Plenary talks
Yann Brenier (CNRS)
Magnetic relaxation of incompressible Euler equations and convection in porous media
The magnetic relaxation of the Euler equations of incompressible fluids can be seen as a dissipative version of the ideal MHD equations leading, for large time, to stationary solutions of the Euler equations. A similar structure appears in the theory of convection in porous media, leading to the "incompressible porous medium" equation. I will discuss several concepts of solutions in close connection with optimal transport theory.
Dongho Chae (Chung-Ang Univ.)
On the Liouville type theorems for the stationary Navier-Stokes equations in R^3
The Liouville type problem for the stationary Navier-Stokes equations in R^3 is a wide open problem in the mathematical fluid mechanics. In this talk we introduce the problem, and review various partial results. After that we discuss some of the recent progresses.
Anne-Laure Dalibard (Sorbonne Université)
Long-time behavior of the Stokes transport system
This talk will be devoted to the analysis of the Stokes-transport system in the domain Tx(0,1), with no-slip conditions for the velocity on the boundaries of the domain. We first prove that linearly stratified density profiles are orbitally stable: small smooth perturbations of such profiles remain small for all times, and the total density converges for long times towards a rearrangement of the initial density.
We also show that boundary layers are formed in the vicinity of z=0 and z=1 for large times, and hinder the rate of decay of the solution. Decomposing the solution as the sum of a boundary layer and an interior part, we prove that the interior part enjoys a higher decay.
This is a joint work with Julien Guillod and Antoine Leblond.
David Gérard-Varet (Université Paris Cité)
Mathematical study of suspensions of non-spherical particles
A popular model for suspensions of non-spherical particles in fluids is the so-called Doi model, which couples a Stokes equation for the fluid velocity $u(t,x)$ together with a transport equation for the distribution of particles in space and orientation $f(t,x,p)$. The Doi model comes from a formal mean-field limit of a system of particles interacting inside a Stokes flow. We will show in this talk that this formal limit is not accurate and will rigorously derive a correction to the model, under natural assumptions on the initial distribution of the particles. This is joint work with R. Höfer (Regensburg university).
François Golse (Ecole Polytechnique)
A Model of Gas-Surface Interaction
This talk presents a model of gas-surface interaction involving (a) a Lennard-Jones type physisorption potential near the surface and (b) a simple relaxation model for the collisions between gas molecules and phonons corresponding to the fluctuations of the crystalline structure of the solid surface. This model leads to a boundary condition for the Boltzmann equation satisfied by the distribution function of gas molecules, which is reminiscent of (but different from) Maxwell’s well-known accommodation condition. A salient feature of this new model is an explicit formula for the accommodation coefficient in terms of the physisorption potential and the relaxation time in the collision between gas molecules and phonons. At variance with Maxwell's condition, the accommodation coefficient so obtained depends on the kinetic energy of gas molecules hitting the solid surface. The part of this boundary condition that is not expressed in terms of specular reflection is not explicit and
is studied by means of numerical simulations. All these results have been obtained in collaboration with K. Aoki, V. Giovangigli and S. Kosuge.
Seung-Yeal Ha (Seoul National Univ.)
Weak flocking of the spatially extended kinetic Cucker-Smale model
In this talk, we discuss the emergent behaviors of the weak solutions to the kinetic Cucker-Smale (in short KCS) model in a non-compact spatial-velocity support setting. Unlike the compact support situation, non-compact support of a weak solution can cause a communication weight to have zero lower bound. This causes previous approach based on the nonlinear functional approach for spatial and velocity diameters to break down. To overcome this difficulty, we derive refined estimates on the upper bounds for the second-order spatial-velocity moments and show the uniqueness of the weak solution using the estimate on the deviation of particle trajectories. For the estimate of emergent dynamics, we consider two classes of distributions functions with decaying properties (an exponential decay or polynomial decay) in phase space, and then verify that the second moment for the velocity deviation from an average velocity tends to zero asymptotically fast, while the second moment for spatial deviation from the center of mass remains bounded uniformly in time. This illustrates the robustness of the mono-cluster flocking dynamics of the KCS model even for non-compact support settings in phase space and generalizes earlier results on flocking dynamics in a compact support setting. This is a joint work with Dr. Xinyu Wang (Seoul National University).
