Seminars and Colloquia by Series

Steady waves in flows over periodic bottoms

Series
CDSNS Colloquium
Time
Friday, April 30, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Zoom (see additional notes for link)
Speaker
Carlos Garcia AzpeitiaUNAM

Please Note: Zoom link: https://zoom.us/j/97732215148?pwd=Z0FBNXNFSy9mRUx3UVk4alE4MlRHdz09

In this talk we present the formation of steady waves in two-dimensional fluids under a current with mean velocity $c$ flowing over a periodic bottom. Using a formulation based on the Dirichlet-Neumann operator, we establish the unique continuation of a steady solution from the trivial solution for a flat bottom, with the exception of a sequence of velocities $c_{k}$.  Furthermore, we prove that at least two steady solutions for a near-flat bottom persist close to a non-degenerate $S^1$-orbit of steady waves for a flat bottom. As a consequence, we obtain the persistence of at least two steady waves close to a non-degenerate $S^1$-orbit of Stokes waves bifurcating from the velocities $c_{k}$ for a flat bottom. This is a joint work with W. Craig.

Normal form and existence time for the Kirchhoff equation

Series
CDSNS Colloquium
Time
Friday, April 23, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Zoom (see additional notes for link)
Speaker
Emanuele HausUniversity of Roma Tre

Please Note: Zoom link: https://zoom.us/j/97732215148?pwd=Z0FBNXNFSy9mRUx3UVk4alE4MlRHdz09

In this talk I will present some recent results on the Kirchhoff equation with periodic boundary conditions, in collaboration with Pietro Baldi.

Computing the first step of quasilinear normal form, we erase from the equation all the cubic terms giving nonzero contribution to the energy estimates; thus we deduce that, for small initial data of size $\varepsilon$ in Sobolev class, the time of existence of the solution is at least of order $\varepsilon^{-4}$ (which improves the lower bound $\varepsilon^{-2}$ coming from the linear theory).

In the second step of normal form, there remain some resonant terms (which cannot be erased) that give a non-trivial contribution to the energy estimates; this could be interpreted as a sign of non-integrability of the equation. Nonetheless, we show that small initial data satisfying a suitable nonresonance condition produce solutions that exist over a time of order at least $\varepsilon^{-6}$.

Symbolic dynamics and oscillatory motions in the 3 Body Problem

Series
CDSNS Colloquium
Time
Friday, April 16, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Zoom (see additional notes for link)
Speaker
Pau MartinUPC

Please Note: Zoom link: https://zoom.us/j/97732215148?pwd=Z0FBNXNFSy9mRUx3UVk4alE4MlRHdz09

Consider the three body problem with masses $m_0,m_1,m_2>0$. Take units such that $m_0+m_1+m_2 = 1$. In 1922 Chazy classified the possible final motions of the three bodies, that is the behaviors the bodies may have when time tends to infinity. One of them are what is known as oscillatory motions, that is, solutions of the three body problem such that the positions of the bodies $q_0, q_1, q_2$ satisfy
\[
\liminf_{t\to\pm\infty}\sup_{i,j=0,1,2, i\neq j}\|q_i-q_j\|<+\infty \quad \text{ and }\quad 
\limsup_{t\to\pm\infty}\sup_{i,j=0,1,2, i\neq j}\|q_i-q_j\|=+\infty.
\] At the time of Chazy, all types of final motions were known, except the oscillatory ones. We prove that, if all three masses $m_0,m_1,m_2>0$ are not equal to $1/3$, such motions exist. In fact, we prove more, since our result is based on the construction of a hyperbolic invariant set whose dynamics is conjugated to the Bernouilli shift of infinite symbols, we prove (if all masses are not all three equals to $1/3$) 1) the existence of chaotic motions and positive topological entropy for the three body problem, 2) the existence of periodic orbits of arbitrarily large period in the 3BP. Reversing time, Chazy's classification describes ``starting'' motions and then, the question if starting and final motions need to coincide or may be different arises.  We also prove that one can construct solutions of the three body problem whose starting and final motions are of different type.

Abelian Livshits Theorem

Series
CDSNS Colloquium
Time
Friday, April 9, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Zoom (see additional notes for link)
Speaker
Andrey GogolevThe Ohio State University

Please Note: Zoom link: https://zoom.us/j/97732215148?pwd=Z0FBNXNFSy9mRUx3UVk4alE4MlRHdz09

The classical Livshits theorem characterizes coboundaries over a transitive Anosov flow as precisely those functions which integrate to zero over all periodic orbits of the flow. I will present a variant of this theorem which uses a weaker assumption and gives a weaker conclusion that the function is an ``abelian coboundary.” Such weaker version corresponds to studying the cohomological equation on infinite abelian covers e.g., for geodesic flows on abelian covers of hyperbolic surfaces. I will also discuss a connection to the marked length spectrum rigidity of Riemannian metrics. Joint work with Federico Rodriguez Hertz.

