Schedule for: 23w5059 - Women in Nonlinear Dispersive PDEs

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We will discuss the derivative nonlinear Schrodinger equation, how some inherent instabilities have hindered the study of this equation, and how we were able to demonstrate global well-posedness in the natural scale-invariant space. This is joint work with Ben Harrop-Griffiths, Rowan Killip, and Maria Ntekoume.

The aim of this joint work with Davide Barilari, Isabelle Gallagher and Matthieu Léautaud consists to establish a spectral summability for the quartic oscillator, then to apply it to study the Engel group G which is a step-3 stratified Lie group. In particular, the influence of the geometry of G on dispersion phenomena for the Schrodinger equation is investigated"

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the PDC front desk for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

The talk concerns new well-posedness results for the dispersion generalised KP-I equations in $\mathbb{R}^2$ with initial data in anisotropic Sobolev spaces. For strong dispersion, we show global well-posedness in $L^2(\mathbb{R}^2}$. This is achieved by exploiting transversality in the resonant case via bilinear Strichartz estimates and nonlinear Loomis-Whitney inequality. For small dispersion, the equations cannot be solved by Picard iteration and we use frequency-dependent time localisation. The talk is based on joint work with Robert Schippa (Karlsruhe Institute of Technology, Germany).

This talk focuses on the well-posedness of the derivative nonlinear Schrödinger equation on the line. This model is known to be completely integrable and $L^2$-critical with respect to scaling. However, until recently not much was known regarding the well-posendess of the equation below $H^{\frac 1 2}$. In this talk we prove that the problem is well-posed in the critical space $L^2$ on the line, highlighting several recent results that led to this resolution. This is joint work with Benjamin Harrop-Griffiths, Rowan Killip, and Monica Visan.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will present recent results concerning examples of loss of regularity for solutions to linear transport equations with advecting field in Sobolev spaces below the Lipscitz class. I will discuss how this loss is generic and can be made instantaneous and total (that is, there exists smooth initial data for which the solution leaves instantaneously any Sobolev space of positive order). This is joint work with Giovanni Alberti, Gianluca Crippa, Gautam Iyer, and Tarek Elgindi

I will show how to construct a solution to the Keller-Segel-Navier-Stokes system in critical spaces via weighted maximal regularity. One of the problem in doing so is the fact that we work in Lipschitz domains (in dimensions 2 or 3) in which the Stokes operator has regularising properties only in a small range of spaces. This is a joint work with Matthias Hieber (TU Darmstadt, Germany), Hideo Kozono (Waseda University, Japan) and Patrick Tolksdorf (Johannes Gutenberg-Uni Mainz, Germany).

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk I will discuss necessary and sufficient conditions on the regularity of the external force for energy balance to hold for weak solutions of the 2D incompressible Euler equations. The problem is motivated by turbulence modeling and the result should be contrasted with the existence of wild solutions in 3D. This is joint work with Fabian Jin (ETH-Zurich), Samuel Lanthaler (CalTech), Milton C Lopes Filho (UFRJ) and Siddhartha Mishra (ETH-Zurich).

When applying Hamiltonian operator splitting methods for the time integration of multi-species Vlasov-Maxwell-Landau systems, the reliable and efficient numerical approximation of the Landau equation represents a fundamental component of the entire algorithm. Substantial computational issues arise from the treatment of the physically most relevant three-dimensional case with Coulomb interaction. This talk is concerned with the introduction and numerical comparison of novel approaches for the evaluation of the Landau collision operator. In the spirit of collocation, common tools are the identification of fundamental integrals, series expansions of the integral kernel and the density function on the main part of the velocity domain, and interpolation as well as quadrature approximation nearby the singularity of the kernel. Focusing on the favourable choice of the Fourier spectral method, their practical implementation uses the reduction to basic integrals, fast Fourier techniques, and summations along certain directions. Moreover, an important observation is that a significant percentage of the overall computational effort can be transferred to precomputations which are independent of the density function. For the purpose of exposition and numerical validation, the cases of constant, regular, and singular integral kernels are distinguished, and the procedure is adapted accordingly to the increasing complexity of the problem. With regard to the time integration of the Landau equation, the most expedient approach is applied in such a manner that the conservation of mass is ensured.