Kyungkeun Kang (Yonsei Univ.)
Singular solutions of the incompressible flow near boundary in the half space
Local boundary regularity is discussed for solutions for Stokes and the Navier-Stokes equations in the half space. Singular solutions have been constructed near boundary for the Stokes system away from the support of non-regular either boundary data or external forces. We have also investigated signs of each velocity components. Similar singular solution can be also established as well for the Navier-Stokes equations via the method of perturbation.
This is joint works partly with T. Chang and C.-H. Min and partly with C.-C. Lai, B. Lai and T.-P. Tsai.
Denis Serre (ENS de Lyon)
Compensated Integrability in Bounded domains. Dynamical applications.
Alexis Vasseur (UT-Austin)
Stability of discontinuous flow for compressible inviscid fluid
The compressible Euler equation can lead to the emergence of shock discontinuities in finite time, notably observed behind supersonic planes. A very natural way to justify these singularities involves studying solutions as inviscid limits of Navier-Stokes solutions with evanescent viscosities. The mathematical study of this problem is however very difficult because of the destabilization effects of the viscosity.
Bianchini and Bressan proved the inviscid limit to small BV solutions using the so-called artificial viscosities (Annals of Math. 2005). However, achieving this limit with physical viscosities remained an open question up to our recent result together with Geng Chen and Moon-Jin Kang.
In this presentation, we will provide a basic overview of classical mathematical theories of compressible fluid mechanics and introduce the recent method of a-contraction with shifts. We will describe the basic ideas and difficulties involved in the study of physical inviscid limits in the context of the barotropic Euler.
Invited talks
Paul Blochas (UT-Austin)
a-contraction applied to a class of hyperbolic-parabolic systems
In this talk I will discuss the application of the theory of a-contraction up to shift for a class of hyperbolic-parabolic systems in one-dimension satisfying some conditions. This theory aims at deriving some control on the distance between a solution to the system and a translation of a given viscous shock. In particular, I will discuss how to derive such estimates for these systems, and how they can be combined with Kawashima estimates to prove some stability results of viscous shocks in Sobolev spaces. This is based on a joint work with Alexis Vasseur.
Geng Chen (University of Kansas)
Poiseuille flow of nematic liquid crystals via Ericksen-Leslie model
In this talk, we will discuss the global existence, singularity formation and regularity of solution on the Poiseuille flow of nematic liquid crystals via full Ericksen-Leslie model. The existing results on the Ericksen-Leslie model for liquid crystals mainly focused on the parabolic and elliptic type models by omitting the kinetic energy term. Currently, there are still not many results on the full (parabolic-hyperbolic type) Ericksen-Leslie model. Two types of singularity formation results will also be discussed, together with the global existence result showing that the solution will in general live in the Holder continuous space. This direction includes joint works with Weishi Liu, Tao, Huang, Majed Sofiani, Xiang Xu and Qingtian Zhang. The earlier related result joint with Alberto Bressan on the stability of variational wave equation using the optimal transport method, will also be briefly discussed.
Kyudong Choi(UNIST)
Existence and stability of a Sadovskii dipole as a maximizer of kinetic energy
The Sadovskii vortex patch is a traveling wave for the two-dimensional incompressible Euler equations consisting of an odd symmetric pair of vortex patches touching the symmetry axis. Its existence was first suggested by numerical computations of Sadovskii in [J. Appl. Math. Mech., 1971], and has gained significant interest due to its relevance in the inviscid limit of planar flows via Prandtl--Batchelor theory and as the asymptotic state for vortex ring dynamics.In this talk, I will sketch a proof of the existence of such a vortex and stability in the class using an energy maximization approach under the exact impulse condition and an upper bound on the circulation. (For reference, a completely different proof of the same existence result with more information via a fixed point method appeared around the same time by Huang and Tong. The uniqueness of such a vortex remains open.) This talk is based on joint work with In-Jee Jeong(SNU), Youngjin Sim(UNIST), and Kwan Woo(SNU).
