Schedule for: 20w5013 - Dynamics in Geometric Dispersive Equations and the Effects of Trapping, Scattering and Weak Turbulence

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.

The asymptotic stability analysis of one-dimensional topological solitons such as the well-known “kink” in the \$phi^4$ model requires an understanding of the asymptotic behavior of small solutions to 1D Klein-Gordon equations with variable coefficient quadratic and cubic nonlinearities. In this talk I will first describe the difficulties caused by variable coefficients to deal with the long-range nature of such nonlinearities. Then I will present a new result on sharp decay estimates and asymptotics for small solutions to 1D Klein-Gordon equations with constant and variable coefficient cubic nonlinearities. The main novelty of our approach is the use of pointwise-in-time local decay estimates to deal with the variable coefficient nonlinearity. If time permits, I will also discuss work in progress on the variable coefficient quadratic case, which exhibits a striking resonant interaction between the spatial oscillations of the variable coefficient and the temporal oscillations of the solutions. This is joint work with Hans Lindblad and Avy Soffer.

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

I will present joint work with Idan Regev and Daniel Tataru on global well-posedness and scattering for the defocusing Davey-Stewartson equation in the mass critical case. Our solution of this problem involves a proof of the Plancherel Theorem for the corresponding scattering transform in two dimensions. The latter had also been a challenging open problem. On the way, there will be a new pointwise estimate for classical fractional integrals, and a new theorem on L^2 boundedness of pseudodifferential operators with non-smooth symbols. Time permitting, I'll also describe a very different application of our Plancherel Theorem, to the inverse boundary value problem of Calderón for a class of unbounded conductivities.

Meet in the Corbett Hall Lounge 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 binormal flow is a model for the dynamics of a vortex filament in a 3-D inviscid incompressible fluid. The flow is also related to the classical continuous Heisenberg model in ferromagnetism, and to the 1-D cubic Schrödinger equation. We consider a class of solutions at the critical level of regularity that generate singularities in finite time. One of our main results presented in this talk is to prove the existence of a natural energy associated to these solutions. This energy remains constant except at the time of the formation of the singularity when it has a jump discontinuity. When interpreting this conservation law in the framework of fluid mechanics, it involves the amplitude of the Fourier modes of the variation of the direction of the vorticity. This is a joint work with Luis Vega.

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.

It is well-known from numerical experiments that in energy supercritical wave equations self-similar blowup solutions may appear as intermediate attractors close to the threshold for singularity formation. For wave maps and the Yang-Mills equation, such critical solutions have been observed in the equivariant setting by Bizoń et al. From an analytic point of view, this phenomenon is still poorly understood. In this talk, we discuss the radial wave equation with a cubic and a quadratic power nonlinearity, respectively, which can be viewed as toy models for co-rotational wave maps into the sphere and the equivariant Yang-Mills equation. For both nonlinearities, we found explicit self-similar blowup solutions, which we conjecture to be critical solutions in the above sense. We prove their co-dimension one stability in d=7 and d=9, respectively, and discuss the main challenges in the generalisation of our approach. This is joint work with Irfan Glogić (Vienna).

In this talk, we will consider the low regularity well-posedness problem for the two dimensional gravity water waves. This quasilinear dispersive system admits an interesting structure which we exploit to prove a new class of energy estimates, which we call balanced cubic estimates. This yields a key improvement over the earlier energy estimates of Hunter-Ifrim-Tataru. These results allow us to significantly lower the regularity threshold for local well-posedness, even without using dispersive properties. This is joint work with Mihaela Ifrim and Daniel Tataru.

I will present a sample of results on the long-term dynamics and blowup of critical wave maps. In particular, I will discuss an ongoing project with Krieger and Miao on the rigidity of the blowup solutions from 2006 constructed by Krieger, Tataru, and myself under sufficiently regular perturbations.

