1pm on Wednesdays (sometimes Mondays or Fridays), in PMA 10.176

Please email Stefania Patrizi (*spatrizi AT math.utexas.edu* or Benjamin Seeger (*seeger AT math.utexas.edu*) if you would like to be added to the mailing list.

Occtober 4, 2023
#### Aaron Nung Kwan Yip, Purdue University

TBA
Occtober 25, 2023
#### Jeff Calder, University of Minnesota

TBA

TBA

TBA

January 11, 2023
#### Jonas Lührmann, Texas A&M

On co-dimension one stability of the soliton for the 1D focusing cubic
Klein-Gordon equation
January 18, 2023
#### Levon Nurbekyan, UCLA (Special Zoom talk)

On Mean-Field Games
January 25, 2023
#### Nestor Guillen, Texas State

Approximate Perron Methods
February 3, 2023 (Friday)
#### Rocio Diaz-Martin, Vanderbilt University

Transport as an embedding tool
February 22, 2023
#### Pablo Raúl Stinga, Iowa State

Surfaces of minimum curvature variation
March 1, 2023
#### Federico Glaudo, IAS

On the stability of the fractional Caffarelli-Kohn-Nirenberg inequality
March 8, 2023 (Zoom talk)
#### Irene Fonseca, Carnegie Mellon

Phase Separation in Heterogeneous Media
March 22, 2023 (Zoom talk)
#### Pierre-Emmanuel Jabin, Penn State

A new approach to the mean-field limit of Vlasov-Fokker-Planck equations
March 29, 2023
#### Andrzej Święch, Georgia Tech

Finite dimensional approximations of Hamilton-Jacobi-Bellman equations in spaces of probability measures and stochastic optimal control of particle systems
March 31, 2023 (Friday)
#### Michele Coti Zelati, Imperial College

Diffusion and mixing for two-dimensional Hamiltonian flows
April 5, 2023
#### Juhi Jang, University of Southern California

Dynamics of Newtonian stars
April 12, 2023
#### Mihaela Ignatova, Temple University

TBA
April 14, 2023 (Friday)
#### Toan T. Nguyen, Penn State

The survival threshold for plasma oscillations
April 19, 2023
#### Yu Gu, University of Maryland

A PDE hierarchy for directed polymer in random environment

Solitons are particle-like solutions to dispersive evolution equations whose shapes persist as time goes by. In some situations, these solitons appear due to the balance between nonlinear effects and dispersion, in other situations their existence is related to topological properties of the model. Broadly speaking, they form the building blocks for the long-time dynamics of dispersive equations.

In this talk I will present forthcoming joint work with W. Schlag on long-time decay estimates for perturbations of the soliton for the 1D focusing cubic Klein-Gordon equation (up to exponential time scales), and I will discuss our previous work on the asymptotic stability of the sine-Gordon kink under odd perturbations. While these two problems are quite similar at first sight, we will see that they differ by a subtle cancellation property, which has significant consequences for the long-time dynamics of the perturbations of the respective solitons.

Mean-field games (MFG) is a framework for modeling and analysis of vast populations of agents that play differential games. It is an actively growing field with numerous applications in economics, finance, industrial engineering, crowd motion, swarm control, and recently machine learning and data science. In this talk, I will discuss several computational, applied, and theoretical aspects of MFG. In particular, I will focus on nonlocal models, applications in machine learning, and MFG PDE.

First introduced in the context of potential theory a century ago, and later expanded to its full generality by Ishii in the 1980's, the Perron method is an essential theoretical tool used to construct viscosity solutions to nonlinear PDE. In work with Stan Osher and Alex Tong Ling (UCLA), we show the Perron method provides us with a simple principle for the numerical computation of solutions to Hamilton-Jacobi (HJ) equations -- as well as any problem admitting a comparison principle, such as obstacle problems or Hele-Shaw flows. Applications include grid-less methods for HJ equations using artificial neural networks or linear bases such as multidimensional Fourier series.