Jan Giesselmann (TU Darmstadt)
Theory of shifts and its connections to a posteriori error analysis of numerical schemes for hyperbolic problems
We discuss the development of reliable a posteriori error estimates for numerical approximations of scalar hyperbolic conservation laws in one space dimension. Our methods have no inherent small-data limitations and are a step towards error control of numerical schemes for systems. We do not appeal to the Kruzhkov theory for scalar conservation laws. Instead, we derive novel quantitative stability estimates that extend the theory of shifts, and in particular, the framework for proving stability first developed by Krupa and Vasseur. We work entirely within the context of the theory of shifts and a-contraction, techniques which adapt well to systems.
Our theoretical findings are complemented by numerical experiments.
Katrin Grunert (NTNU)
Uniqueness for the Camassa--Holm equation
Solutions of the Camassa--Holm (CH) equation might enjoy wave breaking in finite time. This means that even classical solutions, in general, do not exist globally, but only locally in time, since their spatial derivative might become unbounded from below pointwise in finite time, while the solution itself remains continuous and bounded. Furthermore, energy concentrates on sets of measure zero when wave breaking occurs. Thus the prolongation of solutions beyond wave breaking is non-unique and depends heavily on how the concentrated energy is manipulated. The two most prominent ones are called dissipative, i.e., the concentrated energy is taken out, and conservative, i.e., the energy is given back into the system. The existence of both types of solutions has been established with the help of a generalized method of characteristics, which allows to rewrite the Camassa--Holm equation as a system of ODEs in Lagrangian coordinates.
Here we will focus on the uniqueness of dissipative solutions, for which all of the concentrated energy is removed upon wave breaking. The overall idea is to establish a bijection between the properties satisfied by each dissipative solution and the solution operator defined via a generalized method of characteristics. The main ingredients are measure transport equations, ideas from sub- and supersolutions as well as a good understanding of the characterisation of equivalence classes in Lagrangian coordinates. Furthermore, we will compare the necessary properties guaranteeing uniqueness for dissipative solutions to those for the conservative solutions.
Jonas Hirsch (University of Leipzig)
A Nash-Kuiper theorem for isometric embeddings beyond Borisov’s exponent
This joint work with Wentao Cao, and Dominik Inauen.
For any given short embedding from an n-dimensional region into (n + 1)-dimensional Euclidean space, and for any Hölder exponent \alpha < 1/(1+n^2-n), a C^{1,\alpha}-isometric embedding is built within any C^0 neighbourhood of the given short embedding through convex integration, which refines the classical Nash-Kuiper the- orem and extends the flexibility of C^{1,\alpha} isometric embedding beyond Borisov’s ex- ponent. Notably, when n=2, we attain the Onsager exponent 1/3 for isometric embeddings. This convex integration scheme is performed through new construc- tion and leveraging iterative “integration by parts” to effectively transfer large-scale errors to smaller ones. In my talk, I would like to give some ideas for the “integration by parts” proce- dure. Furthermore, I will highlight the differences between the schemes that were previously used."
Jinwoo Jang (Postech)
Boltzmann Equation and the Vanishing Angular Singularity Limit
In this talk, we study the Boltzmann equation for hard spheres and for inverse-power-law potentials. We first discuss the derivation of the Boltzmann equation from N-body dynamics, introducing the BBGKY hierarchy, the Boltzmann-Grad limit, and Lanford's theorem. We then consider a general collisional kinetic equation associated with inverse-power-law interactions and establish a connection between the corresponding Boltzmann equation and the hard-sphere Boltzmann equation through the vanishing angular singularity limit. Precise asymptotic formulas describing the behavior of the singular layer in the limit are also provided. This is joint work with B. Kepka, A. Nota, and J. J. L. Velázquez.
Jeongho Kim (Kyung Hee Univ.)
