Jaeheang Bang (Westlake)
Title: recent developments on ill-posedness of the Euler equation
Abstract: There were various recent developments on ill-posedness of the three-dimensional Euler equations. I will discuss some of our recent results on this topic. This is in collaboration with Alexey Cheskidov.
Myeongju Chae (Hankyung National University)
Title: Nonlinear Instability of Roll Patterns in the Generalized Swift–Hohenberg Equation
Abstract: We present a framework for analyzing nonlinear instability in dissipative equations, using the generalized Swift–Hohenberg equation as a model. We focus on roll patterns, namely spatially periodic steady solutions, and establish their linear and nonlinear instability. The linearized operator is self-adjoint, and Bloch–Floquet theory combined with Lyapunov–Schmidt reduction provides a detailed characterization of its unstable spectrum. Building on this spectral analysis, we apply Grenier’s method to prove nonlinear instability. Finally, we illustrate how this approach applies to Kolmogorov flow in the two-dimensional incompressible Euler equations.
Alexey Cheskidov (Westlake)
Title: Energy cascades for the Navier-Stokes equations: from instantaneous blow-up to global existence of smooth solutions for distributional initial data
Abstract: In this talk we will discuss the role of the forward and inverse energy cascades for non-uniqueness, blow-up, or global existence of smooth solutions of the unforced 3D Navier-Stokes equations. For smooth initial data, inverse energy cascade can result in an instantaneous blow-up and produce smooth for positive time solutions that enjoy the same regularity properties as the Koch-Tataru solutions. For distributional initial data, forward energy cascade can lead to enhanced dissipation and produce global in time very weak solutions. We characterize the class of distributional initial data for which classical weak solutions exist. In particular, any divergence-free initial datum in W^{s,p}, gives rise to a global weak solution, smooth for positive time, if and only if s>=-1. For instance, take any divergence-free initial datum that belongs to W^{-1,p} for some finite p>1 but not H^{-1}. Then the solution to the heat equation is not in L^2_{x,t}. On the other hand, the constructed weak solution is in L^2_{x,t}, as well as C([0,infinity);W^{-1,p}), and smooth for positive time. Hence, the NSE solutions can enjoy better integrability than the solutions of the heat equation. The nonlinear term regularizes the solution via a forward energy cascade, which also results in enhanced dissipation and hence better integrability than the heat flow.
Yasuhide Fukumoto (Institute of Mathematics for Industry, Kyushu University & Osaka Central Advanced Mathematical Institute)
Title: Global time evolution of a counter-rotating vortex pair of finite thickness in a viscous fluid
Abstract: Vortex pairs act to produce lift and thrust of aircrafts and animals. They move by themselves, entraining surrounding fluid, and are capable of transporting neutrally buoyant materials over long distances. We develop a method for calculating self-induced motion of a counter-rotating vortex pair over the entire time range, including the decaying stage, with focus on the influence of viscosity. Initial-value problems are solved for a vortex pair embedded in a viscous fluid. For high-Reynolds-number motion, the method of matched asymptotic expansions in powers of the ratio of the core radius to the distance between vortices, a small parameter, is used to derive the traveling speed of the pair at early times [1]. This gives the upper bound on the traveling speed in an early stage.
As an alternative means, we adapt, to the two-dimensional vortical flow, Helmholtz-Lamb's formula for an axisymmetric vortex ring. This method saves, to a large extent, manipulations of asymptotic expansions, and is widely extensible to magnetic vortex rings say [2]. In the large-Reynolds-number limit, a topological method (Kelvin-Benjamin's kinematic variational principle) achieves a further simplification [3, 4].
For low-Reynolds-number motion, the vorticity obeying the Stokes equation is substituted into the 2D-version of Helmholtz-Lamb's formula. At early times, this gives the lower bound on the traveling speed. An analysis at large times reveals the decaying behavior of the traveling speed, being distinct from the axisymmetric case [5].
References
[1] U. Habibah, H. Nakagawa and Y. Fukumoto, "Finite-thickness effect on speed of a counter-rotating viscous vortex pair", Fluid Dyn. Res. Vol.50, 031401 (2018).
