Special Session 117: Patterns and Attractors in Nonlinear Dynamics

Traveling waves in the reaction-diffusion equation with discontinuity

Junsik Bae Kyungpook National University Korea Co-Author(s):    Wonhyung Choi, Yong-Jung Kim

Abstract: We classify traveling waves and stationary solutions of a reaction-diffusion equation arising in population dynamics with Allee-type effects. The reaction term is given by a quadratic polynomial with a discontinuity at zero, which captures finite-time extinction for sub-threshold populations. This discontinuity induces a free boundary in the wave profile, a phenomenon that distinguishes the model from the classical logistic or Allen-Cahn equations. A complete scenario is presented that connects monostable and bistable traveling waves through the wave speed parameter. This is a joint work with Wonhyung Choi (Hankyong National Univ.) and Yong-Jung Kim (KAIST).

Asymptotic behavior for the incompressible Navier-Stokes system revisited

Noboru Chikami Nagoya Institute of Technology Japan Co-Author(s):

Abstract: We study the asymptotic behavior of small global solutions to the incompressible Navier-Stokes system. It is known that a suitable spatial decay allows the global solution to behave like self-similar solutions as time goes to infinity (Kato(1984), Planchon(1998)}). More precisely, if the spatial decay of the initial data is critical (i.e., $\varphi(x) = O(|x|^{-1})$) then the solution is asymptotic to a nonlinear self-similar solution, and if not, then the solution is asymptotic a linear self-similar solution. Under minimal assumptions on the initial data, we revisit this classical result on asymptotically self-similar solutions, and derive the second asymptotic expansion. The result can be extended to multiplier-type weighted Lorentz spaces.

Transition Routes to Synchronization in the Classical Kuramoto Model

Jia-Yuan Dai National Tsing Hua University Taiwan Co-Author(s):    Bernold Fiedler, Alejandro L\`{o}pez Nieto

Abstract: Most forward orbits of the classical Kuramoto model with identical frequencies lead to stable totally synchronized states. Besides, there is a plethora of saddle partially synchronized states. In this talk, we rigorously describe the transition routes from partial synchronization to total synchronization. By the gradient structure of the classical Kuramoto model, the global attractor consists of (circles of) equilibria and heteroclinic orbits between them. By using the permutation symmetry of indices, we determine when two distinct equilibria can be connected by heteroclinic orbits. This yields a connection graph ordered by inclusion. Moreover, we show that, surprisingly, the invariant manifolds of partially synchronized equilibria are linear. As a consequence, the classical Kuramoto model is structurally stable Morse-Smale system. In particular, the connection graph persists under any small perturbations of the classical model. This is joint work with Bernold Fiedler and Alejandro L\`{o}pez Nieto.

Detecting and quantifying random bifurcations in collapsing attractors

Maximilian Engel University of Amsterdam Netherlands Co-Author(s):    Alexandra Blessing, Alex Blumenthal, Maxime Breden

Abstract: In many situations, deterministic dynamical systems may exhibit non-trivial attractors which collapse to a random point under the addition of Gaussian noise. This phenomenon is widely known as synchronization by noise and seems to make bifurcations of attractors to higher-dimensional objects disappear in the stochastic case. This point of view was first challenged by Callaway et al. 2017 with respect to the pitchfork bifurcation in the presence of additive noise, showing a transition from uniform to non-uniform snychronization mirroring the original bifurcation. In this talk, I will give a summary on results from co-authors and myself that have extended these observations to multiple and infinite dimensions and have provided a quantitative analysis via large deviation estimates for finite-time Lyapunov exponents. I will also mention current efforts to fully transfer this framework to SPDEs.

$L^2$-contraction of viscous-dispersive shocks for KdV-Burgers equation

Namhyun NE Eun Korea Institute for Advanced Study Korea Co-Author(s):    Geng Chen, Moon-Jin Kang, Yannan Shen

Abstract: The Korteweg--de Vries--Burgers (KdVB) equation is a fundamental model capturing the interplay of nonlinearity, viscosity (dissipation), and dispersion, with broad physical relevance. It is well known that the KdVB equation admits traveling wave solutions, called viscous-dispersive shocks. These shock profiles are monotone in the viscosity-dominant regime, while they exhibit infinitely many oscillation when dispersion dominates. In this talk, we study the stability of such viscous-dispersive shocks, focusing on an $L^2$-contraction property under arbitrarily large perturbations, up to a time-dependent shift. We begin with viscous shocks of the viscous Burgers equation (i.e., the KdVB equation without dispersion), then treat monotone viscous-dispersive shocks and finally address oscillatory shocks. We also present detailed structural properties of the oscillatory profiles. This is joint work with Geng Chen (University of Kansas), Moon-Jin Kang (KAIST), and Yannan Shen (University of Kansas).

