Title & Abstract

Monday (15th)

• Ohkitani, Koji (RIMS, Kyoto University)

  • Title: Interpolation between and regularity criteria for the Navier-Stokes and the solenoidal Burgers equations 

  • Abstract: The multi-dimensional Burgers equations are integrable in that they can be reduced to a heat equation under the assumption of potential flows via the so-called Cole-Hopf linearisation. On the other hand, it is believed that the Navier-Stokes equations are not.

In two spatial dimensions, by rotating the velocity gradient by 90 degrees we can obtain an equation which is equivalent to the Burgers equation. Rotating the velocity gradient continuously we introduce a generalised system of incompressible fluid equations that allows us to compare an integrable system with non-integrable ones. Based on direct numerical experiments of the generalised system, we find that the Burgers equation is the most singular in terms of enstrophy.

We then discuss preliminary results on a similar extension in three spatial dimensions using vector potentials. We compare numerically the solenoidal Burgers and the Navier-Stokes equations with a regularity criterion integral involving enstrophy. 

• Ko, Seungchan (Inha University)

  • Title: Analysis and Approximation of Incompressible Chemically Reacting Generalized Newtonian Fluid

  • Abstract: We consider a system of nonlinear partial differential equations modeling the steady motion of an incompressible non-Newtonian fluid, which is chemically reacting. The governing system consists of a steady convection-diffusion equation for the concentration and the generalized steady Navier–Stokes equations, where the viscosity coefficient is a power-law type function of the shear-rate, and the coupling between the equations results from the concentration-dependence of the power-law index. This system of nonlinear partial differential equations arises in mathematical models of the synovial fluid found in the cavities of moving joints. We construct a finite element approximation of the model and perform the mathematical analysis of the numerical method. Key technical tools include discrete counterparts of the Bogovskii operator, De Giorgi’s regularity theorem in two dimensions, and the Acerbi-Fusco Lipschitz truncation of Sobolev functions, in function spaces with variable integrability exponents.

• Michele Dolce (EPFL)

  • Title: Vortex pairs in the large Reynolds number regime

  • Abstract: The evolution of two point vortices in a 2D inviscid fluid in the whole plane, whether counter-rotating or co-rotating, is explicitly determined by the Helmholtz-Kirchhoff system. One translates on a straight line with a constant speed, while the latter rigidly rotates around each other. At a large but finite Reynolds number, vortex core sizes grow due to diffusion, revealing immensely rich phenomena stemming from this simple initial configuration.

The goal is to validate an asymptotic expansion in terms of small adimensional parameters, isolating inviscid and viscous effects. Moreover, at a given order of the expansion, we identify corrections to the Helmholtz-Kirchhoff motion, in agreement with observations in the applied literature. We rigorously justify the approximation by controlling the remainder with an energy method, where Arnold's variational principle for the stability of vortices plays a fundamental role in our analysis. This is an ongoing project with T. Gallay.

• Abe, Ken (Osaka Metropolitan University)

  • Title: Existence of homogeneous Euler flows of degree −α ∉ [−2, 0]

  • Abstract: Shvydkoy (2018) demonstrated the nonexistence of (−α)-homogeneous solutions to the stationary incompressible Euler equations in R³{0} in the range 0 ≤ α ≤ 2 for the Beltrami and axisymmetric flows. In this talk, I will discuss the existence of axisymmetric (−α)-homogeneous solutions in the complementary range α ∈ R[0, 2].

• Del Zotto Augusto (Imperial College London)

  • Title: The role of internal oscillations in the 3D Boussinesq equations around a stably stratified Couette flow

  • Abstract: The Boussinesq equations are a fundamental tool for describing the behavior of stratified fluids. When considering a stably stratified Couette flow, the oscillations generated significantly contribute to the stability of the steady state. In this presentation, we will discuss how this oscillating structure enters the analysis of linear inviscid damping and enhanced dissipation. Additionally, we will address how oscillations effectively remove the 3D lift-up effect. Finally, we will comment on the nonlinear transition threshold of this problem, which utilizes the oscillations through dispersive estimates. This is joint work with Michele Coti Zelati and Klaus Widmayer.