[2] Y. Fukumoto, S. Oshiro and T. Ruzi, "Motion of magnetic vortex rings subject to Hall effect", Phys. Rev. Fluids Vol.10, 124703 (2025).
[3] Y. Fukumoto and H. K. Moffatt, "Kinematic variational principle for motion of vortex rings", Physica D Vol.237, pp.2210-2217 (2008).
[4] Y. Fukumoto, "Global time evolution of viscous vortex rings", Theor. Comp. Fluid Dyn. Vol.24, pp.335-347 (2010).
[5] Y. Fukumoto and F. Kaplanski, "Global time evolution of an axisymmetric vortex ring at low Reynolds numbers", Phys. Fluids Vol.20, 053103 (2008).
Mitsuo Higaki (Kobe University)
Title: An Arnold-type stability analysis of vortex filaments
Abstract: I will present ongoing work with Masashi Aiki (Tokyo Univ. Sci.) on the shape stability of vortex filaments evolving under the localized induction equation. Our approach is based on Arnold’s variational method [Arnold 1965] and provides a unified framework for the stability of arc-shaped and circular filaments.
A central step in Arnold’s method is the construction of a suitable Lagrange functional from conserved quantities. A key ingredient in our setting is the conserved fluid impulse, studied in [Arms and Hama 1965] and [Fukumoto 1987].
Dong-ha Kim (Seoul National University)
Title: A Lagrangian Approach to Uniqueness for Active Scalars in Generalized Mean Oscillation Spaces
Abstract: We present a unified Lagrangian framework for the uniqueness of weak solutions to a broad class of two-dimensional inviscid active scalar equations in generalized mean oscillation spaces. Under common structural assumptions on the Biot--Savart kernels, we identify mean oscillation spaces that ensure an Osgood condition for the velocity, and establish uniqueness in these classes. For the two-dimensional Euler equations, we establish uniqueness for vorticities whose mean oscillation grows mildly at small scales, extending beyond $\mathrm{BMO}$. For the surface quasi-geostrophic (SQG) and incompressible porous media (IPM) equations, we obtain uniqueness in spaces logarithmically rougher than $C^{0,1}$. We also obtain what are (to our knowledge) the strongest known uniqueness results for the $\alpha$-SQG equation, $0<\alpha<1$, in spaces logarithmically rougher than critical H\"older spaces $C^\alpha$. The framework also applies to various logarithmically perturbed models, yielding uniqueness in the corresponding sharp regularity classes. The proof combines a sharp regularity estimate for the velocity with a stability estimate for integral operators under measure-preserving rearrangements. This provides a single uniqueness argument for the entire family under common structural assumptions on the kernels.
Bongsuk Kwon (UNIST)
Title: Hidden Coercivity and (L^2)-Stability of Cahn–Hilliard Fronts
Abstract: The Cahn–Hilliard equation is a classical model for phase separation, admitting stationary transition fronts connecting two stable phases. In this talk, we first review some earlier stability results obtained using pointwise semigroup methods, and then present a new approach to the stability of one-dimensional Cahn–Hilliard fronts based on $L^2$-energy estimates with a dynamical spatial shift.
The main difficulty is that the quadratic form arising from the linearized equation is not manifestly positive. Inspired by recent $L^2$-contraction methods for viscous shocks, we use the front profile itself as a spatial coordinate and reveal a hidden coercive structure through a second-order weighted Poincaré-type inequality.
This approach yields $L^2$-contraction for sufficiently small perturbations and, combined with a far-field semigroup argument, asymptotic orbital stability of the front in the unweighted $L^2$ topology. Notably, the analysis requires only $L^2$-smallness of Hölder-continuous initial perturbations, without higher-order smallness, spatial localization, or moment assumptions.
This is joint work with Wanyong Shim.
