ENG 
ESP Session S22 - Deterministic and probabilistic aspects of nonlinear evolution equations
Talks
Wednesday, July 14, 16:00 ~ 16:30 UTC-3
RIEMANN’S NON-DIFFERENTIABLE FUNCTION AND THE BINORMAL CURVATURE FLOW
Luis Vega
BCAM- University of the Basque Country, Spain - This email address is being protected from spambots. You need JavaScript enabled to view it.
We make a connection between a famous analytical object introduced in the 1860s by Riemann, as well as some variants of it, and a nonlinear geometric PDE, the binormal curvature flow. As a consequence this analytical object has a non-obvious non- linear geometric interpretation. We recall that the binormal flow is a standard model for the evolution of vortex filaments. We prove the existence of solutions of the binormal flow with smooth trajectories that are as close as desired to curves with a multifractal behavior. Finally, we show that this behavior falls within the multifractal formalism of Frisch and Parisi, which is conjectured to govern turbulent fluids.
Joint work with Valeria Banica (Sorbonne U., Paris, France).
Wednesday, July 14, 16:35 ~ 17:05 UTC-3
Solitary waves in generalized KdV and HBO-type equations
Svetlana Roudenko
Florida International University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
We discuss solitary wave solutions and soliton resolution in the generalized KdV equation with nonlinearities $|u|^{p-1} \partial_x u$, $p>1$, including also the critical and supercritical cases ($p \geq 5$). We then look at its higher-dimensional, fractional generalization in 2d, such as the HBO equation, and discuss behavior of solutions in various cases as well as stability of solitary waves.
Joint work with Oscar Riaño (Florida International University, USA) and Kai Yang (Florida International University, USA).
Wednesday, July 14, 17:10 ~ 17:40 UTC-3
On derivation of the wave kinetic equation
Yu Deng
University of Southern California, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
The wave kinetic equation is a central topic in the theory of wave turbulence, which studies the mesoscopic limits of microscopic systems of interacting waves. Since its birth in the 1920s, this theory has been extensively studied in the physics literature, however the mathematical justification has long been open. In this talk we will summarize recent developments in this area, and present our contribution to this problem, which is joint work with Zaher Hani.
Joint work with Zaher Hani (University of Michigan).
Wednesday, July 14, 17:45 ~ 18:15 UTC-3
Global well-posedness for the fractional NLS on the unit disk
Xueying Yu
Massachusetts Institute of Technology, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk, we discuss the cubic nonlinear Schr\"odinger equation with the fractional Laplacian on the unit disk. We show the global well-posedness for certain radial initial data below the energy space and establish a polynomial bound of the global solution. The result is proved by extending the I-method in the fractional nonlinear Schr\"odinger equation setting.
Joint work with Mouhamadou Sy (University of Virginia).
Wednesday, July 14, 18:20 ~ 18:50 UTC-3
On long time behavior of solutions of the Schrödinger-Korteweg-de Vries system
Argenis Mendez
Centro de Modelamiento Matemático, Universidad de Chile, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk we are concerned with the decay of long time solutions of the initial value problem associated to the Schrödinger-Korteweg-de Vries system. We use recent techniques in order to show that solutions of this system decay to zero in the energy space. The result is independent of the integrability of the equations involved and it does not require any size assumptions on the initial data.
Joint work with Felipe Linares (Instituto Nacional de Matemática Pura e Aplicada-IMPA, Brazil).
Friday, July 16, 16:00 ~ 16:30 UTC-3
On the uniqueness of excited states of semilinear evolution equations.
Wilhelm Schlag
Yale University, United States of America - This email address is being protected from spambots. You need JavaScript enabled to view it.
We will discuss the problem of existence and uniqueness of nonzero solutions of finite energy to semilinear elliptic PDEs, which characterize stationary solutions of nonlinear evolution equations. The uniqueness question, which is often delicate, has consequences for the spectral properties of the linearized operators. This in turn is of essence for the long-term dynamics of solutions. In particular, I will describe recent work with Alex Cohen and Kevin Li at Yale on the uniqueness of the first few excited states for the cubic problem in three dimensions.
