Schedule for: 19w5233 - Advances in Dispersive Equations: Challenges & Perspectives

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the 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.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

In this joint work with Massimilano Gubinelli and Hiro Oh we prove local wellposedness for a renormalized wave equation with additive white noise. Key ingredients are an ansatz going back to Da Prato and Debussche, a paraproduct decomposition, and an analysis of various stochastic fields and of a stochastic operator.

We consider the non linear wave equation (NLW) on the d-dimensional torus with an analytic nonlinearity of order at least two at the origin. We prove that, for almost all mass, small smooth solutions of high Sobolev indices are stable up to arbitrary long times with respect to the size of the initial data. To prove this result we use a normal form transformation decomposing the dynamics into low and high frequencies with weak interactions. While the low part of the dynamics can be put under classical Birkhoff normal form, the high modes evolves according to a time dependent linear Hamiltonian system. We then control the global dynamics by using polynomial growth estimates for high modes and the preservation of Sobolev norms for the low modes. Our general strategy applies to any semi-linear Hamiltonian PDEs whose linear frequencies satisfy a very general non resonance condition. In particular it also applies straightforwardly to a Whitham-Boussinesq system in water waves theory Joint work with Joackim Bernier and Erwan Faou.

Previously, we showed that for all continuations of NLS blowup solutions, the phase is lost after the singularity. In this talk I will show that ``loss of phase'' can occur even if the NLS solution does not collapse. Therefore, if two NLS solutions travel a sufficiently long distance (time) before interacting, it is not possible to predict whether they would intersect in- or out-of-phase. Hence, a deterministic prediction of the interaction outcome becomes impossible. ``Fortunately'', because the relative phase between the two solutions becomes uniformly distributed in $[0,2\pi]$, the statistics of the interaction outcome is universal. The statistics can be efficiently computed using a novel Uncertainty-Quantification method, even when the distribution of the noise source is unknown. I will end by arguing that although the NLS has a time-reversal symmetry, its solutions can experience a loss of reversibility.

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 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!

In joint work with Andrea Nahmod and Haitian Yue, we prove local well-posedness for the derivative nonlinear Schrodinger's equation in Fourier-Lebesgue space which has the same scaling as H^s for any s>0. This closes the gap left open by the work of Grunrock-Herr where s>1/4. Here there is no trilinear estimate in any standard function space, instead we will construct the solution in a nonlinear submanifold (of a function space) by exploiting its structure. This is somehow inspired by the theory of para-controlled distributions that Gubinelli et al. developed for stochastic PDEs, but our arguments are purely deterministic.

It is well-known that peakons in the Camassa-Holm equation and other integrable generalizations of the KdV equation are $H^1$-orbitally stable thanks to the presence of conserved quantities and properties of peakons as constrained energy minimizers. By using the method of characteristics, we prove that piecewise $C^1$ perturbations to peakons grow in time in spite of their stability in the $H^1$-norm. We also show that the linearized stability analysis near peakons contradicts the $H^1$-orbital stability result for the Camassa-Holm equation, hence the passage from the linear to nonlinear theory is false.

We will present an overview of the multilinear restriction theory from the perspective of applications in dispersive PDE's with emphasis on particular examples.

A surprisingly large number of physically relevant dispersive partial differential equations are integrable. Using the connection between the spectrum and the eigenfunctions of the associated Lax pair and the linear stability problem, we investigate the stability of the spatially periodic traveling wave solutions of such equations, extending the results to orbital stability in those case where solutions are linearly stable. The talk will emphasize recent results for the focusing NLS equation, as this situation is more complicated than that of other equations previously studied, for which the Lax pair is self adjoint.

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

In this talk I'm going to present a series of recent results about the long time complex dynamics of the water waves equations for a bi-dimensional fluid under the action of gravity and eventually capillary forces, with space periodic boundary conditions. This is an infinite dimensional Hamiltonian system with a quasi-linear nonlinearity. We shall discuss both long time existence results based on Birkhoff normal form arguments as well as bifurcation of small amplitude time quasi-periodic solutions (which are in particular global in time).

