We provide an asymptotic analysis of a nonlinear integro-differential equation which describes the evolutionary dynamics of a population which reproduces sexually and which is subject to selection and competition. The sexual reproduction is modeled via a nonlinear integral term, known as the 'infinitesimal model'. We consider a regime of small segregational variance, where a parameter in the infinitesimal operator, which measures the deviation between the trait of the offspring and the mean parental trait, is small. We prove that in this regime the phenotypic distribution remains close to a Gaussian profile with a fixed small variance and we characterize the dynamics of the mean phenotypic trait via an ordinary differential equation. We also briefly discuss the extension of the method to the study of steady solutions and their stability.[-]

We consider parabolic equations of the form $u_t = u_{xx} + f (u)$ on the real line. Unlike their counterparts on bounded intervals, these equations admit bounded solutions whose large-time dynamics is not governed by steady states. Even with respect to the locally uniform convergence, the solutions may not be quasiconvergent, that is, their omega-limit sets may contain nonstationary solutions.
We will start this lecture series by exhibiting several examples of non-quasiconvergent solutions, discussing also some entire solutions appearing in their omega-limit sets. Minimal assumptions on the nonlinearity are needed in the examples, which shows that non-quasiconvergent solutions occur very frequently in this type of equations. Our next goal will be to identify specific classes of initial data that lead to quasiconvergent solutions. These include localized initial data (joint work with Hiroshi Matano) and front-like initial data. Finally, in the last part of these lectures, we take a more global look at the solutions with such initial data. Employing propagating terraces, or stacked families of traveling fronts, we describe their entire spatial profile at large times.[-]

In this lecture I will describe a framework for the Fredholm analysis of non-elliptic problems both on manifolds without boundary and manifolds with boundary, with a view towards wave propagation on Kerr-de-Sitter spaces, which is the key analytic ingredient for showing the stability of black holes (see Peter Hintz' lecture). This lecture focuses on the general setup such as microlocal ellipticity, real principal type propagation, radial points and generalizations, as well as (potentially) normally hyperbolic trapping, as well as the role of resonances.[-]

In this lecture I will discuss Kerr-de Sitter black holes, which are rotating black holes in a universe with a positive cosmological constant, i.e. they are explicit solutions (in 3+1 dimensions) of Einstein's equations of general relativity. They are parameterized by their mass and angular momentum.
I will discuss the geometry of these black holes, and then talk about the stability question for these black holes in the initial value formulation. Namely, appropriately interpreted, Einstein's equations can be thought of as quasilinear wave equations, and then the question is if perturbations of the initial data produce solutions which are close to, and indeed asymptotic to, a Kerr-de Sitter black hole, typically with a different mass and angular momentum. In this talk, I will emphasize geometric aspects of the stability problem, in particular showing that Kerr-de Sitter black holes with small angular momentum are stable in this sense.[-]

We consider an acoustic waveguide modeled as follows:

$ \left \{\begin {matrix}
\Delta u+k^2(1+V)u=0& in & \Omega= \mathbb{R} \times]0,1[\\
\frac{\partial u}{\partial y}=0& on & \partial \Omega
\end{matrix}\right.$

where $u$ denotes the complex valued pressure, k is the frequency and $V \in L^\infty(\Omega)$ is a compactly supported potential.
It is well-known that they may exist non trivial solutions $u$ in $L^2(\Omega)$, called trapped modes. Associated eigenvalues $\lambda = k^2$ are embedded in the essential spectrum $\mathbb{R}^+$. They can be computed as the real part of the complex spectrum of a non-self-adjoint eigenvalue problem, defined by using the so-called Perfectly Matched Layers (which consist in a complex dilation in the infinite direction) [1].
We show here that it is possible, by modifying in particular the parameters of the Perfectly Matched Layers, to define new complex spectra which include, in addition to trapped modes, frequencies where the potential $V$ is, in some sense, invisible to one incident wave.
Our approach allows to extend to higher dimension the results obtained in [2] on a 1D model problem.[-]

The cubic Szegö equation has been introduced as a toy model for totally non dispersive evolution equations. It turned out that it is a complete integrable Hamiltonian system for which we built a non linear Fourier transform giving an explicit expression of the solutions.
This explicit formula allows to study the dynamics of the solutions. We will explain different aspects of it: almost-periodicity of the solutions in the energy space, uniform analyticity for a large set of initial data, turbulence phenomenon for a dense set of smooth initial data in large Sobolev spaces.
From joint works with Patrick Gérard.[-]

In this talk we discuss the convergence to equilibrium in conservative-dissipative ODE-systems, kinetic relaxation models (of BGK-type), and Fokker-Planck equation. This will include symmetric, non-symmetric and hypocoercive evolution equations. A main focus will be on deriving sharp decay rates.
We shall start with hypocoercivity in ODE systems, with the ”hypocoercivity index” characterizing its structural complexity.
BGK equations are kinetic transport equations with a relaxation operator that drives the phase space distribution towards the spatially local equilibrium, a Gaussian with the same macroscopic parameters. Due to the absence of dissipation w.r.t. the spatial direction, convergence to the global equilibrium is only possible thanks to the transport term that mixes various positions. Hence, such models are hypocoercive.
We shall prove exponential convergence towards the equilibrium with explicit rates for several linear, space periodic BGK-models in dimension 1 and 2. Their BGK-operators differ by the number of conserved macroscopic quantities (like mass, momentum, energy), and hence their hypocoercivity index. Our discussion includes also discrete velocity models, and the local exponential stability of a nonlinear BGK-model.
The third part of the talk is concerned with the entropy method for (non)symmetric Fokker-Planck equations, which is a powerful tool to analyze the rate of convergence to the equilibrium (in relative entropy and hence in L1). The essence of the method is to first derive a differential inequality between the first and second time derivative of the relative entropy, and then between the entropy dissipation and the entropy. For hypocoercive Fokker-Planck equations, i.e. degenerate parabolic equations (with drift terms that are linear in the spatial variable) we modify the classical entropy method by introducing an auxiliary functional (of entropy dissipation type) to prove exponential decay of the solution towards the steady state in relative entropy. The obtained rate is indeed sharp (both for the logarithmic and quadratic entropy). Finally, we extend the method to the kinetic Fokker-Planck equation (with nonquadratic potential).[-]

We are concerned with deriving sharp exponential decay estimates (i.e. with maximum rate and minimum multiplicative constant) for linear, hypocoercive evolution equations. Using a modal decomposition of the model allows to assemble a Lyapunov functional using Lyapunov matrix inequalities for each Fourier mode.
We shall illustrate the approach on the 1D Goldstein-Taylor model, a2-velocity transport-relaxation equation. On the torus the lowest Fourier modes determine the spectral gap of the whole equation in $L^{2}$. By contrast, on the whole real line the Goldstein-Taylor model does not have a spectral gap, since the decay rate of the Fourier modes approaches zero in the small mode limit. Hence, the decay is reduced to algebraic.
In the final part of the talk we consider the Goldstein-Taylor model with non-constant relaxation rate, which is hence not amenable to a modal decomposition. In this case we construct a Lyapunov functional of pseudodifferential nature, one that is motivated by the modal analysis in the constant case.The robustness of this approach is illustrated on a multi-velocity GoldsteinTaylor model, yielding explicit rates of convergence to the equilibrium.
This is joint work with J. Dolbeault, A. Einav, C. Schmeiser, B. Signorello, and T. Wöhrer.[-]