Friday, July 16, 16:35 ~ 17:05 UTC-3
Full derivation of the wave kinetic equation
Zaher Hani
University of Michigan, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
I will be discussing a recent joint work with Yu Deng (USC) in which we give the full derivation of the wave kinetic equation from the NLS equation. The result is the wave analog of Lanford's theorem justifying the derivation of Boltzmann equation from particle system dynamics. This resolves a central problem in wave turbulence theory.
Joint work with Yu Deng (University of Southern California).
Friday, July 16, 17:10 ~ 17:40 UTC-3
Global well-posedness and blow up for NLS systems with quadratic-type nonlinearities
Ademir Pastor
Universidade Estadual de Campinas, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
We report some recents results concerning NLS systems with quadratic-type nonlinearities. We consider both systems under resonance and nonresonance conditions. Special attention will be turned to establish the existence of global and blow up solutions.
Joint work with Norman Noguera (Universidad de Costa Rica, Costa Rica).
Friday, July 16, 17:45 ~ 18:15 UTC-3
Random tensors, propagation of randomness, and nonlinear Schrödinger equations.
Haitian Yue
University of Southern California, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk, we will discuss recent developments on the propagation of randomness, in the setting of random data problems for nonlinear Schrödinger equations. In particular, I will present a new framework called the theory of random tensors that proves almost-sure local well-posedness in the optimal (”probabilistic subcritical”) range of regularity.
Joint work with Yu Deng (University of Southern California) and Andrea Nahmod (UMass Amherst).
Friday, July 16, 18:20 ~ 18:50 UTC-3
Properties of the Support of Solutions of a Class of Nonlinear Evolution Equations
José Manuel Jiménez
Universidad Nacional de Colombia- Sede Medellín, Colombia - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this work we consider equations of the form $$\partial_t u+P(\partial_x) u+G(u,\partial_xu,\dots,\partial_x^l u)=0,$$ where $P$ is any polynomial without constant term, and $G$ is any polynomial without constant or linear terms. We prove that if $u$ is a sufficiently smooth solution of the equation, such that $\supp u(0),\supp u(T)\subset (-\infty,B]$ for some $B>0$, then there exists $R_0>0$ such that $\supp u(t)\subset (-\infty,R_0]$ for every $t\in[0,T]$. Then, as an example of the application of this result, we employ it to show unique continuation properties for some nonlinear dispersive models$$
Joint work with Eddye Bustamante (Universidad Nacional de Colombia-Sede Medellín).
Thursday, July 22, 16:00 ~ 16:30 UTC-3
Normal form transformations and Dysthe's equation for the nonlinear modulation of deep-water gravity waves
Catherine Sulem
University of Toronto, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.
I will present a new Hamiltonian version of Dysthe's equation for weakly modulated gravity waves on deep water. A key ingredient in this derivation is a Birkhoff normal form transformation that eliminates all non-resonant cubic terms and allows for a non-perturbative reconstruction of the free surface. This modulational approximation is tested against numerical solutions of the classical Dysthe's equation and against direct numerical simulations of Euler's equations for nonlinear water waves. An alternate spatial form is proposed and tested against laboratory experiments on short-wave packets. (joint work with W. Craig, P. Guyenne, A. Kairzhan, B. Xu)
Thursday, July 22, 16:35 ~ 17:05 UTC-3
Instability of soliton-type profiles on metric graphs
Jaime Angulo Pava
Department of Mathematics, IME-USP, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk we shed new light on the mathematical studies of nonlinear dispersive evolution equations on metric graphs. This trend has been mainly motivated by the demand of reliable mathematical models for diferent phenomena in branched systems which, in meso- or nano-scales, resemble a thin neighborhood of a graph, such as Josephson junction networks, electric circuits, blood pressure waves in large arteries, or nerve impulses in complex arrays of neurons, just to mention a few examples. Our dynamic problems here will be essentially related to the sine-Gordon model on a $\mathcal Y$-junction graph. We establish a general linear instability criterium for solitons profiles associated to nonlinear dispersive evolution equations on metric graphs. In particular, we see that some kink or kink/anti-kink soliton profiles for the sine-Gordon model are linearly (and nonlinearly) unstable.
The arguments presented in this talk have prospects for the study of the instability of soliton-profiles solutions of other nonlinear evolution equations on branched systems.
Joint work with Ramón G. Plaza (IIMAS, UNAM, Cd. de México, México).