Tuesday (16th)

• Cheskidov, Alexey (Westlake)

  • Title: Dissipation anomaly for long time averages

  • Abstract: In turbulent flows, the energy injected at forced low modes (large scales) cascades to small scales through the inertial range where viscous effects are negligible, and only dissipates above Kolmogorov’s dissipation wavenumber. The persistence of the energy flux through the inertial range is what constitutes dissipation anomaly for viscous fluid flows as well as anomalous dissipation for the limiting inviscid flows. We first analyze these intrinsically linked phenomena on a finite time interval and prove the existence of various scenarios in the limit of vanishing viscosity, ranging from the total dissipation anomaly to a pathological one where anomalous dissipation occurs without dissipation anomaly, as well as the existence of infinitely many limiting solutions of the Euler equations in the limit of vanishing viscosity. Finally, expanding on the obtained total dissipation anomaly construction, we show the existence of dissipation anomaly for long time averages, relevant for turbulent flows, proving that the Doering-Foias upper bound is sharp.

• Dai, Mimi (UIC)

  • Title: Singularity formation for fluid models

  • Abstract: Finite time singularity formation for fluid equations will be discussed. Built on extensive study of approximating models, breakthroughs on this topic have emerged recently for Euler equation. Inspired by the progress for pure fluids, we attempt to understand this challenging issue for magnetohydrodynamics (MHD). Finite time singularity scenarios are discovered for some reduced models of MHD. The investigation also reveals connections of MHD with Euler equation and surface quasi-geostrophic equation.

• Coti Zelati, Michele (Imperial College London)

  • Title: Entropy maximization in the two-dimensional Euler equations

  • Abstract: We investigate certain questions arising in two-dimensional statistical hydrodynamics, by relying on principles of entropy maximization for the vorticity of a two-dimensional perfect fluid in a disc. In analogy with the entropy functions used in statistical mechanics and thermodynamics, we show that similar concavity properties hold for the 2d Euler equations when maximizing entropies at fixed energy levels. The proofs rely on rearrangement inequalities, a modification of the classical min-max principle, and the properties of the Euler-Lagrange equations for the corresponding constrained optimization.

• Choi, Kyudong (UNIST)

• • Title: Large growth in vorticity maximum for anti-parallel axisymmetric flows

• • Abstract: We consider anti-parallel axisymmetric  incompressible inviscid  flows  without swirl. When  the axial vorticity is non-positive in the upper half space and odd in the last coordinate, we may expect a head-on collision of anti-parallel vortex rings. By establishing monotonicity and infinite growth of the vorticity impulse on the upper half-space,  we obtain infinite growth of vorticity maximum  for certain classical/smooth and compactly supported vorticity solutions in R^3. This talk is based on joint work with In-Jee Jeong(SNU).

• Lee, Donghyun (POSTECH)

  • Title: Recent developments on geometric effects on regularity of a Boltzmann solution 

  • Abstract : Regularity and singularity of the solution according to the shape of domains is a challenging research theme in the Boltzmann theory. In this talk, we discuss some recent developments in this research theme which contains non-convex boundary effects on the regularity of a Boltzmann solution. In particular, (optimal) C1/2− regularity of the hard-sphere Boltzmann equation exterior of a uniformly convex obstacle and C1/4− regularity in concentric cylinder will be studied under the specular reflection boundary condition. This talk is based on joint works with Chanwoo Kim and Gayoung An.

• Yun, Seok-Bae (SKKU)

  • Title: Weak solutions to the Boltzmann-BGK model in a slab

  • Abstract: We consider stationary flows between two condensed phases in the framework of the stationary BGK model in a slab. Under the physically minimum conditions on the inflow functions, namely the finite mass flux, energy flux and entropy flux, the existence of weak solution is derived. The main difficulties are, among others, (1) the impossibility of truncation of the relaxation operator in the construction of the approximate scheme, and (2) the control of the velocity distribution functions using the macroscopic fields near vanishing velocity region. This is a joint work with Stephane Brull.