Time-asymptotic stability of composite waves for compressible Navier-Stokes-Korteweg and Navier-Stokes-Fourier-Korteweg systems

Sungho Han KAIST Korea Co-Author(s):    Jeongho Kim and Moon-Jin Kang

Abstract: In this talk, we discuss the long-time behavior of one-dimensional compressible fluids with internal capillarity. For the barotropic Navier-Stokes-Korteweg system, we establish the time-asymptotic stability of a composite wave consisting of a rarefaction wave and a shifted viscous-dispersive shock. Furthermore, we extend this framework to the non-barotropic Navier-Stokes-Fourier-Korteweg system to prove the stability of composite waves involving a rarefaction wave, a viscous contact wave, and a shifted viscous-dispersive shock. Our analysis is based on the method of $a$-contraction with shifts, which combines weighted relative entropy estimates with dynamical shifts applied to the shock profiles. These results are based on joint works with Jeongho Kim and Moon-Jin Kang.

Stable Blow-up Profiles in Transport-Type Equations

Bongsuk Kwon Ulsan National Institute of Science and Technology Korea Co-Author(s):    Yunjoo Kim, Wanyong Shim

Abstract: We investigate $C^1$ blow-up phenomena for a class of transport-type equations. The classical method of characteristics provides only limited information on singularity formation, typically capturing the temporal blow-up rate along characteristic curves. While this approach has been successfully applied to the Burgers equation and various models of wave breaking in water waves, it does not fully describe the structure of singularities. It is well known that singularities in the Burgers equation exhibit $C^{1/3}$ regularity, a feature also observed in certain fluid models such as the Euler equations. This has led to the common belief that all $C^1$ blow-up solutions arising in transport equations share the same regularity profile. In this work, we show that this paradigm is incomplete. We construct self-similar blow-up profiles for a general class of transport-type equations-often arising as leading-order models in water wave dynamics-that exhibit singular behaviors distinct from those of the Burgers equation. In particular, these profiles may possess different H\older regularities. Our framework applies to several important models, including the Camassa--Holm equation, the Hunter--Saxton equation, and the $b$-family of fluid transport equations. We present the construction of explicit stable blow-up profiles and outline the proof of their stability in a suitable functional setting. This yields sharp regularity results and a refined description of blow-up dynamics.If time permits, we will also discuss more exotic self-similar blow-up profiles arising within a broader scaling framework.

Sharp interfaces in singular reaction diffusion systems

Tommaso Lamma Leiden University Netherlands Co-Author(s):    Arjen Doelman, Frits Veerman, Paul Carter

Abstract: There are systems of reaction diffusion equations in which the interplay of bistability and spatial scale separation leads to the emergence of sharp interfaces separating two asymptotic homogeneous states. Multiple sharp interfaces interact and in some cases evolve into a stationary multi-front pattern, a derivation of interaction equations will be presented as a dimensionality reduction procedure.

The dynamics of global attractors for fully nonlinear parabolic equations in 1d

Phillipo Lappicy Universidad Complutense de Madrid Spain Co-Author(s):

Abstract: We explicitly construct global attractors for fully nonlinear parabolic equations in one spatial dimension with two types of phenomena. First, in case the semiflow is dissipative, the attractor is compact and it can be decomposed as equilibria and heteroclinic orbits. Second, in case the semiflow is not dissipative, there are blow-up solutions and the attractor is unbounded - which can be compactified using an appropriate Poincar\`e projection with an induced flow at infinity. In this setting, we view blow-up solutions as heteroclinics orbits to infinity. In both cases of dissipative and non-dissipative nonlinearities, we state necessary and sufficient conditions for the occurrence of heteroclinics between hyperbolic equilibria. The prototype examples are a bounded and unbounded version of the Chafee-Infante attractor.