Thursday, July 22, 17:10 ~ 17:40 UTC-3
Beyond binary interactions of particles
Natasa Pavlovic
The University of Texas at Austin , USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk we shall discuss dynamics of systems of particles that allow interactions beyond binary, and their behavior as the number of particles goes to infinity. In particular, an example of such a system of bosons leads to a quintic nonlinear Schrodinger equation, which we rigorously derived in a joint work with Thomas Chen. An example of a system of classical particles that allows instantaneous ternary interactions leads to a new kinetic equation that can be understood as a step towards modeling a dense gas in non-equilibrium. We call this equation a ternary Boltzmann equation and we rigorously derived it with Ioakeim Ampatzoglou. Time permitting, we will also discuss the recent work with Ampatzoglou on a derivation of a binary-ternary Boltzmann equation describing the kinetic properties of a dense hard spheres gas, where particles undergo either binary or ternary instantaneous interactions, while preserving momentum and energy. An important challenge we overcome in deriving this equation is related to providing a mathematical framework that allows us to detect both binary and ternary interactions. Furthermore, this work introduces new algebraic and geometric techniques in order to eventually decouple binary and ternary interactions and understand the way they could succeed one another in time.
Joint work with Thomas Chen (The University of Texas at Austin) and Ioakeim Ampatzoglou (New York University).
Thursday, July 22, 17:45 ~ 18:15 UTC-3
Nonlinear Matrix Schrödinger Equation with a potential on the half line.
Ivan Naumkin
National University of Mexico (UNAM), Mexico - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk, we consider the nonlinear matrix Schrödinger equation with an external potential on the half line. We prove the existence of linear scattering for this model. Our approach is based on the spectral theorem for the perturbed linear Schrödinger operator and a factorization technique. We present new methods that allow us to consider both generic and exceptional cases without any additional symmetries.
Joint work with Ricardo Weder (UNAM, Mexico).
Thursday, July 22, 18:20 ~ 18:50 UTC-3
The quartic integrability and long time existence of steep water waves in 2d
Sijue Wu
University of Michigan, Ann Arbor, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
It is known since the work of Dyachenko & Zakharov in 1994 that for the weakly nonlinear 2d infinite depth water waves, there are no 3-wave interactions and all of the 4-wave interaction coefficients vanish on the non-trivial resonant manifold. In this talk I will present a recent result that proves this partial integrability from a different angle. We construct a sequence of energy functionals $ \mathcal E_j(t)$, directly in the physical space, which are explicit in the Riemann mapping variable and involve material derivatives of order $j$ of the solutions for the 2d water wave equation, so that $\frac d{dt} \mathcal E_j(t)$ is quintic or higher order. We show that if some scaling invariant norm, and a norm involving one spacial derivative above the scaling of the initial data are of size no more than $\varepsilon$, then the lifespan of the solution for the 2d water wave equation is at least of order $O(\epsilon^{-3}\ )$, and the solution remains as regular as the initial data during this time. If only the scaling invariant norm of the data is of size $\epsilon$, then the lifespan of the solution is at least of order $O( \epsilon^{-5/2}\ )$. Our long time existence results do not impose size restrictions on the slope of the initial interface and the magnitude of the initial velocity, they allow the interface to have arbitrary large steepnesses and initial velocities to have arbitrary large magnitudes.
Thursday, July 22, 18:55 ~ 19:25 UTC-3
Blow-up solutions of the intercritical inhomogeneous NLS equation
Luiz Gustavo Farah
Universidade Federal de Minas Gerais, Brasil - This email address is being protected from spambots. You need JavaScript enabled to view it.
We consider the inhomogeneous nonlinear Schr\"odinger (INLS) equation \[ i u_t +\Delta u+|x|^{-b}|u|^{2\sigma} u = 0, \,\,\, x\in \mathbb{R}^N, \] with $N\geq 3$ and $b\in (0,2)$. We focus on the intercritical case, where the scaling invariant Sobolev index $s_c=\frac{N}{2}-\frac{2-b}{2\sigma}$ satisfies $s_c\in (0,1)$. In this talk, for initial data in $\dot H^{s_c}\cap \dot H^1$, we discuss the existence of blow-up solutions and also a lower bound for the blow-up rate in the radial and non-radial settings.
Joint work with Mykael Cardoso (Universidade Federal do Piauí, Brasil).