arXiv subjects

Stephen Pankavich

Publications and source records attributed to Stephen Pankavich.

At least 19 recordsLinked to original sources

arXiv2026

Modified scattering for the Vlasov-Riesz system with long-range interactions

We study the long-time asymptotic behavior of small-data solutions to the three-dimensional Vlasov--Riesz system with the inverse power-law potential $\lambda |x|^{-\alpha}$ in the strictly long-range regime ($0 < \alpha < 1$). By introducing finite- and infinite-time modified wave operators for the characteristic flows, we describe the asymptotic dynamics via convergence to an effective profile along a suitably modified reference flow, and establish modified scattering of solutions. Our proof relies mainly on ODE techniques for the characteristic flows, while also using PDE methods for weighted $W^{1,\infty}$-bounds. Compared with the earlier result (of Huang and Kwon), our Lagrangian approach extends modified scattering to the broader regime $\frac{1}{2}<\alpha<1$ and provides a distinct and more robust argument.

math.AP↗

arXiv2025

The non-relativistic limit of scattering states for the Vlasov equation with short-range interaction potentials

We study the relativistic and non-relativistic Vlasov equation driven by short-range interaction potentials and identify the large time dynamics of solutions. In particular, we construct global-in-time solutions launched from small initial data and prove that they scatter along the forward free flow to well-behaved limits as $t \to \infty$. Moreover, we prove the existence of wave operators for such a regime and, upon constructing the aforementioned time asymptotic limits, use the wave operator formulation to prove for the first time that the relativistic scattering states converge to their non-relativistic counterparts as $c \to \infty$.

math.AP↗

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↗

arXiv2023

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↗

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↗

arXiv2022

Global Sensitivity Analysis of Plasma Instabilities via Active Subspaces

Active subspace analysis is a useful computational tool to identify and exploit the most important linear combinations in the space of a model's input parameters. These directions depend inherently on a quantity of interest, which can be represented as a function from input parameters to model outputs. As the dynamics of many plasma models are driven by potentially uncertain parameter values, the utilization of active subspaces to perform global sensitivity analysis represents an important tool to understand how certain physical phenomena depend upon fluctuations in the values of these parameters. In the current paper, we construct and implement new computational methods to quantify the induced uncertainty within the growth rate generated by perturbations in a collisionless plasma modeled by the one-dimensional Vlasov-Poisson system near an unstable, spatially-homogeneous equilibrium in the linear regime.

physics.plasm-ph↗

arXiv2022

Scattering and Asymptotic Behavior of Solutions to the Vlasov-Poisson System in High Dimension

We consider the repulsive Vlasov-Poisson system in dimension $d \geq 4$. A sufficient condition on the decay rate of the associated electric field is presented that guarantees the scattering and determination of the complete asymptotic behavior of large data solutions as $t \to \infty$. More specifically, we show that under this condition the spatial average of the particle distribution function converges, and we establish the precise asymptotic profiles of the electric field and macroscopic densities. An $L^\infty$ scattering result for the particle distribution function along the associated trajectories of free transport is also proved. Finally, we construct small data solutions that display this asymptotic behavior. These solutions do not require smallness of $\|f_0\|_\infty$ or derivatives, as only a condition on integrated moments of the distribution function is imposed.

math.AP↗

arXiv2021

Asymptotic Dynamics of Dispersive, Collisionless Plasmas

A multispecies, collisionless plasma is modeled by the Vlasov-Poisson system. Assuming that the electric field decays with sufficient rapidity as $t \to\infty$, we show that the velocity characteristics and spatial averages of the particle distributions converge as time grows large. Using these limits we establish the precise asymptotic profile of the electric field and its derivatives, as well as, the charge and current densities. Modified spatial characteristics are then shown to converge using the limiting electric field. Finally, we establish a modified $L^\infty$ scattering result for each particle distribution function, namely we show that they converge as $t \to \infty$ along the modified spatial characteristics. When the plasma is non-neutral, the estimates of these quantities are sharp, while in the neutral case they may imply faster rates of decay.

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

Exact Large Time Behavior of Spherically-Symmetric Plasmas

We consider the classical and relativistic Vlasov-Poisson systems with spherically-symmetric initial data and prove the optimal decay rates for all suitable $L^p$ norms of the charge density and electric field, as well as, the optimal growth rates for the largest particle position and momentum on the support of the distribution function. Though a previous work \cite{Horst} established upper bounds on the decay of the supremum of the charge density and electric field, we provide a slightly different proof, attain optimal rates, and extend this result to include all other norms. Additionally, we prove sharp lower bounds on each of the aforementioned quantities and establish the time-asymptotic behavior of all spatial and momentum characteristics. Finally, we investigate the limiting behavior of the spatial average of the particle distribution as $t \to \infty$. In particular, we show that it converges uniformly to a smooth, compactly-supported function that preserves the mass, angular momentum, and energy of the system and depends only upon limiting particle momenta.

math.AP↗

arXiv2020

Nonparametric, data-based kernel interpolation for particle-tracking simulations and kernel density estimation

Traditional interpolation techniques for particle tracking include binning and convolutional formulas that use pre-determined (i.e., closed-form, parameteric) kernels. In many instances, the particles are introduced as point sources in time and space, so the cloud of particles (either in space or time) is a discrete representation of the Green's function of an underlying PDE. As such, each particle is a sample from the Green's function; therefore, each particle should be distributed according to the Green's function. In short, the kernel of a convolutional interpolation of the particle sample "cloud" should be a replica of the cloud itself. This idea gives rise to an iterative method by which the form of the kernel may be discerned in the process of interpolating the Green's function. When the Green's function is a density, this method is broadly applicable to interpolating a kernel density estimate based on random data drawn from a single distribution. We formulate and construct the algorithm and demonstrate its ability to perform kernel density estimation of skewed and/or heavy-tailed data including breakthrough curves.