This is a joint work with T.Elgindi and N.Masmoudi. Recently Elgindi proved the existence of a continuum of $C^{1,\alpha}$ self similar solutions to 3D axisymmetric Euler equations with vanishing swirl. The aim of the talk will be to show that those solutions are stable under perturbations with swirl. I will in a first part of my presentation talk about recent result we obtained on one dimensional models for which we proved the stability of smooth and $C^{1,\alpha}$ selfsimilar solutions. In a second part I will explain how those one dimensional models helped us understanding the full 3D axisymmetric Euler.

I talk about the nonexistence of solutions in the Sobolev space and the Gevrey class and the norm inflation of the data-solution map at the origin under slightly different conditions, respectively for the NLS with third order dispersion and Raman scattering term. Physicists sometimes propose models which have strong instability from a mathematical point of view. The equation under consideration is such a kind of example and it is not very clear what role the mathematical ill-posedness plays in physics. I also talk about the local unique existence of solutions in the analytic function space. This talk is based on the joint work with Nobu Kishimoto, RIMS, Kyoto University.

We describe recent progress on sharp well-posedness for two completely integrable systems: the cubic NLS and the mKdV equations. This is based on joint work with B. Harrop-Griffiths and R. Killip.

We discuss the defocusing energy-critical nonlinear wave equation in four dimensions. For deterministic and smooth initial data, solutions exist globally and scatter. In contrast, since deterministic and rough initial data can lead to norm inflation, the energy-critical NLW is ill-posed at low regularities. In this talk, we show that the global existence and scattering behavior persists under random and rough perturbations of the initial data. In particular, norm inflation only occurs for exceptional sets of rough initial data. As part of the argument, we discuss techniques from restriction theory, such as wave packet decompositions and Bourgain's bush argument.

In this talk I will report some recent results and work in progress on the long-time behaviour of the time-dependent density functional theory. Specifically, I will outline main ingredients of the proofs of global existence, local decay and scattering for small data. This is a joint work with Fabio Pusateri.

In this talk, I will discuss a two-dimensional nonlocal, nonlinear dispersive PDE arising in the study of surface water waves called the Davey-Stewartson (DS) system. The first part of the talk will focus on recent work on global well-posedness and scattering for a particular case of the system at the scaling-critical regularity, which is inspired by the program for the mass-critical nonlinear Schr\"{o}dinger equation. We will also discuss open questions and possible research directions for the remaining cases of the system. The second part of the talk will focus on the physical relevance of the DS system as an approximate description of the evolution of wave packets on the surface of an incompressible, irrotational fluid (e.g. water) with a flat bottom. In particular, we will focus on the validity of the DS system as a long-time approximation for wave packet solutions of this water waves problem.

Prandtl's equations arise in the description of boundary layers in fluid dynamics. Solutions might form singularities in finite time, with the first reliable numerical studies performed by Van Dommelen and Shen in the early eighties, and a rigorous proof done later in the nineties in the seminal work of E and Engquist in two dimensions. This singularity formation is intimately linked with a phenomenon: the separation of the boundary layer. The precise structure of the singularity has however not been confirmed yet mathematically. This talk will first describe the dynamics of the inviscid model. We will describe how to compute the maximal time of existence of a solution, study certain self-similar profiles, and show that one in particular gives rise to the generic formation of the van Dommelen and Shen singularity. Then, for the original viscous model, the second part of the talk will focus on the obtention of detailed asymptotics for the solution at a relevant particular location. This is a collaboration with T.-E. Ghoul, S. Ibrahim and N. Masmoudi.

In a joint work with S. Ibrahim, we use the idea of ground states and excited states in nonlinear dispersive equations (e.g. Klein-Gordon and Schr\"odinger equations) to characterize solutions in the N-body problem with strong force under some energy constraints. Indeed, relative equilibria of the N-body problem play a similar role as solitons in PDE. I will introduce the ground state and excited energy for the general N-body problem and give a conditional dichotomy of the global existence and singularity below the excited energy. This dichotomy is given by the sign of a threshold function. I will also talk about a restricted 3-body problem (Hill's lunar type problem) that has a very nice analogy to the nine-set theorem studied by Nakanishi-Schlag on NLKG.