Applications of Optimal Transport (OT) theory have gained popularity in several fields such as machine learning and signal processing. In this seminar, we will address this point by introducing embeddings or transforms based on OT. First, we will present the Cumulative Distribution Transform (CDT), its version for signed signals/measures, and the Linear Optimal Transport Embedding (LOT). Here, the underlying conservation of mass law will be a benefit. Since these tools are new signal representations based on OT, they have suitable properties for decoding information related to certain signal displacements. We will demonstrate this by describing a Wasserstein-type metric in the embedding space and showing applications in classifying (detecting) signals under random displacements, parameter estimation problems for certain types of generative models, and interpolation. Moreover, these techniques allow faster computation of the classical Wasserstein between pairs of probability measures. However, even though the balanced mass requirement from classical OT is crucial, it also limits the performance of these transforms/embeddings. Therefore, we will finally move to Optimal Partial Transport (OPT) theory and propose a new (linear) embedding.

Surfaces whose curvature minimizes the Dirichlet energy are central in applications such as surface design in industry and architecture and are generally constructed by using computer-aided design (CAD). We present the system of equations and prove the first results on existence of classical and weak solutions. This is joint work with Luis A. Caffarelli (UT Austin) and Hern\’an Vivas (Universidad Nacional de Mar del Plata, Argentina).

I will present a general framework to establish the stability of inequalities of the form $\langle Lu, u \rangle \ge F(u)$; where $L$ is a positive linear operator and $F$ is a $2$-homogeneous nonlinear functional. We will then see how this framework can be employed to obtain some stability results concerning the fractional Caffarelli-Kohn-Nirenberg inequality. This is joint work with N. De Nitti and T. Konig.

Modern technologies and biological systems, such as temperature-responsive polymers and lipid rafts, take advantage of engineered inclusions, or natural heterogeneities of the medium, to obtain novel composite materials with specific physical properties. To model such situations by using a variational approach based on the gradient theory, the potential and the wells may have to depend on the spatial position, even in a discontinuous way, and different regimes should be considered.

In the critical case case where the scale of the small heterogeneities is of the same order of the scale governing the phase transition and the wells are fixed, the interaction between homogenization and the phase transitions process leads to an anisotropic interfacial energy. The supercritical case for fixed wells is also addressed, now leading to an isotropic interfacial energy. In the subcritical case with moving wells, where the heterogeneities of the material are of a larger scale than that of the diffuse interface between different phases, it is observed that there is no macroscopic phase separation and that thermal fluctuations play a role in the formation of nanodomains.

This is joint work with Riccardo Cristoferi (Radboud University, The Netherlands) and Likhit Ganedi (Aachen University, Germany), USA), based on previous results also obtained with Adrian Hagerty (USA) and Cristina Popovici (USA).

We introduces a novel approach to the mean-field limit of stochastic systems of interacting particles, leading to the first ever derivation of the mean-field limit to the Vlasov-Poisson-Fokker-Planck system for plasmas in dimension $2$ together with a partial result in dimension $3$. The method is broadly compatible with second order systems that lead to kinetic equations and it relies on novel estimates on the BBGKY hierarchy. By taking advantage of the diffusion in velocity, those estimates bound weighted $L^p$ norms of the marginals or observables of the system, uniformly in the number of particles. This allows to qualitatively derive the mean-field limit for very singular interaction kernels between the particles, including repulsive Poisson interactions, together with quantitative estimates for a general kernel in $L^2$.

We will discuss recent results about a class of Hamilton-Jacobi-Bellman (HJB) equations in spaces of probability measures that arise in the study of stochastic optimal control problems for systems of $n$ particles with common noise, interacting through their empirical measures. We will present a procedure to show that the value functions $u_n$ of $n$ particle problems, when converted to functions of the empirical measures, converge as $n\to\infty$ uniformly on bounded sets in the Wasserstein space of probability measures to a function $V$, which is the unique viscosity solution of the limiting HJB equation in the Wasserstein space. The limiting HJB equation is interpreted in its "lifted" form in a Hilbert space, a technique introduced by P.L. Lions. The proofs of the convergence of $u_n$ to $V$ use PDE viscosity solution techniques. An advantage of this approach is that the lifted function $U$ of $V$ is the value function of a stochastic optimal control problem in the Hilbert space. We will discuss how, using Hilbert space and classical stochastic optimal control techniques, one can show that $U$ is regular and there exists an optimal feedback control. We then characterize $V$ as the value function of a stochastic optimal control problem in the Wasserstein space. The talk will also contain an overview of existing works and various approaches to partial differential equations in abstract spaces, including spaces of probability measures and Hilbert spaces.