physics.data-an↗

arXiv2019

Bistable Dynamics and Hopf Bifurcation in a Refined Model of Early Stage HIV Infection

Recent clinical studies have shown that HIV disease pathogenesis can depend strongly on many factors at the time of transmission, including the strength of the initial viral load and the local availability of CD4+ T-cells. In this article, a new within-host model of HIV infection that incorporates the homeostatic proliferation of T-cells is formulated and analyzed. Due to the effects of this biological process, the influence of initial conditions on the proliferation of HIV infection is further elucidated. The identifiability of parameters within the model is investigated and a local stability analysis, which displays additional complexity in comparison to previous models, is conducted. The current study extends previous theoretical and computational work on the early stages of the disease and leads to interesting nonlinear dynamics, including a parameter region featuring bistability of infectious and viral clearance equilibria and the appearance of a Hopf bifurcation within biologically relevant parameter regimes.

q-bio.PE↗

arXiv2019

Entropy: The former trouble with particles (including a new numerical model computational penalty for the Akaike information criterion)

Traditional random-walk particle-tracking (PT) models of advection and dispersion do not track entropy, because particle masses remain constant. Newer mass-transfer particle tracking (MTPT) models have the ability to do so because masses of all compounds may change along trajectories. Additionally, the probability mass functions (PMF) of these MTPT models may be compared to continuous solutions with probability density functions, when a consistent definition of entropy (or similarly, the dilution index) is constructed. This definition reveals that every numerical model incurs a computational entropy. Similar to Akaike's entropic penalty for larger numbers of adjustable parameters, the computational complexity of a model (e.g., number of nodes) adds to the entropy and, as such, must be penalized. The MTPT method can use a particle-collision based kernel or an SPH-derived adaptive kernel. The latter is more representative of a locally well-mixed system (i.e., one in which the dispersion tensor equally represents mixing and solute spreading), while the former better represents the separate processes of mixing versus spreading. We use computational means to demonstrate the viability of each of these methods.

physics.comp-ph↗

arXiv2018

On the separate treatment of mixing and spreading by the reactive-particle-tracking algorithm: An example of accurate upscaling of reactive Poiseuille flow

The Eulerian advection-dispersion-reaction equation (ADRE) suffers the well-known scale-effect of reduced apparent reaction rates between chemically dissimilar fluids at larger scales (or dimensional averaging). The dispersion tensor in the ADRE must equally and simultaneously account for both solute mixing and spreading. Recent reactive-particle-tracking (RPT) algorithms can, by separate mechanisms, simulate 1) smaller-scale mixing by inter-particle mass transfer, and 2) mass spreading by traditional random walks. To test the supposition that the RPT can accurately track these separate mechanisms, we upscale reactive transport in Hagen-Poiseuille flow between two plates. The simple upscaled 1-D RPT model with one velocity value, an upscaled Taylor macro-dispersivity, and the local molecular diffusion coefficient matches the results obtained from a detailed 2-D model with fully described velocity and diffusion. Both models use the same thermodynamic reaction rate, because the rate is not forced to absorb the loss of information upon upscaling. Analytic and semi-analytic upscaling is also performed using volume averaging and ensemble streamtube techniques. Volume averaging does not perform as well as the RPT, while ensemble streamtubes (using an effective dispersion coefficient along with macro-dispersion) perform almost exactly the same as RPT.

physics.flu-dyn↗

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

On the spatially homogeneous and isotropic Einstein-Vlasov-Fokker-Planck system with cosmological scalar field

The Einstein-Vlasov-Fokker-Planck system describes the kinetic diffusion dynamics of self-gravitating particles within the Einstein theory of general relativity. We study the Cauchy problem for spatially homogeneous and isotropic solutions and prove the existence of both global solutions and solutions that blow-up in finite time depending on the size of certain functions of the initial data. We also derive information on the large-time behavior of global solutions and toward the singularity for solutions which blow-up in fine time. Our results entail the existence of a phase of decelerated expansion followed by a phase of accelerated expansion, in accordance with the physical expectations in cosmology.

math.AP↗

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↗

arXiv2017

Estimating the reproductive number, total outbreak size, and reporting rates for Zika epidemics in South and Central America

As South and Central American countries prepare for increased birth defects from Zika virus outbreaks and plan for mitigation strategies to minimize ongoing and future outbreaks, understanding important characteristics of Zika outbreaks and how they vary across regions is a challenging and important problem. We developed a mathematical model for the 2015 Zika virus outbreak dynamics in Colombia, El Salvador, and Suriname. We fit the model to publicly available data provided by the Pan American Health Organization, using Approximate Bayesian Computation to estimate parameter distributions and provide uncertainty quantification. An important model input is the at-risk susceptible population, which can vary with a number of factors including climate, elevation, population density, and socio-economic status. We informed this initial condition using the highest historically reported dengue incidence modified by the probable dengue reporting rates in the chosen countries. The model indicated that a country-level analysis was not appropriate for Colombia. We then estimated the basic reproduction number, or the expected number of new human infections arising from a single infected human, to range between 4 and 6 for El Salvador and Suriname with a median of 4.3 and 5.3, respectively. We estimated the reporting rate to be around 16% in El Salvador and 18% in Suriname with estimated total outbreak sizes of 73,395 and 21,647 people, respectively. The uncertainty in parameter estimates highlights a need for research and data collection that will better constrain parameter ranges.

q-bio.PE↗