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## Geometric Analysis of the Linear Boltzmann Equation I. Trend to Equilibrium

### Annals of PDE (2015-12-01) 1: 1-84 , December 01, 2015

This work is devoted to the analysis of the linear Boltzmann equation on the torus, in the presence of a force deriving from a potential. The collision operator is allowed to be degenerate in the following two senses: (1) the associated collision kernel may vanish in a large subset of the phase space; (2) we do not assume that it is bounded below by a Maxwellian at infinity in velocity. We study how the association of transport and collision phenomena can lead to convergence to equilibrium, using concepts and ideas from control theory. We prove two main classes of results. On the one hand, we show that convergence towards an equilibrium is equivalent to an almost everywhere geometric control condition. The equilibria (which are not necessarily Maxwellians with our general assumptions on the collision kernel) are described in terms of the equivalence classes of an appropriate equivalence relation involving transport and collisions. On the other hand, we characterize the exponential convergence to equilibrium in terms of the Lebeau constant, which involves some averages of the collision frequency along the flow of the transport. We also explain how to handle the case of linear Boltzmann equations posed on the phase space associated to a compact Riemannian manifold without boundary.

## From Hard Sphere Dynamics to the Stokes–Fourier Equations: An Analysis of the Boltzmann–Grad Limit

### Annals of PDE (2017-02-03) 3: 1-118 , February 03, 2017

We derive the linear acoustic and Stokes–Fourier equations as the limiting dynamics of a system of *N* hard spheres of diameter
$${\varepsilon }$$
in two space dimensions, when
$$N\rightarrow \infty $$
,
$${\varepsilon }\rightarrow 0$$
,
$$N{\varepsilon }=\alpha \rightarrow \infty $$
, using the linearized Boltzmann equation as an intermediate step. Our proof is based on Lanford’s strategy (Time evolution of large classical systems, Springer, Berlin, 1975), and on the pruning procedure developed in Bodineau et al. (Invent Math 203:493–553, 2016) to improve the convergence time to all kinetic times with a quantitative control which allows us to reach also hydrodynamic time scales. The main novelty here is that uniform
$$L^2$$
a priori estimates combined with a subtle symmetry argument provide a weak version of chaos, in the form of a cumulant expansion describing the asymptotic decorrelation between the particles. A refined geometric analysis of recollisions is also required in order to discard the possibility of multiple recollisions.

## On the Global Stability of the Wave-map Equation in Kerr Spaces with Small Angular Momentum

### Annals of PDE (2015-09-15) 1: 1-78 , September 15, 2015

This paper is motivated by the problem of the nonlinear stability of the Kerr solution for axially symmetric perturbations. We consider a model problem concerning the axially symmetric perturbations of a wave map $$\Phi $$ defined from a fixed Kerr solution $${\mathcal K}(M,a)$$ , $$0\le a \le M $$ , with values in the two dimensional hyperbolic space $${\mathbb H}^2$$ . A particular such wave map is given by the complex Ernst potential associated to the axial Killing vector-field $$\mathbf{Z}$$ of $${\mathcal K}(M,a)$$ . We conjecture that this stationary solution is stable, under small axially symmetric perturbations, in the domain of outer communication (DOC) of $${\mathcal K}(M,a)$$ , for all $$0\le a<M$$ and we provide preliminary support for its validity, by deriving convincing stability estimates for the linearized system.

## Prandtl Boundary Layer Expansions of Steady Navier–Stokes Flows Over a Moving Plate

### Annals of PDE (2017-04-11) 3: 1-58 , April 11, 2017

This paper concerns the validity of the Prandtl boundary layer theory in the inviscid limit for steady incompressible Navier–Stokes flows. The stationary flows, with small viscosity, are considered on $$[0,L]\times \mathbb {R}_{+}$$ , with a no-slip boundary condition over a moving plate at $$y=0$$ . We establish the validity of the Prandtl boundary layer expansion and its error estimates.

## Landau Equation for Very Soft and Coulomb Potentials Near Maxwellians

### Annals of PDE (2017-01-31) 3: 1-65 , January 31, 2017

This work deals with the Landau equation for very soft and Coulomb potentials near the associated Maxwellian equilibrium. We first investigate the corresponding linearized operator and develop a method to prove strong asymptotical (but not uniformly exponential) stability estimates of its associated semigroup in large functional spaces. We then deduce existence, uniqueness and fast decay of the solutions to the nonlinear equation in a close-to-equilibrium framework. Our result drastically improves the set of initial data compared to the one considered by Guo and Strain who established similar results in Guo (Commun Math Phys 231:391–434, 2002) and Strain and Guo (Commun Partial Differ Equ 31(1–3):417–429, 2006; Arch Ration Mech Anal 187(2):287–339, 2008). Our functional framework is compatible with the non perturbative frameworks developed by Villani (Arch Ration Mech Anal 143(3):273–307 1998), Desvillettes and Villani (Invent Math 159(2):245–316, 2005), Desvillettes (J Funct Anal 269(5):1359–1403, 2015) and Carrapatoso et al. ( arXiv:1510.08704 , 2016), and our main result then makes possible to improve the speed of convergence to the equilibrium established therein.