We consider general two-dimensional autonomous velocity fields and prove that their mixing and dissipation features are limited to algebraic rates. As an application, we consider a standard cellular flow on a periodic box, and explore potential consequences for the long-time dynamics in the two-dimensional Euler equations.

A classical model to describe the dynamics of Newtonian stars is the gravitational Euler-Poisson system. The Euler-Poisson system admits a wide range of star solutions that are in equilibrium or expand for all time or collapse in a finite time or rotate. In this talk, I will discuss some recent progress on those star solutions with focus on expansion and collapse. If time permits, I will also discuss the non-radial stability of self-similarly expanding Goldreich-Weber star solutions.

TBA

Plasmas in a nonequilibrium state experience complex behavior at the large time due to their collective meanfield interaction, including phase mixing, Landau damping, dispersion, and oscillations also known as Langmuir's oscillatory waves. Part of the talk is to overview a complete linear theory of the dynamics of charged particles near spatially homogeneous equilibria, focusing on the classical collision-less models including Vlasov-Poisson and Vlasov-Maxwell systems, while the other part surveys some recent nonlinear results on the subject. Amusingly, a threshold of the wavenumber is provided for the survival of oscillations: namely, below the threshold pure oscillatory waves that obey a Klein-Gordon’s type dispersion relation are found, at the threshold waves are damped by the classical Landau damping (i.e. the faster electrons decay or vanish, the weaker Landau damping is), and above the threshold waves decay polynomially or exponentially fast via the phase mixing mechanism, the full picture of which is deeply linked to the resonant interaction between waves and particles.

I will discuss a prototypical model in nonequilibrium statistical mechanics, the directed polymer in a random environment. It is a model of random paths interacting with a disordered environment, and the goal is to understand how the random environment affects the large-scale behavior of the paths and how some magical exponents show up in $1+1$ dimension. I will focus on the so-called endpoint distribution and discuss an infinite system of partial differential equations related to it. This is based on a joint work with Chris Henderson.

September 14, 2022
#### Rene Cabrera, UT Austin

An optimal transportation principle for interacting paths
September 28, 2022
#### Jessica Lin, McGill University

Quantitative Homogenization of the Invariant Measure for Nondivergence Form Elliptic Equations
October 5, 2022
#### Geng Chen, University of Kansas

Poiseuille flow of nematic liquid crystals via Ericksen-Leslie model
October 12, 2022
#### Kevin Zumbrun, Indiana University Bloomington

Large-amplitude modulation of periodic traveling waves (joint with G. Metivier)
October 14, 2022 (Friday)
#### Chiara Saffirio, University of Basel, Switzerland

Semiclassical limits in plasma physics: towards Vlasov-Poisson and Vlasov-Maxwell equations
October 19, 2022
#### Giuseppe Genovese, University of Zurich, Switzerland

Quasi invariance of Gaussian measures for Hamiltonian PDEs
October 26, 2022
#### Scott Smith, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences

Large $N$ Limits via Stochastic Quantization
November 2, 2022
#### Lucas Bouck, University of Maryland

Finite Element Approximation of a Membrane Model for Liquid Crystal Polymeric Networks
November 9, 2022
#### Jacob Bernstein, Johns Hopkins University

Density lower bounds for topologically nontrivial minimal cones
November 14, 2022 (Monday)
#### Xuwen Chen, University of Rochester

Well/ill-posedness bifurcation for the Boltzmann equation
November 16, 2022
#### Cyril Imbert, École Normale Supérieure

Global regularity estimates for the Boltzmann equation without cutoff

In this talk I will briefly discuss the Monge-Kantorovich optimal transportation problem. This problem involves optimally mapping one mass distribution onto another, where optimality is measured against a given cost function $c(x, y)$. I will review some important results, such as Brenier’s theorem, and then discuss a new transportation problem involving path-dependence and interaction effects.

In this talk, I will first give an overview of stochastic homogenization for nondivergence form elliptic equations, from both the PDE perspective and the probability perspective. I will then present new quantitative homogenization results on the parabolic Green Function and large-scale averages of the unique, ergodic, mutually absolutely continuous, invariant measure. This invariant measure is a solution of the adjoint equation in doubly divergence form satisfying certain integrability conditions. Time permitting, I will present a large-scale $C^{0,1}$-regularity result for the invariant measure. This talk is based on joint work with Scott Armstrong and Benjamin Fehrman.