Convergence to superposition of boundary layer, rarefaction, and shock for the 1D Navier-Stokes equations
We establish the asymptotic stability of solutions to the inflow problem for the one-dimensional barotropic Navier-Stokes equations in half space. When the boundary value is located at the subsonic regime, all the possible thirteen asymptotic patterns are classified in [11]. We consider the most complicated pattern, the superposition of the boundary layer solution, the 1-rarefaction wave, and the viscous 2-shock waves. In this superposition, the boundary layer is degenerate and large. We prove that, if the strengths of the rarefaction wave and shock wave are small, and if the initial data is a small perturbation of the superposition, then the solution asymptotically converges to the superposition up to a dynamical shift for the shock. As a corollary, our result implies the asymptotic stability for the simpler case where the superposition consists of the degenerate boundary layer solution and the viscous 2-shock. Therefore, we complete the study of the asymptotic stability of the inflow problem for the 1D barotropic Navier-Stokes equations for subsonic boundary values.
Bongsuk Kwon (UNIST)
Finite-time blow-up in transport-type equations
We investigate $C^1$ blow-up phenomena in various transport-type equations. The classical method of characteristics provides only limited insights into blow-up solutions, such as the temporal blow-up rate along characteristic curves. While this approach has been successfully applied to the Burgers equation and several wave-breaking models in water waves, it offers an incomplete picture of singularity formation. It is well known that the singularity in the Burgers equation exhibits $C^{1/3}$ regularity at the blow-up time, resembling the singularities observed in the Euler equations. A prevailing conjecture suggests that all $C^1$ blow-up solutions in transport equations share the same regularity properties as those in the Burgers equation. However, our recent findings challenge this conventional belief. We demonstrate that $C^1$ blow-up solutions can exhibit singularities distinct from those of the Burgers equation. To substantiate this claim, we construct self-similar blow-up profiles for a general transport-type equation, which frequently serves as the leading-order correction to well-known water wave models. These profiles, which may differ significantly from those of the Burgers equation, lead to distinct Holder regularities. The models under consideration include the Camassa-Holm equation, the Hunter-Saxton model, and the b-family of fluid transport equations. We will discuss the construction of specific stable blow-up profiles and outline the proof of their stability in an appropriate functional topology. This analysis establishes the precise regularity of blow-up solutions and offers detailed insights into their dynamics. If time permits, we will also introduce more exotic blow-up profiles, derived within the framework of self-similar variables.
This is joint work with Yunjoo Kim at UNIST.
Hyunju Kwon (ETH Zurich)
Non-Conservation of Generalized Helicity in 3D Euler Flows
Recently, there has been significant research into the non-conservation of total kinetic energy in Euler flows, which has led to Onsager’s theorem and its intermittent version. In this talk, I will discuss an analogous question for another conserved quantity: helicity. I will present the first example of a weak solution to the 3D Euler equations in C^0_t(H^{1/2-}\cap L^{\infty-}) for which the helicity, defined in a generalized sense, is not conserved in time. The talk will be based on recent collaboration with Matthew Novack and Vikram Giri.
Donghyun Lee (Postech)
Optimal C^{1/2} regularity of the Boltzmann equation in non-convex domains
The regularity of Boltzmann-type equations, particularly the hard-sphere Boltzmann equation, with respect to the shape of domains presents a challenging research theme. A crucial aspect in determining the regularity of a Boltzmann solution lies in the intricate interaction between characteristics and the boundary. For uniformly convex domains, weighted C^1 and W^{1,p} type regularity results have been established for specular reflection and diffuse boundary conditions (BC) respectively. In contrast, for non-convex domains with diffuse BC, propagation of discontinuity and BV regularity results are known. However, the problem concerning specular reflection BC in non-convex domains remained outstanding due to the severe singularity near the grazing set, where the trajectory map exhibits only C^{1/2} regularity. This challenge was recently addressed affirmatively by the second author and his collaborator, who demonstrated (weighted) C^{1/2−} regularity outside a uniformly convex obstacle through sharp singularity averaging in the billiard map. In this paper, we introduce a novel 'dynamical singular regime integration' method to obtain optimal (weighted) C^{1/2} regularity for the Boltzmann solution past a convex obstacle. This is joint work with Gayoung An.