## Logarithmic Local Energy Decay for Scalar Waves on a General Class of Asymptotically Flat Spacetimes

### Annals of PDE (2016-04-27) 2: 1-124 , April 27, 2016

This paper establishes that on the domain of outer communications of a general class of stationary and asymptotically flat Lorentzian manifolds of dimension $$d+1$$ , $$d\ge 3$$ , the local energy of solutions to the scalar wave equation $$\square _{g}{\uppsi }=0$$ decays at least with an inverse logarithmic rate. This class of Lorentzian manifolds includes (non-extremal) black hole spacetimes with no restriction on the nature of the trapped set. Spacetimes in this class are moreover allowed to have a small ergoregion, but are required to satisfy an energy boundedness statement. Without making further assumptions, this logarithmic decay rate is shown to be sharp. Our results can be viewed as a generalisation of a result of Burq, dealing with the case of the wave equation on flat space outside compact obstacles, and results of Rodnianski–Tao for asymptotically conic product Lorentzian manifolds. The proof will bridge ideas from Rodnianski and Tao (see [58]) with techniques developed in the black hole setting by Dafermos and Rodnianski (see [21, 22]). As a soft corollary of our results, we will infer an asymptotic completeness statement for the wave equation on the spacetimes considered in the case where no ergoregion is present.

## Global Well-Posedness and Scattering for the Defocusing, Mass-Critical Generalized KdV Equation

### Annals of PDE (2017-02-15) 3: 1-35 , February 15, 2017

In this paper we prove that the defocusing, mass-critical generalized KdV initial value problem is globally well-posed and scattering for $$u_{0} \in L^{2}(\mathbf {R})$$ . To prove this, we combine the profile decomposition of Killip et al. (Discrete Contin Dyn Syst Ser A 32(1):191–221, 2012) with an interaction Morawetz estimate constructed from the monotonicity formula of Tao (Discrete Contin Dyn Syst Ser A 18(1):1–14, 2007).

## Edge States in Honeycomb Structures

### Annals of PDE (2016-12-16) 2: 1-80 , December 16, 2016

An edge state is a time-harmonic solution of a conservative wave system, *e.g.* Schrödinger, Maxwell, which is propagating (plane-wave-like) parallel to, and localized transverse to, a line-defect or “edge”. Topologically protected edge states are edge states which are stable against spatially localized (even strong) deformations of the edge. First studied in the context of the quantum Hall effect, protected edge states have attracted huge interest due to their role in the field of topological insulators. Theoretical understanding of topological protection has mainly come from discrete (tight-binding) models and direct numerical simulation. In this paper we consider a rich family of *continuum* PDE models for which we rigorously study regimes where topologically protected edge states exist. Our model is a class of Schrödinger operators on
$${\mathbb {R}}^2$$
with a background two-dimensional honeycomb potential perturbed by an “edge-potential”. The edge potential is a domain-wall interpolation, transverse to a prescribed “rational” edge, between two distinct periodic structures. General conditions are given for the bifurcation of a branch of topologically protected edge states from *Dirac points* of the background honeycomb structure. The bifurcation is seeded by the zero mode of a one-dimensional effective Dirac operator. A key condition is a spectral no-fold condition for the prescribed edge. We then use this result to prove the existence of topologically protected edge states along zigzag edges of certain honeycomb structures. Our results are consistent with the physics literature and appear to be the first rigorous results on the existence of topologically protected edge states for continuum 2D PDE systems describing waves in a non-trivial periodic medium. We also show that the family of Hamiltonians we study contains cases where zigzag edge states exist, but which are not topologically protected.

## Finite Time Blowup for Lagrangian Modifications of the Three-Dimensional Euler Equation

### Annals of PDE (2016-12-01) 2: 1-79 , December 01, 2016

In the language of differential geometry, the incompressible inviscid Euler equations can be written in vorticity-vector potential form as
$$\begin{aligned} \partial _t \omega + {\mathcal {L}}_u \omega&= 0\\ u&= \delta \tilde{\eta }^{-1} \Delta ^{-1} \omega \end{aligned}$$
where
$$\omega $$
is the vorticity 2-form,
$${\mathcal {L}}_u$$
denotes the Lie derivative with respect to the velocity field *u*,
$$\Delta $$
is the Hodge Laplacian,
$$\delta $$
is the codifferential (the negative of the divergence operator), and
$$\tilde{\eta }^{-1}$$
is the canonical map from 2-forms to 2-vector fields induced by the Euclidean metric
$$\eta $$
. In this paper we consider a generalisation of these Euler equations in three spatial dimensions, in which the vector potential operator
$$\tilde{\eta }^{-1} \Delta ^{-1}$$
is replaced by a more general operator *A* of order
$$-2$$
; this retains the Lagrangian structure of the Euler equations, as well as most of its conservation laws and local existence theory. Despite this, we give three different constructions of such an operator *A* which admits smooth solutions that blow up in finite time, including an example on
$$\mathbb {R}^3$$
which is self-adjoint and positive definite. This indicates a barrier to establishing global regularity for the three-dimensional Euler equations, in that any method for achieving this must use some property of those equations that is not shared by the generalised Euler equations considered here.

## Landau Damping: Paraproducts and Gevrey Regularity

### Annals of PDE (2016-04-21) 2: 1-71 , April 21, 2016

We give a new, simpler, but also and most importantly more general and robust, proof of nonlinear Landau damping on $${\mathbb {T}}^d$$ in Gevrey $$-\frac{1}{s}$$ regularity ( $$s > 1/3$$ ) which matches the regularity requirement predicted by the formal analysis of Mouhot and Villani [67]. Our proof combines in a novel way ideas from the original proof of Landau damping Mouhot and Villani [67] and the proof of inviscid damping in 2D Euler Bedrossian and Masmoudi [10]. As in Bedrossian and Masmoudi [10], we use paraproduct decompositions and controlled regularity loss along time to replace the Newton iteration scheme of Mouhot and Villani [67]. We perform time-response estimates adapted from Mouhot and Villani [67] to control the plasma echoes and couple them to energy estimates on the distribution function in the style of the work Bedrossian and Masmoudi [10]. We believe the work is an important step forward in developing a systematic theory of phase mixing in infinite dimensional Hamiltonian systems.