arXiv subjects

Jonathan Ben-Artzi

Publications and source records attributed to Jonathan Ben-Artzi.

At least 19 recordsLinked to original sources

arXiv2025

Arbitrary Polynomial Decay Rates of Neutral, Collisionless Plasmas

A multispecies, collisionless plasma is modeled by the Vlasov-Poisson system. Assuming the plasma is neutral and the electric field decays with sufficient rapidity as $t \to\infty$, we show that solutions can be constructed with arbitrarily fast, polynomial rates of decay, depending upon the properties of the limiting spatial average and its derivatives. In doing so, we establish, for the first time, a countably infinite number of asymptotic profiles for the charge density, electric field, and their derivatives, each of which is necessarily realized by a sufficiently smooth solution and exceeds the established dispersive decay rates. Finally, in each case we establish a linear $L^\infty$ scattering result for every particle distribution function, namely we show that they converge as $t \to \infty$ along the transported spatial characteristics at increasingly faster rates.

math.AP↗

arXiv2024

Modified Scattering of Solutions to the Relativistic Vlasov-Maxwell System Inside the Light Cone

We consider the relativistic Vlasov-Maxwell system in three dimensions and study the limiting asymptotic behavior as $t \to \infty$ of solutions launched by small, compactly supported initial data. In particular, we prove that such solutions scatter to a modification of the free-streaming asymptotic profile. More specifically, we show that the spatial average of the particle distribution function converges to a smooth, compactly-supported limit and establish the precise, self-similar asymptotic behavior of the electric and magnetic fields, as well as, the macroscopic densities and their derivatives in terms of this limiting function. Upon constructing the limiting fields, a modified $L^\infty$ scattering result for the particle distribution function along the associated trajectories of free transport corrected by the limiting Lorentz force is then obtained. When the limiting charge density does not vanish, our estimates are sharp up to a logarithmic correction. However, when this quantity is identically zero in the limit, the limiting current density and fields may also vanish, which gives rise to decay rates that are faster than those attributed to the dispersive mechanisms in the system.

math.AP↗

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↗

arXiv2023

On the complexity of the inverse Sturm-Liouville problem

This paper explores the complexity associated with solving the inverse Sturm-Liouville problem with Robin boundary conditions: given a sequence of eigenvalues and a sequence of norming constants, how many limits does a universal algorithm require to return the potential and boundary conditions? It is shown that if all but finitely many of the eigenvalues and norming constants coincide with those for the zero potential then the number of limits is zero, i.e. it is possible to retrieve the potential and boundary conditions precisely in finitely many steps. Otherwise, it is shown that this problem requires a single limit; moreover, if one has a priori control over how much the eigenvalues and norming constants differ from those of the zero-potential problem, and one knows that the average of the potential is zero, then the computation can be performed with complete error control. This is done in the spirit of the Solvability Complexity Index. All algorithms are provided explicitly along with numerical examples.

math.NA↗

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↗

arXiv2021

A toy model for the relativistic Vlasov-Maxwell system

The global-in-time existence of classical solutions to the relativistic Vlasov-Maxwell (RVM) system in three space dimensions remains elusive after nearly four decades of mathematical research. In this note, a simplified ``toy model'' is presented and studied. This toy model retains one crucial aspect of the RVM system: the phase-space evolution of the distribution function is governed by a transport equation whose forcing term satisfies a wave equation with finite speed of propagation.

math.AP↗

arXiv2021

Universal algorithms for computing spectra of periodic operators

Schrödinger operators with periodic (possibly complex-valued) potentials and discrete periodic operators (possibly with complex-valued entries) are considered, and in both cases the computational spectral problem is investigated: namely, under what conditions can a "one-size-fits-all" algorithm for computing their spectra be devised? It is shown that for periodic banded matrices this can be done, as well as for Schrödinger operators with periodic potentials that are sufficiently smooth. In both cases implementable algorithms are provided, along with examples. For certain Schrödinger operators whose potentials may diverge at a single point (but are otherwise well-behaved) it is shown that there does not exist such an algorithm, though it is shown that the computation is possible if one allows for two successive limits.

math.SP↗

arXiv2021

Computing the Sound of the Sea in a Seashell

The question of whether there exists an approximation procedure to compute the resonances of any Helmholtz resonator, regardless of its particular shape, is addressed. A positive answer is given, and it is shown that all that one has to assume is that the resonator chamber is bounded and that its boundary is $\mathcal C^2$. The proof is constructive, providing a universal algorithm which only needs to access the values of the characteristic function of the chamber at any requested point.

math.SP↗

arXiv2021

Global Strichartz estimates for the Dirac equation on symmetric spaces

In this paper we study global-in-time, weighted Strichartz estimates for the Dirac equation on warped product spaces in dimension $n\geq3$. In particular, we prove estimates for the dynamics restricted to eigenspaces of the Dirac operator on the compact spin manifolds defining the ambient manifold under some explicit sufficient condition on the metric, and estimates with loss of angular derivatives for general initial data in the setting of spherically symmetric and asymptotically flat manifolds.

