arXiv subjects

Baptiste Morisse

Publications and source records attributed to Baptiste Morisse.

7 recordsLinked to original sources

arXiv2023

A uniform ergodic theorem for degenerate flows on the annulus

Motivated by the well-known phase-space portrait of the nonlinear pendulum, the purpose of this paper is to obtain convergence rates in the ergodic theorem for flows in the plane that have arbitrarily slow trajectories. Considering bounded periodic trajectories near the heteroclinic orbits, it is shown that despite lacking a spectral gap, there exists a functional space (which is a strict subset of $L^2$) on which time averages converge uniformly to spatial averages (with an explicit rate). The main ingredient of the proof is an estimate of the density of the spectrum of the generator of the flow near zero.

math.DS↗

arXiv2022

Asymptotic Growth and Decay of Two-Dimensional Symmetric Plasmas

We study the large time behavior of classical solutions to the two-dimensional Vlasov-Poisson (VP) and relativistic Vlasov-Poisson (RVP) systems launched by radially-symmetric initial data with compact support. In particular, we prove that particle positions and momenta grow unbounded as $t \to \infty$ and obtain sharp rates on the maximal values of these quantities on the support of the distribution function for each system. Furthermore, we establish nearly sharp rates of decay for the associated electric field, as well as upper and lower bounds on the decay rate of the charge density in the large time limit. We prove that, unlike (VP) in higher dimensions, smooth solutions do not scatter to their free-streaming profiles as $t \to \infty$ because nonlinear, long-range field interactions dominate the behavior of characteristics due to the exchange of energy from the potential to the kinetic term. In this way, the system may "forget" any previous configuration of particles.

math.AP↗

arXiv2020

Uniform convergence in von Neumann's ergodic theorem in the absence of a spectral gap

Von Neumann's original proof of the ergodic theorem is revisited. A uniform convergence rate is established under the assumption that one can control the density of the spectrum of the underlying self-adjoint operator when restricted to suitable subspaces. Explicit rates are obtained when the bound is polynomial, with applications to the linear Schrödinger and wave equations. In particular, decay estimates for time-averages of solutions are shown.

math.DS↗

arXiv2019

On hyperbolicity and Gevrey well-posedness. Part three: a class of weakly hyperbolic systems

We consider a class of weakly hyperbolic systems of first-order, nonlinear PDEs. Weak hyperbolicity means here that the principal symbol of the system has a crossing of eigenvalues, and is not uniformly diagonalizable. We prove the well-posedness of the Cauchy problem in the Gevrey regularity for all Gevrey indices greater than $1/2$. The proof is based on the construction of a suitable approximate symmetrizer of the principal symbol and an energy estimate in Gevrey spaces. We discuss both the generality of the assumption on the structure of the principal symbol and the sharpness of the lower bound of the Gevrey index.

math.AP↗

arXiv2018

On hyperbolicity and Gevrey well-posedness. Part two: Scalar or degenerate transitions

For first-order quasi-linear systems of partial differential equations, we formulate an assumption of a transition from initial hyperbolicity to ellipticity. This assumption bears on the principal symbol of the first-order operator. Under such an assumption, we prove a strong Hadamard instability for the associated Cauchy problem, namely an instantaneous defect of Hölder continuity of the flow from $G^σ$ to $L^2$, with $0 < σ< σ_0$, the limiting Gevrey index $σ_0$ depending on the nature of the transition. We restrict here to scalar transitions, and non-scalar transitions in which the boundary of the hyperbolic zone satisfies a flatness condition. As in our previous work for initially elliptic Cauchy problems [B. Morisse, \textit{On hyperbolicity and Gevrey well-posedness. Part one: the elliptic case}, arXiv:1611.07225], the instability follows from a long-time Cauchy-Kovalevskaya construction for highly oscillating solutions. This extends recent work of N. Lerner, T. Nguyen, and B. Texier [\textit{The onset of instability in first-order systems}, to appear in J. Eur. Math. Soc.].

math.AP↗

arXiv2017

On the action of pseudo-differential operators on Gevrey spaces

In this paper we study the action of pseudo-differential operators acting on Gevrey spaces. We introduce classes of classical symbols with spatial Gevrey regularity. As the spatial Gevrey regularity of a symbol $p(\cdot,ξ)$ may depend on the frequency $ξ$, the action of the associated pseudo-differential operator $op(p)$ may induce a loss of regularity. The proof is based on a para-product decomposition.

math.AP↗

arXiv2017

On hyperbolicity and Gevrey well-posedness. Part one: the elliptic case

In this paper we prove that the Cauchy problem for first-order quasi-linear systems of partial differential equations is ill-posed in Gevrey spaces, under the assumption of an initial ellipticity. The assumption bears on the principal symbol of the first-order operator. Ill-posedness means instability in the sense of Hadamard, specifically an instantaneous defect of Hölder continuity of the flow from $G^σ$ to $L^2$, where $σ\in(0,1)$ depends on the initial spectrum. Building on the analysis carried out by G. Métivier [\textit{Remarks on the well-posedness of the nonlinear Cauchy problem}, Contemp. Math. 2005], we show that ill-posedness follows from a long-time Cauchy-Kovalevskaya construction of a family of exact, highly oscillating, analytical solutions which are initially close to the null solution, and which grow exponentially fast in time. A specific difficulty resides in the observation time of instability. While in Sobolev spaces, this time is logarithmic in the frequency, in Gevrey spaces it is a power of the frequency. In particular, in Gevrey spaces the instability is recorded much later than in Sobolev spaces.

math.AP↗