Dispersive Equations, Solitons, and Blow-up

Monday, September 4

08:30 - 09:00 Self-Registration
09:00 - 10:30 Yvan Martel: Strongly interacting solitons for NLS - Part I
10:30 - 11:00 Coffee Break
11:00 - 12:30 Hiro Oh: Stochastic dispersive dynamics - Part I
12:30 - 14:00 Lunch Break
14:00 - 15:30 Yvan Martel: Strongly interacting solitons for NLS - Part II
15:30 - 16:00 Coffee Break
16:00 - 17:00 Xian Liao: Conserved energies for the one dimensional Gross-Pitaevskii equation

Tuesday, September 5

09:00 - 10:30 Piotr Bizon: Global dynamics of nonlinear waves - Part I
10:30 - 11:00 Coffee Break & Group Photo
11:00 - 12:30 Yvan Martel: Strongly interacting solitons for NLS - Part III
12:30 - 14:00 Lunch Break
14:00 - 15:30 Piotr Bizon: Global dynamics of nonlinear waves - Part II
15:30 - 16:00 Coffee Break
16:00 - 17:00 Timothy Candy: Bilinear Restriction Estimates for General Phases

Wednesday, September 6

09:00 - 10:30 Hiro Oh: Stochastic dispersive dynamics - Part II
10:30 - 11:00 Coffee Break
11:00 - 12:30 Yvan Martel: Strongly interacting solitons for NLS - Part IV
12:30 Lunch Break and Excursion
Boat Trip to Königwinter, starting either at the Mathematics Center or at the landing place "Alter Zoll" (1:45 pm). Walk/hike to the Drachenfels. Return by SWB tram line no. 66

Friday, September 8

09:00 - 10:30 Hiro Oh: Stochastic dispersive dynamics - Part IV
10:30 - 11:00 Coffee Break
11:00 - 12:30 Piotr Bizon: Global dynamics of nonlinear waves - Part IV
12:30 Lunch Break / End of School

Abstracts

Piotr Bizon: Global dynamics of nonlinear waves

In my mini-course I will discuss the Cauchy problem for semilinear wave equations using two toy models, wave equations with a power nonlinearity and  equivariant wave maps, to illustrate basic concepts and types of evolution.

A tentative plan of lectures is as follows:

    1. Toy models, their properties and special solutions.

    2. Global-in-time evolution of small initial data; asymptotic convergence to an

        equilibrium on unbounded domains.

    3. Singularity formation for large initial data; various types of blowup, its stability,

        and critical behavior at the threshold of blowup.

    4. Quasiperiodic and weakly turbulent dynamics on compact manifolds.

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Timothy Candy: Bilinear Restriction Estimates for General Phases

Bilinear restriction estimates were originally developed to make progress on the question of L^p estimates for the Fourier transform of compact hypersurfaces. However bilinear restriction estimates can also be stated in terms of products of transverse waves, and thus are closely connected to problems in dispersive PDE. In this talk we present new bilinear restriction estimates for general phases at multiple scales, which extend recent estimates of Bejenaru, Lee-Vargas, and Tao (among others). These estimates can be extended to hold in the framework of adapted function spaces, and hence have applications to dispersive PDE in scale invariant settings.

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Masahiro Ikeda: Global dynamics below the ground state for the semilinear Schrödinger equations with a linear potential

We study global dynamics of solutions to the Cauchy problem for the focusing semi-linear Schrödinger equation with a linear potential on the real line. The problem is locally well-posed in the energy space under some assumptions on the potential. Our aim in the presentation is to study global behavior of the solution and prove a scattering result in the energy space and a blow-up or grow-up result for the problem, with the data whose mass-energy is less than that of the ground state.

Here the ground state is the unique radial positive solution to the stationary Schrödinger equation without the potential. A scattering result for the defocusing problem was studied by Lafontain '16. However, the focusing problem is more delicate because the energy might be negative

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Xian Liao: Conserved energies for the one dimensional Gross-Pitaevskii equation

In this talk I will establish conserved energies which describe the H^s regularity of the solution of the one dimensional Gross-Pitaevskii equation, in the case where the initial data is of  small perturbation.