Jaemin Park (Yonsei Univ.)
Stability of stratified density under incompressible flow
In this talk, I will discuss asymptotic stability in the incompressible porous media equation in a periodic channel. It is well known that a stratified density, which strictly decreases in the vertical direction, is asymptotically stable under sufficiently small, smooth perturbations. We achieve improvements in the regularity assumptions on the perturbation and in the convergence rate. We apply a similar idea to the Stokes transport system. Instead of relying on the linearized equations, we directly address the nonlinear problem, and the decay of solutions will be obtained from the gradient flow structure of the equation.
Charlotte Perrin (Université Aix Marseille)
Constrained 1D Euler equations
In this talk, I will review recent results on one-dimensional Euler equations with a maximal density constraint. Such a constraint models congestion phenomena in fluid flows and can arise either from microscopic non-overlapping conditions or from geometric restrictions on the flow. I will present results concerning the existence of solutions within this constrained framework, highlighting the mathematical challenges posed by the presence of a density threshold and its implications on the dynamics of the system.
Miguel Rodrigues (Univ. Rennes)
Singular waves of hyperbolic systems, a frontier in nonlinear wave stability
After a broad introduction to large-time dynamics, the talk will focus on a part of the frontiers of our current understanding of nonlinear stability of traveling waves of partial differential equations, especially on how spectral stability implies nonlinear stability and which kind of dynamics may be expected. We shall highlight main expected difficulties related to the stability of discontinuous and/or characteristic waves of hyperbolic systems, and show a few significant steps obtained by the speaker with respectively Vincent Duchêne (Rennes), Gregory Faye (Toulouse) and Louis Garénaux (Karslruhe).
Yannan Shen (University of Kansas)
Regularity of a family of quasilinear waves
In this talk, we consider the Cauchy problem of a family of quasilinear wave equations, named as $\lambda$-family equations where $\lambda$ is related to the power of the nonlinear wave speed. The $\lambda$-family equations include conservation laws ($\lambda=1$), Hunter-Saxton and Camassa-Holm equation ($\lambda=1/2$) and Novikov equation ($\lambda=1/4$), a nonlinear wave equation ($\lambda=0$). The solutions generically form finite time cusp singularities for $0<\lambda<1$. We will show the global energy conservative solution is actually H\"older continuous with exponent $1-\lambda$.
Yi Wang (Institute of Applied Mathematics, AMSS, CAS)
On the stability of viscous shock and its composite waves by a-contraction method: from scalar viscous conservation laws to Boltzmann equation
The talk is concerned with our recent developments on the time-asymptotic stability of viscous shock and its composite with rarefaction wave (and even with viscous contact wave) for several kinds of viscous conservation laws, from scalar non-convex viscous conservation laws, to compressible barotropic Navier-Stokes equations, full compressible Navier-Stokes-Fourier equations and non-convex viscoelasticity system with physical viscosities, and to kinetic Boltzmann equation, by using a-contraction method. I will also talk about the time-asymptotic shock location of the time-dependent shift in a-contraction method for the stability of a single viscous shock, which, in particular, shows the compatibility of the classical anti-derivative techniques and a-contraction method for shock stability, and moreover, that the shock stability by a-contraction method is more general than the anti-derivative framework in a broader perturbation space.
Emil Wiedemann (University of Erlangen-Nürnberg)
Selection of Measure-Valued Solutions
Measure-valued solutions for fluid equations have been around for decades but have received renewed interest in recent years. Measure-valued solutions and related concepts (such as dissipative or dissipative measure-valued solutions) can be useful in the study of singular limits and numerical approximation. It is usually not difficult to construct measure-valued solutions, however in many situations they display a vast degree of non-uniqueness. I will present some recent results on possible selection criteria. While there doesn't seem to be a criterion that singles out one unique solution, I shall argue that many undesirable ones are eliminated. Joint work with Dennis Gallenmüller and with Christian Klingenberg and Simon Markfelder.