In this talk, we will discuss a global existence result on the Poiseuille flow of nematic liquid crystals via full Ericksen-Leslie model. The existing results on the Ericksen-Leslie model for the liquid crystals mainly focused on the parabolic and elliptic type models by omitting the kinetic energy term. We established a new method to study the full model. A singularity formation result will also be discussed, together with the global existence result showing that the solution will in general live in the Holder continuous space. The earlier related result on the stability of variational wave equation using the optimal transport method, and the recent result on singularity formation due to geometric effect will also be discussed. The talk is on the joint work with Tao Huang & Weishi Liu.

We introduce a new approach to the study of modulation of high-frequency periodic wave patterns, based on pseudodifferential analysis, multi-scale expansion, and Kreiss symmetrizer estimates like those in hyperbolic and hyperbolic-parabolic boundary-value theory. Key ingredients are local Floquet transformation as a preconditioner removing large derivatives in the normal direction of background rapidly oscillating fronts and the use of the periodic Evans function of Gardner to connect spectral information on component periodic waves to block structure of the resulting approximately constant-coefficient resolvent ODEs. Our main result is bounded-time existence and validity to all orders of large-amplitude smooth modulations of planar periodic solutions of multi-D reaction diffusion systems in the high-frequency/small wavelength limit.

The analysis of many-body interacting systems is a very challenging task due to the high complexity caused by the interaction, that makes an analytical and numerical treatment difficult and often out of reach. However, in my situations it is possible to approximate the many-body dynamics with simplified PDEs, called effective equations, that accurately reflect the empirically relevant features of the system in certain regimes. In this talk I will focus on the mean-field and semiclassical regime to study the approximation of a system of many interacting fermions by the Vlasov-Poisson equation. Furthermore, in presence of a magnetic field governed by the Maxwell equations, we will prove the semiclassical approximation of the dynamics by the Vlasov-Maxwell system. This is more relevant for applications in plasma physics than Vlasov-Poisson, as it provides a more accurate prediction of the dynamics in a tokamak. Based on joint works with J. Chong, L. Lafleche, N. Leopold.

I will review some recent results on the transport of Gaussian and Gibbs measure along the flow of Hamiltonian PDEs, such as nonlinear Schroedinger equations and Benjamin-Ono type equations underlining the main open problems in the field.

I will discuss a vector valued version of the Phi4 model with $N$ components, known as the linear sigma model in quantum field theory. I will review the well posedness theory for the Langevin dynamic and present some new uniform in $N$ estimates. These are applied to describe the large $N$ behavior of the model in both equilibrium and non-equilibrium settings, the corresponding mean-field limit being a singular SPDE of Mckean-Vlasov type. Time permitting, I will also discuss a related continuum random matrix model, where the large $N$ limit of correlation functions leads to a hierarchy of elliptic PDEs. Based on joint work with Hao Shen, Rongchan Zhu, and Xiangchan Zhu.

Liquid crystal polymeric networks are materials where a nematic liquid crystal is coupled with a rubbery material. When actuated with heat or light, the interaction of the liquid crystal with the rubber creates complex shapes. Starting from the classical 3D trace formula energy of Bladon, Warner and Terentjev (1994), we derive a 2D membrane energy as the formal asymptotic limit of the 3D energy. We characterize the zero energy deformations and prove that the energy lacks rank-1 convexity. We propose a finite element method to discretize the problem. To address the lack of convexity of the membrane energy, we regularize with a term that mimics a higher order bending energy. We prove that minimizers of the discrete energy converge to minimizers of the continuous energy using techniques inspired by Frieseke, James, and Müller (2002). For minimizing the discrete problem, we employ a nonlinear gradient flow scheme, which is energy stable. Additionally, we present computations showing the geometric effects that arise from liquid crystal defects. Computations of configurations from nonisometric origami are also presented.

I will discuss how to use mean curvature flow to give nearly optimal lower bounds on the density of topologically nontrivial minimal hypercones in low dimensions. This compliments work of Ilmanen-White who gave analogous bounds for topologically nontrvial area-minimizing hypercones. This is joint work with L. Wang.

