bound states. These ultra-cold Bose-Fermi mixtures are intrinsically different from gases with only bosons or fermions. Namely, they show a fundamental instability due to energetic considerations coming from the Pauli exclusion principle. Inspired by this activity in the physics community, recently we started exploring the mathematical theory of Bose-Fermi mixtures. We will describe results obtained with Cárdenas, Miller and Mitrouskas inspired by recent experiments by DeSalvo et al. on mixtures of light fermionic atoms and heavy bosonic atoms. These experiments reveal the emergence of an attractive fermion-mediated interaction between bosons, as well as a stability-instability transition. We give the first mathematical demonstration of this transition by studying the low-energy spectrum of a many-body interspecies Hamiltonian.

Fabio Pusateri: Asymptotic stability of the degree-one vortex in the abelian Yang-Mills-Higgs model

Abstract: We consider the abelian Yang-Mills-Higgs equations in the plane, a.k.a. the magnetic Ginzburg-Landau equations. This system admits topological vortex solutions. We will discuss a proof of full asymptotic stability of the degree-one vortex at the self-dual coupling for co-rotational perturbations. This is the first asymptotic stability result for a topological soliton (in conservative flows) in dimension larger than 1. This is joint work with J. Lührmann (Koln), J. Palacios (EPFL), W. Schlag (Yale), and S. Shashahani (U. Mass. Amherst).

Sylvia Serfaty: Singular mean-field limits via a multiscale mollification metric

Abstract: We are interested in the question of mean-field limits, or deriving effective evolution equations of PDE type for a system of N points in singular interaction, for instance of Coulomb or Riesz nature, evolving by first order dynamics. The modulated energy method works well for gradient flows or conservative flows of Coulomb/Riesz type energies. We will present here a new method based on a multiscale mollification metric, which works well for up to Coulomb interaction singularity, without much structure assumed. This is joint work with Hung Q. Nguyen (Chinese academy of science).

Minh-Binh Tran: Condensations for 4-wave kinetic equations

Abstract: In this work, we prove that solutions of 4-wave kinetic equations, under very general forms of dispersion relations, develop condensates at the origin in finite time. When the dispersion relation is quadratic $|k|^2$, we prove that condensates develop under much weaker conditions on the initial data than the ones considered previously by Escobedo and Velazquez. We also extend this result to a more complicated situation of finite temperature trapped Bose gases. This is joint work with Gigliola Staffilani.

Sijue Wu: The quartic integrability of water waves and related fluid models

Abstract: I will present a new approach of constructing quartic energy sequences for water waves and other related models.