math.AP↗

arXiv2020

Strichartz estimates for the Klein-Gordon equation in a conical singular space

Consider a conical singular space $X=C(Y)=(0,\infty)_r\times Y$ with the metric $g=\mathrm{d}r^2+r^2h$, where the cross section $Y$ is a compact $(n-1)$-dimensional closed Riemannian manifold $(Y,h)$. We study the Klein-Gordon equations with inverse-square potentials in the space $X$, proving in particular global-in-time Strichartz estimates in this setting.

math.AP↗

arXiv2020

Computing Spectra -- On the Solvability Complexity Index Hierarchy and Towers of Algorithms

This paper establishes some of the fundamental barriers in the theory of computations and finally settles the long-standing computational spectral problem. That is to determine the existence of algorithms that can compute spectra $\mathrm{sp}(A)$ of classes of bounded operators $A = \{a_{ij}\}_{i,j \in \mathbb{N}} \in \mathcal{B}(l^2(\mathbb{N}))$, given the matrix elements $\{a_{ij}\}_{i,j \in \mathbb{N}}$, that are sharp in the sense that they achieve the boundary of what a digital computer can achieve. Similarly, for a Schrödinger operator $H = -Δ+V$, determine the existence of algorithms that can compute the spectrum $\mathrm{sp}(H)$ given point samples of the potential function $V$. In order to solve these problems, we establish the Solvability Complexity Index (SCI) hierarchy and provide a collection of new algorithms that allow for problems that were previously out of reach. The SCI is the smallest number of limits needed in the computation, yielding a classification hierarchy for all types of problems in computational mathematics that determines the boundaries of what computers can achieve in scientific computing. In addition, the SCI hierarchy provides classifications of computational problems that can be used in computer-assisted proofs. The SCI hierarchy captures many key computational issues in the history of mathematics including the insolvability of the quintic, Smale's problem on the existence of iterative generally convergent algorithm for polynomial root finding, the computational spectral problem, inverse problems, optimisation etc.

cs.CC↗

arXiv2020

Computing Scattering Resonances

The question of whether it is possible to compute scattering resonances of Schrödinger operators - independently of the particular potential - is addressed. A positive answer is given, and it is shown that the only information required to be known a priori is the size of the support of the potential. The potential itself is merely required to be $\mathcal{C}^1$. The proof is constructive, providing a universal algorithm which only needs to access the values of the potential at any requested point.

math.SP↗

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

Weak Poincaré inequalities in the absence of spectral gaps

For generators of Markov semigroups which lack a spectral gap, it is shown how bounds on the density of states near zero lead to a so-called "weak Poincaré inequality" (WPI), originally introduced by Liggett [Ann. Probab., 1991]. Applications to general classes of constant coefficient pseudodifferential operators are studied. Particular examples are the heat semigroup and the semigroup generated by the fractional Laplacian in the whole space, where the optimal decay rates are recovered. Moreover, the classical Nash inequality appears as a special case of the WPI for the heat semigroup.

math.FA↗

arXiv2018

Concentrating solutions of the relativistic Vlasov-Maxwell system

We study smooth, global-in-time solutions of the relativistic Vlasov-Maxwell system that possess arbitrarily large charge densities and electric fields. In particular, we construct spherically symmetric solutions that describe a thin shell of equally charged particles concentrating arbitrarily close to the origin and which give rise to charge densities and electric fields as large as one desires at some finite time. We show that these solutions exist even for arbitrarily small initial data or any desired mass. In the latter case, the time at which solutions concentrate can also be made arbitrarily large.

math-ph↗

arXiv2017

Arbitrarily large solutions of the Vlasov-Poisson system

We study smooth, global-in-time solutions of the Vlasov-Poisson system in the plasma physical case that possess arbitrarily large charge densities and electric fields. In particular, we construct two classes of solutions with this property. The first class are spherically-symmetric solutions that initially possess arbitrarily small density and field values, but attain arbitrarily large values of these quantities at some later time. Additionally, we construct a second class of spherically-symmetric solutions that possess any desired mass and attain arbitrarily large density and field values at any later prescribed time.

math.AP↗

arXiv2016

Moment bounds on the corrector of stochastic homogenization of non-symmetric elliptic finite difference equations

We consider the corrector equation from the stochastic homogenization of uniformly elliptic finite-difference equations with random, possibly non-symmetric coefficients. Under the assumption that the coefficients are stationary and ergodic in the quantitative form of a Logarithmic Sobolev inequality (LSI), we obtain optimal bounds on the corrector and its gradient in dimensions $d \geq 2$. Similar estimates have recently been obtained in the special case of diagonal coefficients making extensive use of the maximum principle and scalar techniques. Our new method only invokes arguments that are also available for elliptic systems and does not use the maximum principle. In particular, our proof relies on the LSI to quantify ergodicity and on regularity estimates on the derivative of the discrete Green's function in weighted spaces.

math.AP↗

arXiv2016

Approximations of strongly continuous families of unbounded self-adjoint operators

The problem of approximating the discrete spectra of families of self-adjoint operators that are merely strongly continuous is addressed. It is well-known that the spectrum need not vary continuously (as a set) under strong perturbations. However, it is shown that under an additional compactness assumption the spectrum does vary continuously, and a family of symmetric finite-dimensional approximations is constructed. An important feature of these approximations is that they are valid for the entire family uniformly. An application of this result to the study of plasma instabilities is illustrated.

math.SP↗