We study the well/ill-posedness of the Boltzmann equation with dispersive methods. We take the constant collision kernel case as the first example. We construct a family of special solutions, which are neither near equilibrium nor self-similar, and prove the ill-posedness in $H^s$ Sobolev space for $s<1$, despite the fact that the equation is scale invariant at $s=1/2$. Combining with the previous Chen-Denlinger-Pavlovic result regarding well-posedness, we have found the exact well/ill-posedness threshold.

In this talk, I will present $C^\infty$ a priori estimates for solutions of the inhomogeneous Boltzmann equation without cut-off, conditional to point-wise bounds on their mass, energy and entropy densities. Joint work with Luis Silvestre.

March 2, 2022
#### Mikhail Vishik, UT Austin

Instability and nonuniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid
March 9, 2022
#### Benjamin Seeger, UT Austin

The Neumann problem for fully nonlinear SPDE
March 28, 2022 (Monday)
#### Ioakeim Ampatzoglou, Courant Institute

On the derivation of the inhomogeneous kinetic wave equation from quadratic dispersive equations
March 30, 2022

(9 am, Zoom only)
#### Maria Colombo, EPFL

Non-uniqueness of Leray solutions of the forced Navier-Stokes equations
April 1, 2022 (Friday)
#### Maja Taskovic, Emory University

Moment estimates and global well-posedness of the binary-ternary Boltzmann equation
April 6, 2022
#### Michael Novack, UT Austin

Isoperimetric residues and a mesoscale flatness criterion for hypersurfaces with bounded mean curvature
May 4, 2022
#### Jun Kitagawa, Michigan State University

An alternate optimal transport formulation of prescribed Gauss-Kronecker curvature

We plan to discuss parts of the construction of incompressible flows of an ideal fluid with vorticity in Lebesgue classes in dimension two that imply nonuniqueness in the Cauchy problem.

The notion of pathwise viscosity solutions was developed by Lions and Souganidis to study fully nonlinear stochastic partial differential equations set on the entire space. I will explain how this notion can be generalized to treat equations set on a convex sub-domain with Neumann boundary conditions. A comparison principle is proved for sub and supersolutions, which yields existence and uniqueness of solutions as well as continuity with respect to the driving noise. As an application, I show how the well-posedness theory is used to study the long-time behavior of a stochastically perturbed mean-curvature flow in a cylinder-like domain with right angle contact boundary condition. This is joint work with Paul Gassiat.

The topic of this talk will be the validity of a kinetic description for wave turbulence of a model quadratic dispersive equation. We focus on deriving an inhomogeneous (transport) effective equation for the Wigner transform of the microscopic model solution, up to a small polynomial loss of the kinetic time. We will present nonlinearities where the kinetic description holds and cases where it might fail. We achieve that by examining the convergence of the Dyson series under random data in the weakly nonlinear regime.

(9 am, Zoom only)

In his seminal work, Leray demonstrated the existence of global weak solutions, with nonincreasing energy, to the Navier-Stokes equations in three dimensions. In this talk we exhibit two distinct Leray solutions with zero initial velocity and identical body force. The starting point of our construction is Vishik's answer to another long-standing problem in fluid dynamics, namely whether the Yudovich uniqueness result for the 2D Euler system can be extended to the class of L^p-integrable vorticity. Building on Vishik's work, we construct a `background' solution which is unstable for the 3D Navier-Stokes dynamics in similarity variables; the second solution from the same initial datum is a trajectory on the unstable manifold associated to the background solution, in accordance with the predictions of Jia and Sverak.

The binary-ternary Boltzmann equation was recently rigorously derived by Ampatzoglou and Pavlovic for a dense hard-spheres gas in which particles interact via either binary or ternary interactions. We study analytic properties of the spatially homogeneous version of this equation for a wider range of potentials. In particular, we study polynomial and exponential moment estimates and show that the presence of the ternary part can improve moment estimates of the classical (binary) Boltzmann equation. We will also present a result on global in time existence and uniqueness of solutions which is based on moment estimates and an abstract ODE theory in Banach spaces. This is a joint work with Ioakeim Ampatzoglou, Irene Gamba and Natasa Pavlovic.

