Seminars and Colloquia by Series

A rigorous proof of Batchelor's law for passive scalar turbulence

Series
CDSNS Colloquium
Time
Monday, February 17, 2020 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alex BlumenthalUniversity of Maryland and Georgia Tech

Batchelor's law describes the power law spectrum of the turbulent regime of passive scalars (e.g., temperature or a dilute concentration of some tracer chemical) advected by an incompressible fluid (e.g., the Navier-Stokes equations at fixed Reynolds number), in the limit of vanishingly low molecular diffusivity. Predicted in 1959, it has been confirmed empirically in a variety of experiments, e.g. salinity concentrations among ocean currents. On the other hand, as with many turbulent regimes in physics, a true predictive theory from first principles has been missing (even a non-rigorous one), and there has been some controversy regarding the extent to which Batchelor's law is universal. 

 

In this talk, I will present a program of research, joint with Jacob Bedrossian (UMD) and Sam Punshon-Smith (Brown), which has rigorously proved Batchelor's law for passive scalars advected by the Navier-Stokes equations on the periodic box subjected to Sobolev regular, white-in-time body forces. The proof is a synthesis of techniques from dynamical systems and smooth ergodic theory, stochastics/probability, and fluid mechanics. To our knowledge, this work constitutes the first mathematically rigorous proof of a turbulent power law spectrum. It also establishes a template for predictive theories of passive scalar turbulence in more general settings, providing a strong argument for the universality of Batchelor's law. 

On mixing properties of infinite measure preserving systems

Series
CDSNS Colloquium
Time
Monday, February 10, 2020 - 11:15 for 1.5 hours (actually 80 minutes)
Location
Skiles 006
Speaker
Dmitry DolgopyatUniversity of Maryland

We present several new results concerning mixing properties of
hyperbolic systems preserving an infinite measure making a particular
emphasis on mixing for extended systems. This talk is based on a joint
work with Peter Nandori.

Unique measure of maximal entropy for the finite horizon periodic Lorentz gas

Series
CDSNS Colloquium
Time
Monday, February 3, 2020 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Mark F. DemersFairfield University

While the existence and properties of the SRB measure for the billiard map associated with a periodic Lorentz gas are well understood, there are few results regarding other types of measures for dispersing billiards. We begin by proposing a naive definition of topological entropy for the billiard map, and show that it is equivalent to several classical definitions. We then prove a variational principle for the topological entropy and proceed to construct a unique probability measure which achieves the maximum. This measure is Bernoulli and positive on open sets. An essential ingredient is a proof of the absolute continuity of the unstable foliation with respect to the measure of maximal entropy. This is joint work with Viviane Baladi.
 

Rigidity for expanding maps

Series
CDSNS Colloquium
Time
Monday, January 13, 2020 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Federico Rodriguez HertzPenn State

In a recent work with A. Gogolev we found some new form of rigidity for expanding maps through marching of potentials (also named cocycles). In this talk I plan to discuss these rigidity results and explain how this relates to some old results by Shub and Sullivan and de la Llave.

Asymptotic Homotopical Complexity of a Sequence of 2D Dispersing Billiards

Series
CDSNS Colloquium
Time
Monday, November 25, 2019 - 11:15 for 1 hour (actually 50 minutes)
Location
Skyles 005
Speaker
Nandor SimanyiUniversity of Alabama at Birgminham

We are studying the asymptotic homotopical complexity of a sequence of billiard flows on the 2D unit torus T^2 with n
circular obstacles. We get asymptotic lower and upper bounds for the radial sizes of the homotopical rotation sets and,
accordingly, asymptotic lower and upper bounds for the sequence of topological entropies. The obtained bounds are rather
close to each other, so this way we are pretty well capturing the asymptotic homotopical complexity of such systems.

Note that the sequence of topological entropies grows at the order of log(n), whereas, in sharp contrast, the order of magnitude of the sequence of metric entropies is only log(n)/n.


Also, we prove the convexity of the admissible rotation set AR, and the fact that the rotation vectors obtained from
periodic admissible trajectories form a dense subset in AR.

 

Ergodic properties of low complexity symbolic systems

Series
CDSNS Colloquium
Time
Monday, November 18, 2019 - 11:15 for 1 hour (actually 50 minutes)
Location
Skyles 005
Speaker
van.cyr@bucknell.eduBucknell University

The topological entropy of a subshift is the exponential growth rate of the number of words of different lengths in its language. For subshifts of entropy zero, finer growth invariants constrain their dynamical properties. In this talk we will survey how the complexity of a subshift affects properties of the ergodic measures it carries. In particular, we will see some recent results (joint with B. Kra) relating the word complexity of a subshift to its set of ergodic measures as well as some applications.

Renormalization for the almost Mathieu operator and related skew products.

Series
CDSNS Colloquium
Time
Friday, November 1, 2019 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Hans KochUniv. of Texas, Austin

Considering SL(2,R) skew-product maps over circle rotations,
we prove that a renormalization transformation
associated with the golden mean alpha
has a nontrivial periodic orbit of length 3.
We also present some numerical results,
including evidence that this period 3 describes
scaling properties of the Hofstadter butterfly
near the top of the spectrum at alpha,
and scaling properties of the generalized eigenfunction
for this energy.

Effective bounds for the measure of rotations

Series
CDSNS Colloquium
Time
Monday, October 28, 2019 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 05
Speaker
Alex HaroUniv. de Barcelona

A fundamental question in Dynamical Systems is to identify regions of
phase/parameter space satisfying a given property (stability,
linearization, etc).  In this talk, given a family of analytic circle
diffeomorphisms depending on a parameter, we obtain effective (almost
optimal) lower bounds of the Lebesgue measure of the set of parameters
for which that diffeomorphism is conjugate to a rigid rotation.
We estimate this measure using an a-posteriori KAM
scheme that relies on quantitative conditions that
are checkable using computer-assistance. We carefully describe
how the hypotheses in our theorems are reduced to a finite number of
computations, and apply our methodology to the case of the
Arnold family, in the far-from-integrable regime.

This is joint work with Jordi Lluis Figueras and Alejandro Luque.

 

Mixing and Explosions for the Generalized Recurrent Set

Series
CDSNS Colloquium
Time
Monday, October 21, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skyles 006
Speaker
Jim WisemanAgnes Scott

We consider the strong chain recurrent set and the generalized recurrent set for continuous maps of compact metric spaces.  Recent work by Fathi and Pageault has shown a connection between the two sets, and has led to new results on them.  We discuss a structure theorem for transitive/mixing maps, and classify maps that permit explosions in the size of the recurrent sets.

New mechanisms of instability in Hamiltonian systems

Series
CDSNS Colloquium
Time
Monday, October 21, 2019 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 06
Speaker
Tere M. SearaUniv. Politec. de Catalunya

In this talk we present some recent results which allow to prove
instability in near integrable Hamiltonian systems. We will show how
these mechanisms are suitable to apply to concrete systems but also are
useful to obtain Arnold diffusion in a large set  of Hamiltonian systems.

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