Time-asymptotic behavior of solutions to the 3D Barotropic Navier-Stokes equations toward viscous shock in outflow problems

Hobin Lee Korea Advanced Institute of Science and Technology Korea Co-Author(s):    Hyeonseop Kim, Moon-Jin Kang

Abstract: We study an initial-boundary value problem for the 3D barotropic Navier-Stokes equations. More specifically, we consider the outflow problem on the spatial domain $\mathbb{R}_+ \times \mathbb{T}^2$, where the far-field condition is given by two different constant states. In this paper, our main result is the orbital stability of the viscous shock wave in the subsonic case, assuming that both the shock amplitude and the initial perturbation are small. The core of our proof consists of two main steps: obtaining a priori zeroth-order estimates and higher-order estimates. In the first step, we utilize the method of $a$-contraction with shifts, originally introduced by Kang and Vasseur in 2017, which is highly effective for controlling the effects of the viscous shock. In the second step, we employ the classical energy method to derive sharp estimates for controlling the boundary terms.

On the optimal rate of vortex stretching for axisymmetric Euler flows without swirl

Deokwoo Lim Korea Institute for Advanced Study Korea Co-Author(s):    Deokwoo Lim and In-Jee Jeong

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. This is a joint work with In-Jee Jeong (SNU).

Nonautonomous differential equations in the presence of bounded noise

Iacopo P. Longo University of Exeter England Co-Author(s):    Konstantinos Kourliouros, Martin Rasmussen

Abstract: Nonautonomous systems subject to noise arise naturally in many applied contexts. Particularly relevant are those featuring time-dependent parameters and uncertainties, which may drive the emergence of tipping points. In this talk, we focus on the case of bounded noise, where the dynamics can be captured topologically via deterministic set-valued dynamical systems: initial conditions are evolved under all admissible noise realisations, abstracting from probabilistic details. However, as these systems operate on the space of nonempty compact subsets---a space lacking Banach structure---they pose substantial analytical and numerical challenges. In particular, bifurcation analysis of attractors remains a significant obstacle. To address this, we derive a higher-dimensional single-valued determinisitic dynamical system which characterizes the evolution of the boundary of invariant sets (typically attractors) of certain differential inclusions.

Uniqueness and weak-BV stability for the isentropic Euler system: inflow and outflow problems

HyeonSeop Oh Korea Advanced Institute of Science and Technology Korea Co-Author(s):    Moon-Jin Kang, Jiayun Meng, and Alexis Vasseur

Abstract: In this talk, we consider initial-boundary value problems for the one-dimensional isentropic Euler system in the half-space, focusing on both inflow and outflow problems. We prove that small BV solutions in the subsonic region are unique and stable within a wild class of weak solutions satisfying the so-called strong trace property. In particular, we establish quantitative H\older-type stability estimates in the $L^2$ norm. While the small BV solutions, being in the subsonic region, are associated with non-characteristic boundaries, the wild solutions admit characteristic boundaries. Our analysis is based on the method of $a$-contraction with shifts, although the stability results do not depend on the shift. This talk is based on a joint work with Moon-Jin Kang, Jiayun Meng, and Alexis Vasseur.

Stability of Multidimensional Periodic Waves of Parabolic Systems

Luis Miguel Rodrigues Univ Rennes France Co-Author(s):    Benjamin Melinand (Strasbourg, France), Aric Wheeler (Duke, USA).

Abstract: We discuss a few steps towards a complete stability theory for genuinely multidimensional periodic traveling waves of parabolic systems. On the one hand for large classes of parabolic systems we prove that spectral stability implies nonlinear asymptotic stability in a suitable sense. On the other hand, for two-dimensional reaction-diffusion systems we provide large-time asymptotics in terms of solutions to averaged, modulation systems. Thus, for such systems, we extend the comprehensive theory now available for plane periodic waves to the multidimensional context. Concerning the latter, a significant part of the analysis is devoted to obtaining new results on near-constant anisotropic dynamics, to be applied to the averaged systems and including dispersive-diffusive estimes and derivations of artificial viscosity systems.

Stability of shock profiles for the Navier-Stokes-Poisson system

Wanyong Shim Korea Advanced Institute of Science and Technology Korea Co-Author(s):

Abstract: We consider the one-dimensional Navier--Stokes--Poisson (NSP) system, which describes the dynamics of positive ions in a collision-dominated plasma. The NSP system admits a one-parameter family of smooth traveling wave solutions, known as shock profiles. In this talk, we study the stability of these shock profiles. Our analysis is based on the pointwise semigroup method, which combines spectral analysis with Green`s function estimates. We first establish the spectral stability of the shock profiles. Building upon this, we derive pointwise bounds on the Green`s function for the associated linearized problem, which yield linear and nonlinear asymptotic orbital stability.