In this talk we discuss a full resolution result for minimizers in the exterior isoperimetric problem with respect to a compact obstacle in the large volume regime. This is achieved by the study of a Plateau-type problem with obstacle and boundary datum at infinity, which identifies the first obstacle-dependent term in energy expansion for large volumes of the exterior isoperimetric problem, and is therefore called the isoperimetric residue of the obstacle. A crucial tool in the analysis is a new “mesoscale flatness criterion” for hypersurfaces with bounded mean curvature, which we obtain as a development of ideas originating in the theory of minimal surfaces with isolated singularities. This is a joint work with Francesco Maggi.

It is well known that the prescribed Gauss-Kronecker curvature problem for a graph on a convex domain can be described via optimal transport. In this talk, I will discuss an alternate interpretation, again as an optimal transport problem but in a setting different from the traditional approach. This talk is based on joint work with N. Guillen.

November 3, 2021
#### Robin Neumayer, Carnegie Mellon University

Quantitative Faber-Krahn Inequalities and Applications
November 12, 2021 (Friday)
#### Govind Menon, Brown University

Renormalization group flows for nonlinear PDE
November 17, 2021
#### Tristan Buckmaster, Princeton University

Smooth Imploding Solutions for 3D Compressible Fluids
December 1, 2021
#### Mary Vaughan, UT Austin

Harnack inequality for fractional elliptic equations in nondivergence form
December 8, 2021
#### Matthew Novack, Institute for Advanced Study

Turbulent Weak Solutions of the 3D Euler Equations

Among all drum heads of a fixed area, a circular drum head produces the vibration of lowest frequency. The general dimensional analogue of this fact is the Faber-Krahn inequality, which states that balls have the smallest principal Dirichlet eigenvalue among subsets of Euclidean space with a fixed volume. I will discuss new quantitative stability results for the Faber-Krahn inequality on Euclidean space, the round sphere, and hyperbolic space, as well as an application to the Alt-Caffarelli-Friedman monotonicity formula used in free boundary problems. This is based on joint work with Mark Allen and Dennis Kriventsov.

I will discuss a probabilistic approach to the Nash embedding theorems along with several applications. The initial motivation for this work were the results of De Lellis and Szekelyhidi linking Nash embedding with the Euler equations. The main idea in my work is to replace ad hoc construction schemes with a principled choice of stochastic flows, in order that one may answer the question “what does a typical embedding look like?”. More precisely, we ask how one may construct natural probability measures supported on solutions to nonlinear PDE. This idea will be illustrated with examples. This program is still some ways away from completely new proofs of these theorems, however it yields many new models and new insights into Nash-Moser and KAM theorems. Most of the effort at this point is in sharpening the fundamental insight through numerics. In particular, there is a stong interplay with interior point methods for semidefinite programming.

Building upon the pioneering work of Merle-Rodnianski-Szeftel, we construct exact, smooth self-similar imploding solutions to the 3D isentropic compressible Euler equations for ideal gases for all adiabatic exponents. For the particular exponent $7/5$, corresponding to air and akin to the result of Merle-Raphael-Rodnianski-Szeftel, we show the existence of a sequence of smooth, self-similar imploding solutions. In addition, we provide simplified proofs of linear stability and non-linear stability which allows us to construct asymptotically self-similar imploding solutions to the compressible Navier-Stokes equations with density independent viscosity for the adiabatic exponent $7/5$ and density bounded from below.

In this talk, we will define fractional powers of nondivergence form elliptic operators in bounded domains under minimal regularity assumptions and highlight several applications. We will characterize a Poisson problem driven by such operators with a degenerate/singular extension problem. We then develop the method of sliding paraboloids in the Monge–Ampère geometry to prove Harnack inequality for classical solutions to the extension equation. This in turn implies Harnack inequality for solutions to the fractional Poisson problem. This work is joint with Pablo Raúl Stinga (Iowa State University).

The Navier-Stokes and Euler equations are the fundamental models for describing viscous and inviscid fluids, respectively. Based on ideas which date back to Kolmogorov and Onsager, solutions to these equations are expected to dissipate energy, which in turn suggests that such solutions are somewhat rough and thus only weak solutions. At these low regularity levels, however, one may construct wild weak solutions using convex integration methods. In this talk, I will discuss the motivation and methodology behind joint work with Tristan Buckmaster, Nader Masmoudi, and Vlad Vicol in which we construct wild solutions to the Euler equations which deviate from the predictions of Kolmogorov's classical K41 phenomenological theory of turbulence.