Convergence over fractals for the periodic Schrödinger equation

Series
CDSNS Colloquium
Time
Friday, March 26, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Zoom (see add&#039;l notes for link)
Speaker
Daniel EceizabarrenaU Mass Amherst

Please Note: Zoom link: https://zoom.us/j/97732215148?pwd=Z0FBNXNFSy9mRUx3UVk4alE4MlRHdz09 Meeting ID: 977 3221 5148 Passcode: 801074

 

In 1980, Lennart Carleson introduced the following problem for the free Schrödinger equation: when does the solution converge to the initial datum pointwise almost everywhere? Of course, the answer is immediate for regular functions like Schwartz functions. However, the question of what Sobolev regularity is necessary and sufficient for convergence turned out to be highly non-trivial. After the work of many people, it has been solved in 2019, following important advances in harmonic analysis. But interesting variations of the problem are still open. For instance, what happens with periodic solutions in the torus? And what if we refine the almost everywhere convergence to convergence with respect to fractal Hausdorff measures? Together with Renato Lucà (BCAM, Spain), we tackled these two questions. In the talk, I will present our results after explaining the basics of the problem.

The mechanics of finite-time blowup in an Euler flow

Series
CDSNS Colloquium
Time
Friday, March 19, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Zoom (see add&#039;l notes for link)
Speaker
Dwight BarkleyU Warwick

Please Note: Zoom link: https://zoom.us/j/97732215148?pwd=Z0FBNXNFSy9mRUx3UVk4alE4MlRHdz09

One of the most fundamental issues in fluid dynamics is whether or not an initially smooth fluid flow can evolve over time to arrive at a singularity -- a state for which the classical equations of fluid mechanics break down and the flow field no longer makes physical sense.  While proof remains an open question, numerical evidence strongly suggests that a singularity arises at the boundary of a flow like that found in a stirred cup of tea.  The analysis here focuses on the interplay between inertia and pressure, rather than on vorticity.  A model is presented based on a primitive-variables formulation of the Euler equations on the cylinder wall, with closure coming from how pressure is determined from velocity. The model generalizes Burger's equation and captures key features in the mechanics of the blowup scenario. 

Lyapunov exponent of random dynamical systems on the circle

Series
CDSNS Colloquium
Time
Friday, March 12, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Zoom (see add&#039;l notes for link)
Speaker
Dominique MalicetUniversity Paris-Est Marne la vallée

Please Note: Zoom link: https://zoom.us/j/97732215148?pwd=Z0FBNXNFSy9mRUx3UVk4alE4MlRHdz09

We consider a sequence of compositions of orientation preserving diffeomorphisms of the circle chosen randomly with a fixed distribution law. There is naturally associated a Lyapunov exponent, which measures the rate of exponential contractions of the sequence. It is known that under some assumptions, if this Lyapunov exponent is null then all the diffeomorphisms are simultaneously conjugated to rotations. If the Lyapunov exponent is not null but close to 0, we study how well we can approach rotations by a simultaneous conjugation. In particular, our results can apply to random products of matrices 2x2, giving quantitative versions of the classical Furstenberg theorem.

Synchronization in Markov random networks

Series
CDSNS Colloquium
Time
Friday, March 5, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Zoom (see add&#039;l notes for link)
Speaker
Shirou WangU Alberta

Please Note: Zoom link: https://zoom.us/j/97732215148?pwd=Z0FBNXNFSy9mRUx3UVk4alE4MlRHdz09

Many complex biological and physical networks are naturally subject to both random influences, i.e., extrinsic randomness, from their surrounding environment, and uncertainties, i.e., intrinsic noise, from their individuals. Among many interesting network dynamics, of particular importance is the synchronization property which is closely related to the network reliability especially in cellular bio-networks. It has been speculated that whereas extrinsic randomness may cause noise-induced synchronization, intrinsic noises can drive synchronized individuals apart. This talk presents an appropriate framework of (discrete-state and discrete time) Markov random networks to incorporate both extrinsic randomness and intrinsic noise into the rigorous study of such synchronization and desynchronization scenario.  By studying the asymptotics of the Markov perturbed stationary distributions, probabilistic characterizations of the alternating pattern between synchronization and desynchronization behaviors is given.  More precisely, it is shown that if a random network without intrinsic noise perturbation is synchronized, then after intrinsic noise perturbation high-probability synchronization and low-probability desynchronization can occur intermittently and alternatively in time, and moreover, both the probability of (de)synchronization and the proportion of time spent in (de)synchrony can be explicitly estimated.

Computer Assisted Proof of Drift Orbits Along Normally Hyperbolic Manifolds

Series
CDSNS Colloquium
Time
Friday, February 26, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Zoom (see add&#039;l notes for link)
Speaker
Jorge GonzalezGeorgia Tech

Please Note: Zoom link: https://zoom.us/j/97732215148?pwd=Z0FBNXNFSy9mRUx3UVk4alE4MlRHdz09

 

We will discuss a new method for proving the existence of diffusion in some systems with Normally Hyperbolic Invariant Manifolds (NHIM). We apply this approach to the generalized standard map to show the existence of drift orbits for an explicit range of actions.  The method consists of verifying a finite number of conditions on a computer (keywords: NHIM, shadowing, scattering map, Chirikov Standard model, Parameterization Method, Interval Newton Method).  

Symplectic Geometry of Anosov Flows in Dimension 3 and Bi-Contact Topology

Series
CDSNS Colloquium
Time
Friday, February 19, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Zoom (see add&#039;l notes for link)
Speaker
Surena HozooriGeorgia Tech

Please Note: Link: https://zoom.us/j/97732215148?pwd=Z0FBNXNFSy9mRUx3UVk4alE4MlRHdz09

We give a purely contact and symplectic geometric characterization of Anosov flows in dimension 3 and set up a framework to use tools from contact and symplectic geometry and topology in the study of questions about Anosov dynamics. If time permits, we will discuss a characterization of Anosovity based on Reeb flows and its consequences.

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