The propagation of electromagnetic waves in some composite mediums as meta-materials is described as strong dispersion. This can be seen through the propagation of multiple modes one can observe when taking into consideration a constitutive relation of nonlocal type. This later links the exciting electric field to the electric displacement by considering the effect of the surrounding neighbourhood of the observation point. We show that the spatially nonlocal characterisation of the material law serves us for deriving additional effective material parameters. In other words writing the homogenised problem with higher order derivative terms. These effective parameters translate the nonlocal effects produced in the response of meta-materials to the exciting electric field

The Muskat equation models the interaction of two incompressible fluids with equal viscosity propagating in porous medium, governed by Darcy’s law. In this talk, we investigate the small data critical regularity theory for this equation, and in particular, the desingularization of interfaces with small moving corners. This is a joint work with Eduardo Garcia-Juarez, Javier Gomez-Serrano and Benoit Pausader.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The Korteweg – de Vries (KdV) equation, a model that describes shallow water waves in a one dimensional channel, has traveling wave solutions, called solitary waves, or solitons, which have been observed experimentally back in 1834 and studied since late 1800s. In 1960s the inverse scattering transform was introduced to study solutions of KdV, and in particular, show that a sufficiently fast decaying solution will eventually split into a sum of solitons, traveling to the right, and an oscillatory decaying part, traveling to the left (the ``soliton resolution conjecture"). Various generalizations of the KdV with different power nonlinearities (gKdV) have been also studied and in the subcritical cases of gKdV the stability of solitons was shown. In the critical and supercritical cases of gKdV, it is known that solitary waves are not stable, and they may lead to formation of singularities, or blow-up solutions.      If the dispersion operator in KdV is replaced with a fractional derivative, the equation is referred to as the fractional (generalized) KdV equation, some specific cases (e.g., Benjamin-Ono equation) appear in modeling of deep water waves.    The KdV family of equations can also be extended to higher dimensions, appearing in various physical applications in 2d and 3d.      In this talk we discuss the question of solitary waves and their stability or instability in fractional setting, and in higher dimensions.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We consider the cubic 1-D NLS on R and prove a blow-up result for functions that are of borderline regularity, i.e. $H^s$ for any $s<1/2$ for the Sobolev scale and $\mathcal{F}L^\infty$ for the Fourier-Lebesgue scale. This is done by identifying at this regularity a certain functional framework from which solutions exit in finite time. This functional framework allows, after using a pseudo-conformal transformation, to reduce the problem to a large time study of a periodic Schrödinger equation with non-autonomous cubic nonlinearity. The blow-up result corresponds to an asymptotic completeness result for the new equation, that we prove by using Bourgain's method and by exploiting the oscillatory nature of the coefficients involved in the time-evolution of the Fourier modes. Finally, as an application we give conditions on curvature and torsion of a smooth curve to insure the existence of a binormal flow solution that generates several singularities in finite time. This is a joint work with Renato Lucà, Nikolay Tzvetkov and Luis Vega.

In this talk, we consider the Klein--Gordon equation $$ \begin{cases} \frac{\partial ^2}{\partial t^2} \psi(x,t) =\Delta \psi(x,t) - m(x,t)\psi(x,t), \quad t >t_0, \quad x \in \mathbb{T}^d\\ \psi(x,t_0) = \psi_0(x), \quad \partial_t \psi (x,t_0) = \varphi_0(x) \end{cases} $$ equipped with periodic initial and boundary conditions and time and space dependant coefficient $m(x,t)$. The latter assumption was proposed only recently and it allows for dealing with the problems of negative probability density and of violation of Lorenz covariance. Moreover application of time and space dependant coefficients extends application of such a problem to the domain of quantum cosmology, where $m(x,t)$ may bring possibly highly oscillatory form $$ m(x,t) = \sum_n a_n(x,t) e^{i \omega_n t} $$ with frequencies $\omega_n \in \mathbb{R}, \: n \in \mathbb{Z}$. Numerical approximation of such a problem requires various approaches when $m(x,t)$ is non-oscillatory, or highly oscillatory. The most challenging form of coefficient $m(x,t)$, however, is when it includes low and high frequencies, for example $m(x,t)=a_0(x,t)+a_1(x,t){\rm e}^{i t}+a_2(x,t){\rm e}^{i 10^6 t}$. In this talk we will present various approaches to all the kinds of these problems, will present final error estimates and plenty of numerical examples. Results of these investigations were obtained with Karolina Lademann (University of Gdansk), Katharina Schratz (Sorbonne Universite), Mrissa Condon (Dublin City University) and Rafal Perczynski (University of Gdansk).