Deokwoo Lim (KIAS)
Title : Growth of vorticity for axisymmetric Euler flows without swirl and its generalized structures
Abstract : We consider incompressible Euler flows with axisymmetry and without swirl. In $\mathbb{R}^3$, we prove the $t^{4/3}$-upper bound for the growth of the vorticity maximum. This was conjectured by Childress(Phys. D, 2008) and supported by numerical computations from Childress—Gilbert—Valiant(J. Fluid Mech., 2016). The key idea is to estimate the velocity maximum by the kinetic energy and conserved quantities related to the vorticity. Then we extend this idea to Euler flows in $\mathbb{R}^d$, $d\geq 4$ with bi-rotational symmetry, which refers to rotation-invariant solutions fixing either the first $n\geq 2$ or the last $d-n$ variables. This talk is consisted of separate joint works with In-Jee Jeong(KIAS) and Khakim Egamberganov(NUS).
Xiaoran Liu (Westlake University)
Title: Inverse Energy Cascade and Non-uniqueness for the Advection-Diffusion Equation
Abstract: We prove non-uniqueness for the advection-diffusion equation with a drift in a logarithmically near-critical class below $L_t^2L_x^\infty$. The first construction is a diffusion-assisted inverse cascade of scalar energy that diverges as $t \to 0$. The second uses compressed inverse mixing, with diffusion acting perturbatively, and produces a bounded parabolic solution with the energy jump at the initial time. In both constructions the drift and scalar are smooth for positive time. We also consider the drift in $L^{2,\infty}_tL^\infty_x$ and solutions satisfying the energy inequality starting from almost every time. In this setting, we prove uniqueness of solutions with bounded energy, and construct non-unique solution with energy blowing up as time goes to zero.
José Antonio Lucas-Manchón (ICMAT)
Title: Finite-time blow-up for generalized SQG in the Sobolev well-posedness regime
Abstract: We consider the generalized surface quasi-geostrophic (gSQG) equations in the singular range $\gamma\in(0,1)$, where the velocity is more singular than in SQG. Local well-posedness is known in $H^{\kappa}(\mathbb{R}^2)$ for $\kappa>2+\gamma$, also in the presence of a force $F\in L^1_t H^{\kappa}$. In this talk, we construct an initial datum and a force in this well-posedness class such that the unique local solution, which is smooth and compactly supported before the blow-up time, leaves $H^{\kappa}$ in finite time, while the force remains in $L^1([0,1];H^{\kappa})$ up to and including the singular time. The solution is built as an infinite cascade of layers at increasingly small scales: each new layer is stretched by the shear-type velocity generated by the previous ones, which makes its Sobolev norms grow, and it generates in turn a similar shear for the next one. The force collects the small errors of this approximation. This is joint work with Diego Córdoba, Óscar Domínguez and Luis Martínez-Zoroa.
Young-Jin Sim (KIAS)
Title: Geometric Rigidity of Variational Sadovskii Vortices
Abstract: In this talk, I will explain how energy maximization constrains the geometry of Sadovskii vortices, pairs of oppositely rotating vortices that touch along a symmetry axis and travel without changing shape. For the two-dimensional incompressible Euler equations, we study a family of those vortices obtained by maximizing kinetic energy subject to constraints on the impulse and the $L^p$-norm of the vorticity. The exponent $1<p\leq\infty$ parameterizes this family, which includes the classical Sadovskii vortex patch at $p=\infty$ and the Lamb-Chaplygin dipole at $p=2$.
For every exponent in this range, we prove that each upper vortex core is simply connected. Its free boundary has a nondegenerate quadratic tangency at the top and meets the symmetry axis orthogonally, with quantitative estimates on the local boundary behavior. For $2<p\leq\infty$, the core becomes strictly narrower as the height increases. For $2<p<\infty$, every nonempty positive vorticity superlevel set is a symmetric, simply connected region.
Numerical computations will also illustrate the vortex shapes and vorticity distributions across the family, including evidence for a point-vortex limit as $p$ approaches $1$. I will also compare our variational computations with Huang and Tong’s fixed-point computations of the Sadovskii patch. This talk is based on joint work with Guolin Qin (CAS).