Abstract : Building a satisfactory mathematical theory of turbulence remains one of the most significant challenges in the physical sciences, with several fundamental problems still open. Deeply tied to these problems are issues associated with the regularity and well-posedness of fluid flows. Recent developments have shed light on the chaotic behavior of turbulent flows, the role of criticality in ill-posedness, singularity formation in classical solutions, and the relation between irregularity, instability, and non-uniqueness of solutions. The goal of this session is to gather those who have contributed to these developments, promote the exchange of ideas, and inspire new ones.
Organizer(s) : Aseel Farhat, Evelyn Lunasin, Vincent Martinez
[03313] Singularity formation for models of fluids
Format : Talk at Waseda University
Author(s) :
Mimi Dai (University of Illinois at Chicago)
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.
[03731] Vorticity estimates for the 3D incompressible Navier-Stokes equation
Format : Online Talk on Zoom
Author(s) :
Jincheng Yang (University of Chicago)
Abstract : We show some a priori regularity estimates for the vorticity and its trace in the three-dimensional incompressible Navier-Stokes equation. These a priori estimates are obtained via the blow-up method and a novel averaging operator. The averaging operator can be used to provide regularity and trace estimates for PDEs with $\varepsilon$-regularity.
[04943] On criticality of the Navier-Stokes diffusion
Format : Online Talk on Zoom
Author(s) :
zoran grujic (university of virginia)
Abstract : The main purpose of this talk is to present a mathematical evidence of criticality of the Navier-Stokes diffusion. In particular, considering a plausible candidate for a finite time blow-up, a two-parameter family of the dynamically rescaled profiles, we show that as soon as the hyper-diffusion exponent is greater than one, a new region in the parameter space (completely in the super-critical regime) is ruled out. As a matter of fact, the region is a neighborhood (in the parameter space) of the self-similar profile, i.e., the `approximately self-similar' blow-up is ruled out for all hyper-diffusive models.
[05062] Well-posedness of mildly regularized active scalars in Sobolev spaces
Format : Online Talk on Zoom
Author(s) :
Anuj Kumar (Florida State University)
Vincent Ryan Martinez (CUNY Hunter College)
Abstract : In this work, we consider the initial value problem for a family of active scalar equations when perturbed by a logarithmic order regularization in dissipation. These equations, commonly known as generalized surface quasi-geostrophic equations (gSQG) interpolate between the 2D incompressible Euler equation and the 2D SQG equation, and extrapolate beyond SQG to a family with more singular velocities. Ill-posedness at the threshold regularity for the
unperturbed models has been established in the celebrated works of Bourgain and Li, and Elgindi and Masmoudi for the 2D Euler equation, and recently by Cordoba and Zoroa-Martinez, and Jeong and Kim for the 2D SQG equation. In this work, we treat the positive side of well-posedness and consider a minimally dissipative regularization to recover local well-posedness (in the Hadamard sense) in the threshold Sobolev regularity. The proof is based on developing estimates for a suitably identified linear system that preserves the underlying commutator structure of the nonlinearity.
[04347] Bounded weak solutions to the 2D quasi-geostrophic equation
Format : Online Talk on Zoom
Author(s) :
Elaine Cozzi (Oregon State University)
Abstract : We outline a proof of global existence of bounded weak solutions to the 2D quasi-geostrophic equation (SQG), building on a result of Marchand. Our proof utilizes a Littlewood-Paley version of a Serfati type of identity for SQG. This is joint work with David Ambrose and Jim Kelliher.
[02935] On the support of anomalous dissipation measures
Format : Online Talk on Zoom
Author(s) :
Theodore D. Drivas (Stony Brook University)
Abstract : By means of a unifying measure-theoretic approach, we establish lower bounds on
the Hausdorff dimension of the space-time set which can support anomalous dissipation for
weak solutions of fluid equations, both in the presence or absence of a physical boundary.
Boundary dissipation, which can occur at both the time and the spatial boundary, is analyzed
by suitably modifying the Duchon & Robert interior distributional approach. One implication
of our results is that any bounded Euler solution (compressible or incompressible) arising as
a zero-viscosity limit of Navier--Stokes solutions cannot have anomalous dissipation supported
on a set of dimension smaller than that of the space. This is joint work with L. De Rosa and M. Inversi.