Double Beltrami States in Hall magneto-hydrodynamics

Jaeyong Shin Yonsei University Korea Co-Author(s):    Hantaek Bae, Kyungkeun Kang

Abstract: In this talk, we investigate double Beltrami states in the Hall magneto-hydrodynamic (Hall MHD) equations. Initially, we examine the double Beltrami states as a special class of steady solutions to the ideal Hall MHD equations, which are closely related to Beltrami flows in incompressible fluid dynamics. Specifically, we classify the double Beltrami states and show that they can be derived by using the variational method as energy minimizers, subject to the conservation of two helicities. We then extend our analysis to time-dependent double Beltrami states in the viscous and resistive Hall MHD equations, exploring their exact form and stability properties.

Vortex atmospheres of traveling vortices: rigorous definition, existence, and topological classification

Young-Jin Sim KIAS Korea Co-Author(s):    Kyudong Choi, In-Jee Jeong, Young-Jin Sim

Abstract: In incompressible and inviscid fluids, the vortex atmosphere refers to the collection of fluid particles outside the support of a traveling vortex that are nevertheless carried along with it. This phenomenon has been recognized since the nineteenth century, e.g., in the classical works of O. Reynolds [Nature, 1876] and O. Lodge [Lond. Edinb. Dubl. Phil. Mag., 1885], yet rigorous mathematical definitions and proofs have remained largely undeveloped, with most subsequent studies relying on thin-core approximations or asymptotic analyses. In this talk, we give a rigorous definition of a vortex atmosphere and establish its existence and uniqueness. We further compare the planar atmosphere surrounding a 2D vortex dipole with the axisymmetric atmosphere surrounding a 3D vortex ring. In particular, we emphasize and prove the topological distinctions observed by W. Hicks [Lond. Edinb. Dubl. Phil. Mag., 1919]: under natural assumptions, every 2D dipole with its atmosphere forms an oval-shaped region, whereas for 3D rings, both spheroidal and toroidal configurations may occur. Our proof is based on showing that each atmosphere can be characterized precisely as a specific superlevel set of its corresponding stream function.

High-Voltage Ionized Gas with Spherical Cathode Emission

Masahiro Suzuki Nagoya Institute of Technology Japan Co-Author(s):

Abstract: We consider a plasma that is created by a high voltage difference, which is known as a Townsend gas discharge. The plasma is confined to the region between two concentric spheres, one of which is a cathode and the other an anode. Ion-electron pairs are created by collisions inside the plasma. Additional electrons enter the plasma by collisions of ions with the cathode. We prove under certain conditions that there are many steady states exhibiting gas discharge, beginning with a `sparking` voltage. In fact, there is an analytic one-parameter family of them that connects the non-ionized gas to a plasma with arbitrarily high ionization or arbitrarily high potential, or else the family ends at an `anti-sparking` voltage. This talk is based on a joint work with Professor W. A. Strauss (Brown Univ.).

Patterns in miscible displacement in porous media

Sergey Tikhomirov Pontificia Universidade Catolica do Rio de Janeiro - PUC-Rio Brazil Co-Author(s):    Yu. Petrova, Ya. Efendiev

Abstract: We study the motion of miscible liquids in porous media. Injection of a less viscous fluid into a more viscous one produces an instability known as viscous fingering. This phenomenon is described by a multidimensional PDE system consisting of mass conservation, incompressibility, and Darcy law. In heterogeneous media, the displacement pattern forms a regular structure. In homogeneous media, it is more chaotic, and displays an intriguing regime of intermediate concentration. To improve estimates for the mixing-zone size, we study a related model of gravity-driven fingering based on the incompressible porous media equation with diffusion. We represent the medium as a system of vertical tubes. For the simplest case of 2 tubes we were able to find cascades of travelling waves [1]. For the case of infinitely many tubes we demonstrate the existence of an infinite cascade of travelling waves. Speed of travelling waves could be interpreted as speed of viscous fingers and back front propagation. The result in the simplified model suggests that existing estimates for original multi-dimensional problems could be improved. [1] Petrova Yu., Tikhomirov S., Efendiev Ya. Propagating terrace in a two-tubes model of gravitational fingering. SIAM Journal on Mathematical Analysis, 57 (2025), 30-64.