We consider the nonlinear Schrodinger equation on the spatial two dimensional cylinder. The transverse stability consists of studying the stability of the standing waves for 1D NLS under the 2D perturbations. Yamazaki showed that, once the size of the confined direction is normalized, the transverse stability depends on the frequency of the standing wave. More precisely, there exists a frequency threshold that separates stability and instability. In this talk, we will discuss the critical frequency case and extend Yamazaki's result by proving a sharp classification with respect to the exponent of the nonlinearity. This is a joint work with H. Kikuchi and S. Ibrahim.

A tour guide from the Banff Centre will guide participants on a quick walk around the campus and explain the Centre's function and current programs. This tour is optional.

Semilinear conformally invariant wave equations on spheres (cubic on the 3-sphere and quintic on the 2-sphere) are simple toy models for understanding the dynamics of nonlinear waves on compact domains. Small amplitude solutions of these equations are well approximated by the solutions of the corresponding resonant (time-averaged) systems, called conformal flows. In my talk I will present the derivation and some remarkable properties of the cubic and quintic conformal flows.

In order to justify certain model equations proposed in the biophysics literature for charge transport on polymers like DNA and protein, we consider a general class of discrete nonlinear Schroedinger equations on lattices, and prove that in the continuum limit, the limiting dynamics are given by a nonlinear Schroedinger equation (NLS) with a fractional Laplacian. In particular, a range of fractional powers arise from long-range lattice interactions in this limit, whereas the usual NLS with the non-fractional Laplacian arises from short-range interactions. We also obtain equations of motion for the expected position and momentum, the fractional counterpart of the well-known Newtonian equations of motion for the standard Schroedinger equation, and use a numerical method to suggest that the nonlocal Laplacian introduces decoherence, but that effect can be mitigated by the nonlinearity. Joint work with Gigliola Staffilani, Enno Lenzmann, and Yanzhi Zhang. Time permitting, I will talk about recent work defining biophysical machines that out-perform Turing machines, in joint work with Onyema Osuagwu and Daniel Inafuku.

I will discuss recent developments related to a blow up construction which was originally conceived for the critical Wave Maps problem but has since found applications to a wide variety of problems, including Schrodinger type equations as well as very recently quasilinear wave equations. A point of interest are stability properties of these solutions, as well as some conjectures about general classifications.

We consider a vector-valued blow-up solution with values in Rm for the semilinear wave equation with power nonlinearity in one space dimension. We first characterize all the solutions of the associated stationary problem as an m-parameter family. Then, we show that the solution in self-similar variables approaches some particular stationary one in the energy norm, in the non-characteristic cases.

(Joint work with Y. Guo, E. Grenier and M. Suzuki) We consider the 2 fluid Euler-Poisson equation in 3d space and show that, when the mass of electron tends to 0, the solutions can be well approximated by the strong limit which solves the (1 fluid) Euler-Poisson equation for ions and an initial layer which disperses the excess electron density and velocity in short time. This is a singular limit, somewhat akin to the low-Mach number problem studied by Klainerman-Majda, Ukai and Metivier-Schochet, but in this case, the dispersive layer comes from a quasilinear equation involving coefficients depending on space and time (in fact depending on the strong limit), and the analysis relies on a local energy decay.

In this talk I will outline recent work in collaboration with Pierre Germain, Zaher Hani and Jalal Shatah regarding a rigorous derivation of the kinetic wave equation. The proof presented will rely of methods from PDE, statistical physics and number theory.

We'll discuss L^2 dispersive estimates for a general class of time dependent wave equations on manifolds. When the operators are symmetric with slow time variation and a zero energy spectral condition, we establish local decay of energy for solutions modulo finite dimensional dynamics. This is joint work with Jason Metcalfe and Daniel Tataru

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.