Wednesday (17th)

• Bumja, Jin

  • Title: Decay rate of the Stokes flow in the exterior domain with slowly decaying initial data

  • Abstract: TBA

• Hong, Youngjoon

  • Title: Machine Learning for Computational Fluid Dynamics

  • Abstract: Machine learning is rapidly becoming a core technology for scientific computing, with numerous opportunities to advance the field of computational fluid dynamics. This lecture highlights some of the areas of highest potential impact, including to accelerate numerical simulations, to improve efficiency of scientific computing. In each of these areas, it is possible to improve machine learning capabilities by incorporating physics into the process, and in turn, to improve the simulation of fluids to uncover new physical understanding. Despite the promise of machine learning described here, we also note that classical methods are often more efficient for many tasks. We demonstrate that this framework facilitates the rapid discovery of accurate constitutive equations from limited data, and that the learned models may be used to describe more complex systems. This inherent flexibility admits the application of these digital twins to a range of material systems and engineering problems. We illustrate this flexibility by deploying a trained model within a multidimensional computational fluid dynamics simulation.

Thursday (18th)

• Ozanski, Wojciech

  • Title: Instabilities in incompressible inviscid fluids

  • Abstract: We will discuss some recent developments which enable us to rigorously construct unstable perturbations of some families of steady $2$D and $3$D Euler flows, which involve some $2$D shear flows, $2$D vortices, and $3$D vortex columns, that is vector fields of the form $u=V(r)e_{\theta } + W(r) e_z$, where $r$ denotes the distance to the axis of rotation and $e_\theta$ and $e_z$ denote the standard cylindrical unit vectors, for a family of profiles $V,W$.

We will demonstrate the first construction of infinitely many, genuinely three-dimensional modes of instabilities of some vortex columns, which take the form of ``ring modes'', localized around $r=r_0$, for some $r_0>0$. We will describe the relevance of such instabilities and related open problems in the context of experimental and numerical phenomena observed in incompressible fluids. This is joint work with D. Albritton.

• Kwon, Bongsuk

  • Title: Blow-up for the Euler-Poisson system of ion dynamics

  • Abstract: Various plasma phenomena are mathematically studied using a fundamental fluid model for plasmas, called the Euler-Poisson system. Among them, plasma solitary waves are of our interest, for which existence, stability, and the time-asymptotic behavior of the solitary wave will be briefly discussed. On the other hand, to study nonlinear stability, a question of existence of smooth global solution naturally arises, which is completely open, to the best of our knowledge. We introduce the finite-time blow-up results for the Euler-Poisson system, and discuss the related open questions. This talk is based on joint work with Junsik Bae at UNIST.

• Gotoda, Takeshi

  • Title: Vortex dynamics on the 2D filtered Euler flow

  • Abstract: We consider weak solutions of the 2D filtered-Euler equations, which are a regularization of the 2D Euler equations based on a spatial filtering. We introduce the results of the global well-posedness for initial vorticity in the space of finite Radon measures and the convergence to the 2D Euler equations for vortex sheet initial vorticity in the limit of a filtering scale. We also see the energy conservation in the filter scale limit of the 2D filtered-Euler equations and give some examples of filtered weak solutions that dissipate the enstrophy.

• Kang, Moon-Jin (KAIST)

  • Title: Well-posedness of small BV solutions to the isentropic Euler from Navier-Stokes

  • Abstract: The Cauchy problem for compressible Euler system from inviscid limit of Navier-Stokes remains completely open, as a challenging issue in fluid dynamics. In this talk, I will give a first resolution for this problem in the 1D isentropic case. We will show the global well-posedness of entropy solutions with small BV initial data in the class of inviscid limits from the associate Navier-Stokes.