The proof relies on the detailed analysis of the transmission coefficient (associated to the Lax operator) which is conserved by the GP flow. It is a joint work with Herbert Koch and is highly related to the recent work by Koch-Tataru on the conserved energies for the cubic NLS, mKdV and KdV.

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Yvan Martel: Strongly interacting solitons for NLS

Based on works by Yvan Martel-Pierre Raphael and Tien Vinh Nguyen, these lectures will be devoted to study two cases of strong interactions between solitons for the nonlinear Schrödinger equations (NLS). In the sub- and super-critical cases, the existence of multi-solitons with logarithmic distance in time is proved, extending a classical result of the integrable case (1D cubic NLS equation). In the mass-critical case, we will show how to construct a new class of multi-solitary wave solutions: a global (for positive time) solution of (NLS) that decomposes asymptotically into a sum of solitary waves concentrating at a logarithmic rate in large time. This solution blows up in infinite time with logarithmic rate. Using the pseudo-conformal transform, this yields the first example of solution blowing up in finite time with a rate strictly above the pseudo-conformal one. Such solution concentrates several bubbles at a point.

These special behaviours are due to strong interactions between the waves, in contrast with most previous works on multi-solitary waves of (NLS) where interactions do not affect the general behaviour (and in particular the blow up rate) of each bubble.

We will start by a brief introduction to the subject, including the construction of multi-solitary waves in the case of weak interactions.

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Hiro Oh: Stochastic dispersive dynamics

1. Invariant Gibbs measures for Hamiltonian PDE dynamics

By drawing an analogy to the finite-dimensional Hamiltonian dynamics, a Gibbs measure is expect to be invariant under the dynamics of a Hamiltonian PDE.

Such a Gibbs measure is often constructed as a weighted Gaussian measure on an infinite dimensional space of functions/distributions.

We consider  two-dimensional nonlinear wave equations (NLW) as our primary example.

We first go over the construction of Gibbs measures associated to NLW.  The main tools here are  the Wick renormalization, the Wiener chaos estimate, and Nelson's estimate.

Then, we discuss the associated dynamical problem and invariance of the Gibbs measures.

2. On the transport property of Gaussian measures under Hamiltonian PDE dynamics

Invariance of a Gibbs measure, constructed as a weighted Gaussian measure, implies that the underlying Gaussian measure is quasi-invariant. Namely, the original measure and the pushforward measure under the dynamics are equivalent - mutually absolutely continuous.

In this part, we study the transport properties of Gaussian measures supported on smooth functions under the dynamics of the cubic NLW on the two-dimensional torus.

The main difficulty lies in establishing an energy estimate with a certain smoothing property.

We achieve this goal by (i) introducing a renormalized H^s-energy functional and (ii) studying the corresponding energy estimate in the probabilistic setting.

If time permits, we will talk about known results for other equations.

3. Stochastic dispersive PDEs

In this last part, we turn our attention to stochastic dispersive PDEs.

In particular, we take two-dimensional stochastic NLW with additive space-time white noise forcing and study its well-posedness.

For this purpose, we introduce a time-dependent renormalization and perform a stochastic analysis on the Wick powers of the stochastic convolution.

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Dimitrios Roxanas: Long-time dynamics of solutions to the focusing energy-critical heat equation

We study the focusing energy-critical nonlinear heat equation u_t - ∆u - |u|^2 u = 0, in R^4 . We prove that solutions emanating from initial data with energy and kinetic energy below those of the stationary solution are global and decay to zero. We show that global solutions dissipate to zero building on a refined small data theory and L^2−dissipation, expanding on ideas that have previously been applied to the Navier-Stokes system. To rule out the possibility of blow-up we argue by resorting to the "concentration-compactness plus rigidity" approach of Kenig and Merle for dispersive equations. We exploit the dissipation but our proof does not rely on maximum/comparison principles.

This is joint work with Stephen Gustafson.

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