[03059] Kinetic shock profiles for the Landau equation
Format : Talk at Waseda University
Author(s) :
Dallas Albritton (UW-Madison)
Abstract : Compressible Euler solutions develop jump discontinuities known as shocks. However, physical shocks are not, strictly speaking, discontinuous. Rather, they exhibit an internal structure which, in certain regimes, can be represented by a smooth function, the shock profile. We demonstrate the existence of weak shock profiles to the kinetic Landau equation. Joint work with Matthew Novack (Purdue University) and Jacob Bedrossian (UCLA).
[04056] On sharp-crested water waves and finite-time singularity formation
Format : Online Talk on Zoom
Author(s) :
Nastasia Grubic (ICMAT, CSIC)
Abstract : We show that the 2d gravity water waves system is locally wellposed in weighted Sobolev spaces which allow for interfaces with corners. These singular points are not rigid; if the initial interface exhibits a corner, it remains a corner but generically its angle changes. Using a characterization of the asymptotic behavior of the fluid near a corner that follows from our a-priori energy estimates, we show the existence of initial data in these spaces for which the fluid becomes singular in finite time.
[03868] A localized maximum principle and its application to the critical SQG on bounded domain
Format : Talk at Waseda University
Author(s) :
Tsukasa Iwabuchi (Tohoku University)
Abstract : We discuss a spectral localization technique for the Dirichlet Laplacian on smooth bounded domain to deal with the fractional Laplacian of the derivative order one and commutator estimates in the framework of Besov spaces. It corresponds to a generalization of the analysis by the dyadic decomposition of the frequency through the Fourier transform in the Euclidean space. As an application, we show the existence of global solutions of the surface quasi-geostrophic equation with the critical dissipation for small initial data.
[04796] Turbulent solutions of fluid equations
Format : Online Talk on Zoom
Author(s) :
Alexey Cheskidov (University of Illinois at Chicago)
Abstract : In the past couple of decades, mathematical fluid dynamics has been highlighted by numerous constructions of solutions to fluid equations that exhibit pathological or wild behavior. These include the loss of the energy balance, non-uniqueness, singularity formation, and dissipation anomaly. Interesting from the mathematical point of view, providing counterexamples to various well-posedness results in supercritical spaces, such constructions are becoming more and more relevant from the physical point of view as well. Indeed, a fundamental physical property of turbulent flows is the existence of the energy cascade. Conjectured by Kolmogorov, it has been observed both experimentally and numerically, but had been difficult to produce analytically. In this talk I will overview new developments in discovering not only pathological mathematically, but also physically realistic solutions of fluid equations.
[03232] Speeding up Langevin Dynamics by Mixing
Format : Talk at Waseda University
Author(s) :
Yuanyuan Feng (East China Normal University)
Gautam Iyer (Carnegie Mellon University)
Alexei Novikov (Penn State University)
Alexander Christie (Penn State University)
Abstract : We add a drift to the Langevin dynamics (without changing the stationary distribution) and obtain quantitative estimates on the mixing time. We show that an exponentially mixing drift can be rescaled to make the mixing time of the Langevin system arbitrarily small.
[04760] On intermittent strong Onsager conjecture
Format : Online Talk on Zoom
Author(s) :
Hyunju Kwon (ETH Zurich)
Abstract : Smooth solutions to the incompressible 3D Euler equations, which are spatially periodic, are known to conserve kinetic energy in every local region. Turbulent flows, however, exhibit anomalous dissipation of kinetic energy, indicating the existence of a weak solution to the Euler equations with dissipation of kinetic energy in some region, but no creation of energy everywhere in the domain. This motivates the strong Onsager conjecture, which combines the original Onsager conjecture with the local energy inequality. In this talk, I will discuss the flexibility side of the $L^3$-based strong Onsager conjecture, adapting to the intermittent nature of turbulence, and introduce a wavelet-based convex integration scheme. The talk is based on a joint work with Matt Novack and Vikram Giri.