The proof is based on the three main methodologies: the modified front tracking algorithm; the a-contraction with shifts; the method of compensated compactness.

This is a joint work with Geng Chen (U. Kansas) and Alexis Vasseur (UT-Austin).

• Ayman Said (Cambridge)

  • Title: Small scale creation of the Lagrangian flow in 2d perfect fluids

• • Abstract: In this talk I will present a recent result showing that for all solutions of the 2d Euler equations with initial vorticity with finite Sobolev smoothness then an initial data dependent norm  of the associated Lagrangian flow blows up in infinite time at least like $t^{\frac{1}{3}}$. This initial data dependent norm quantifies the exact $L^2$ decay of the Fourier transform of the solution. This adapted norm turns out to be the exact quantity that controls a low to high frequency cascade which we then show to be the quantitative phenomenon behind the Lyapunov construction by Shnirelman.

• Sato Naoki

  • Title: On the Landau collision operator in noncanonical phase space and the resulting modification of the moments equations governing fluid dynamics

  • Abstract: The phase space of a noncanonical Hamiltonian system is partially inaccessible due to the presence of dynamical constraints (Casimir invariants) arising from the kernel of the Poisson operator. When an ensemble of noncanonical Hamiltonian systems is allowed to interact, dissipative processes eventually break the phase space constraints, resulting in a thermodynamic equilibrium described by a Maxwell-Boltzmann distribution. However, the time scale needed to reach such statistics is often much longer than the time scale over which a state of equilibrium is achieved. Examples include collisionless relaxation in magnetized plasmas and stellar systems, where the interval between binary Coulomb or gravitational collisions can be longer than the time scale over which stable structures are self-organized. Here, we study self-organizing phenomena over spacetime scales such that particle interactions do not break the noncanonical Hamiltonian structure, but yet act to create a state of thermodynamic equilibrium. We derive a collision operator for fast spatially localized interactions in general noncanonical Hamiltonian systems which depends both on the interaction exchanged by colliding particles and on the Poisson operator encoding the noncanonical phase space structure. This collision operator is consistent with conservation of particle number and energy as well as with entropy growth, preserves the total value of each Casimir invariant, reduces to the Landau collision operator in the limit of grazing binary Coulomb collisions in canonical phase space, and exhibits a metriplectic structure. We further show how thermodynamic equilibria depart from Maxwell-Boltzmann statistics due to the noncanonical phase space structure, how collisionless relaxation in magnetized plasmas and stellar systems can be described through the derived collision operator, and how the fluid equations governing the moments of the distribution function depart from the standard Navier-Stokes equations.

Friday (19th)

• Yamazaki, Kazuo

  • Title: Recent Developments on Uniqueness and Non-uniqueness of (Singular) Stochastic PDEs

  • Abstract: Singular stochastic PDEs refer to PDEs forced by noise that is so rough that the products within the nonlinear terms become ill-defined. We review recent developments concerning the solution theory of such singular stochastic PDEs (some well-posed and some ill-posed).

• Jo, Min Jun

  • Title: Breakdown of the QG description in certain balance between rotation and stratification

  • Abstract: The QG (quasi-geostrophic) description simplifies the analysis of a large class of geophysical fluids in the asymptotic regime of rapid rotation and strong stratification. In this talk, we show that such well-known approximation is invalid under certain balance between rotation and stratification. More precisely, we prove non-convergence of the rotating stratified 3D incompressible inviscid Boussinesq flows toward the corresponding 3D inviscid QG flows when the rotation-stratification ratio is fixed to be one or tends to one sufficiently slowly. This is a joint work with Junha Kim and Jihoon Lee.

• Kim, Junha

  • Title: On the wellposedness of gSQG equation

  • Abstract: In this talk, we consider the gSQG equation in a half-plane. We prove the local wellposedness of gSQG in an anisotropic Lipschitz space and the instantaneous blow-up of solutions in Hölder spaces. This is a joint work with In-Jee Jeong(SNU) and Yao Yao(NUS).