We prove a series of intimately related results tied to the regularity and geometry of solutions to the three-dimensional compressible Euler equations. The solutions are allowed to have nontrivial vorticity and entropy, and an arbitrary equation of state with positive sound speed. The central theme is that under low regularity assumptions on the initial data, it is possible to avoid, at least for short times, the formation of shocks. Our main result is that the time of classical existence can be controlled under low regularity assumptions on the part of the initial data associated with propagation of sound waves in the fluid. Such low regularity assumptions are in fact optimal. To implement our approach, we derive several results of independent interest: (i) sharp estimates for the acoustic geometry, which in particular capture how the vorticity and entropy interact with the sound waves; (ii) Strichartz estimates for quasilinear sound waves coupled to vorticity and entropy; (iii) Schauder estimates for the transport-div-curl-part of the system. Compared to previous works on low regularity, the main new feature of our result is that the quasilinear PDE system under study exhibits multiple speeds of propagation. In fact, this is the first result of its kind for a system with multiple characteristic speeds. An interesting feature of our proof is the use of techniques that originated in the study of the vacuum Einstein equations in general relativity.

Confinement effects by rigid boundaries in the dynamics of ideal fluids are considered from the perspective of long-wave models and their parent Euler systems, with the focus on the consequences of establishing contacts of material surfaces with the confining boundaries. When contact happens, it can be shown that the model evolution can lead to the dependent variables developing singularities in finite time. The conditions and the nature of these singularities are illustrated in several cases, progressing from a single layer homogeneous fluid with a constant pressure free surface and flat bottom, to the case of a two-fluid system contained between two horizontal rigid plates, and finally, through numerical simulations, to the full Euler stratified system. These illustrate the qualitative and quantitative predictions of the models within a set of examples chosen to illustrate the theoretical results.

In this talk we consider the cubic nonlinear Schrödinger and modified Korteweg-de Vries equations on the real line. We present a proof of global well-posedness for initial data in H^s for any s > - 1/2. This is joint work with Rowan Killip and Monica Visan.

In joint work with D. Tataru we use Bäcklund transforms to add multisolitons to solutions to NLS and mKdV. Of particular interest are multiple solitons with the same spectral parameter. We prove 1) The pure N soliton manifold is uniformly smooth in H^n(R) for any n. 2) If the initial data has distance epsilon to the initial data of a pure N soliton then there is a (generically different) N soliton solution so that the distance is bounded by c epsilon uniformly in t. A key tool are holomorphic families of wave functions for the Lax operator.

We discuss a newly found focusing L2-critical NLS in 1-d with a list of beautiful features (explicit solitons, uniqueness, N-soliton, blowup etc.) This is joint ongoing work with Patrick Gérard.

This is a joint work with Jeremy Marzuola and Daniel Tataru. We explore local well-posedness for quasilinear Schrodinger equations with large initial data. This builds off of preceding works in the small data regime. In the large data regime, the possibility of trapping must be carefully dealt with.

Wave turbulence theory conjectures that the long-time behavior of “generic" solutions of nonlinear dispersive equations is governed (at least over certain long timescales) by the so-called wave kinetic equation (WKE). This approximation is supposed to hold in the limit when the size L of the domain goes to infinity, and the strength \alpha of the nonlinearity goes to 0. We will discuss some recent progress towards settling this conjecture, focusing on a recent joint work with Yu Deng (USC), in which we show that the answer seems to depend on the “scaling law” with which the limit is taken. More precisely, we identify two favorable scaling laws for which we justify rigorously this kinetic picture for very large times that are arbitrarily close to the kinetic time scale (i.e. within $L^\epsilon$ for arbitrarily small $\epsilon$). This is similar to how the Boltzmann-Grad scaling law is imposed in the derivation of Boltzmann's equation. We also give counterexamples showing divergences for the complementary scaling laws.

We consider the 2 fluid Euler-Poisson system and we investigate the behavior of solutions as the ratio electron mass/ion mass goes to 0. On a timescale where the ion move at speed one (and the electrons move infinitely fast), the solutions can be described on a fixed time interval and the solution is described by a simpler one fluid system, the Euler equation for ions. This is a joint work with E. Grenier, Y. Guo and M. Suzuki.

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.