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

This talk is devoted to the study of the equation $u u_x - u_{yy}=f$ in the domain $(x_0,x_1)\times (-1,1)$, in the vicinity of the shear flow profile $u(x,y)=y$. This equation serves as a toy model for more complicated fluid equations such as the Prandtl system. The difficulty lies in the fact that we are interested in changing sign solutions. Hence the equation is forward parabolic in the region where $u>0$, and backward parabolic in the region $u<0$. The line $u=0$ is a free boundary and an unknown of the problem. Unexpectedly, we prove that even when the data (i.e. the source term $f$ or the boundary data) are smooth, existence of strong solutions of the equation fails in general. This phenomenon is already present at the linear level, and linked to the existence of singular profiles for the homogeneous linearized equation. In fact, we prove that strong solutions exist (both for the linearized and for the nonlinear system) if and only if the data satisfy a finite number of orthogonality conditions, whose purpose is to avoid the presence of singular profiles in the solution. A key difficulty of our work is to cope with these orthogonality conditions during the nonlinear fixed-point scheme. In particular, we are led to prove their stability with respect to the underlying base flow. This is a joint work with Frédéric Marbach and Jean Rax.

For $\alpha = 2$ one recovers the classical KP-I equation, which was introduced by Kadomtsev $\&$ Petviashvili as a weakly two-dimensional extension of the Korteweg– de Vries (KdV) equation. Similarly as in the classical case, the fractional KP-I equation is a two-dimensional extension of the fractional KdV equation. Trivially, any solitary solution of the fractional KdV equation is a solution of the fractional KP-I equation – called the line solitary solution. It is known that the line solitary solution for the classical KP-I equation is transversely linear and nonlinear unstable (Zakharov ’73; Alexander, Pego, Sachs ’97; Rousset, Tzvetkov ’09, Rous- set, Tzvetkov ’07). Relying on a simple criterion posed by Rousset $\&$ Tzvetkov in 2013, we extend the result on transverse linear instability for the fractional KP-I equation. Numerical experiments support the instability result for the fractional KP-I equation and suggest transverse stability for the fractional KP-II equation (which is given by equation above when replacing $-u_{yy}$ by $+u_{yy})$. Furthermore, we discuss the existence and properties of fully localized solitary solutions for the fractional KP-I equation. The results are based on joint works with H. Borluk (Istanbul) and D. Nilsson (Lund). 

The Kaluza-Klein theories represent the classical mathematical approach to the unification of general relativity with electromagnetism and more generally with gauge fields. In these theories, general relativity is considered in 1+3+d dimensions and in the simplest case d=1 dimensional gravity is compactified on a circle to obtain at low energies a (3+1)-dimensional Einstein-Maxwell-Scalar systems.  In this talk I will discuss the problem of the classical global stability of Kaluza-Klein theories when d=1. This is a joint work with C. Huneau and Z. Wyatt.

The Landau-Lifshitz-Gilbert equation (LLG) is a continuum model describing the dynamics for the spin in ferromagnetic materials. In the first part of this talk we describe the properties and dynamical behaviour of the self-similar solutions of this model in one dimension. Time permitting, and motivated by the properties of these solutions, we consider the Cauchy problem for the LLG-equation and provide a global well-posedness result provided that the BMO norm of the initial data is small.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The Boussinesq equations are a member of a family of models of incompressible fluid equations, including the 3D Euler equations, for which the problem of global existence of solutions is open. The Boussinesq equations arise in fluid mechanics, in connection to thermal convection and they are extensively studied in that context. Formation of finite time singularities from smooth initial data in ideal (conservative) 2D Boussinesq equations is an important open problem, related to the blow up of solutions in 3D Euler equations. The Voigt Boussinesq is a conservative approximation of the Boussinesq equations which has certain attractive features, including sharing the same steady solutions with the Boussinesq equations. In this talk, after giving a brief description of issues of local and global existence, well-posedness and approximation in the incompressible fluids equations, I will present a global regularity result for critical Voigt Boussinesq equations. Some of the work is joint with Jingyang Shu.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.

In this talk, we will discuss uniqueness properties of solutions to the linear generalized fourth-order Schrödinger equations. We show that a solution with fast enough decay in certain Sobolev spaces at two different times has to be trivial. This is a joint work with Zachary Lee.