arXiv subjects

Irene M. Gamba

Publications and source records attributed to Irene M. Gamba.

At least 19 recordsLinked to original sources

arXiv2026

Analysis of Moment Closures Using $φ$-Divergences for Rarefied Dynamics with Binary Collisions and Their Galerkin Discretizations

This work introduces a robust deterministic framework for approximating solutions of the Boltzmann equation with binary collisions by discretizing their dependence on time, position, and velocity using Galerkin methods. By employing a family of parametric Galerkin closures based on $φ$-divergences in velocity space, we derive rigorous hierarchies of moment equations that govern fluid dynamic variables. Addressing the limitation that these closures alone do not guarantee dissipation of a $φ$-divergence entropy for the true binary collision operator, we restore this property by formulating a compatible approximate collision operator tailored to each closure. This constructed operator intrinsically retains fundamental physical properties essential for high-fidelity flow simulations, including Galilean invariance, exact conservation of mass, momentum, and energy, and strict dissipation of a $φ$-divergence entropy. Furthermore, we show that the resulting closed moment systems are symmetric-dissipative, yielding Cauchy problems that are well-posed locally in time. To translate this mathematical foundation into an efficient computational tool, we discretize the position and time variables with an entropy-stable discontinuous Galerkin (DG) finite element method. The fully implicit, entropy-stable space-time approach enables time steps far beyond typical CFL-limited step sizes and the direct computation of steady states. The robustness and accuracy of the methodology are verified and validated through numerical simulations on the supersonic nozzle flow of argon, mass flow through a channel, and heat transfer between parallel walls, demonstrating agreement with analytical benchmarks, experimental measurements, and stochastic particle simulations.

math.NA↗

arXiv2026

Existence and Smoothing for a Nondivergence-Form Degenerate Diffusion from Plasma-Wave Theory

We prove existence, positive-time smoothing, and physical admissibility of weak solutions to the degenerate parabolic Cauchy-Dirichlet problem $\partial_t u = ρ_λ(x) u \partial_x^2 u + ρ_λ(x) g(x) u$ on the half-line, where $ρ_λ$ vanishes at the boundary and grows at infinity. This scalar problem arises by formally reducing the system of equations given by the quasilinear theory of plasma waves in the one-dimensional case. This theory models a background distribution of electrons $f$ coupled to a spectral energy density $W$ through wave-particle resonance. For the scalar problem we construct weak solutions from weighted $L^p$ initial data and bounded reaction, admitting the unbounded, discontinuous data that the physical model demands, and lying beyond the reach of the continuous-data theories developed for nearby problems. We identify a parabolic smoothing effect for the constructed solution, namely one-sided bounds on $\partial_t u$: from merely integrable data, the solution becomes locally Hölder in space and time and locally Lipschitz in space at positive times. This spatial regularity is shown to be sharp by explicit examples. Finally, we address the quasilinear system itself, whose well-posedness remains open: we prove that the scalar solution induces a particle-wave pair $(f^{\ast}, W^{\ast})$ which is a weak solution of the system. Under nonnegativity and finite-moment hypotheses on the initial data, both components remain nonnegative and the initial mass is conserved. Moreover, the pair inherits positive-time regularity, with $W^{\ast}$ decaying quantitatively at large wavenumber and $f^{\ast}$ smoothing to a locally bounded function even when initially a measure.

math.AP↗

arXiv2026

Stability of the reconstruction of the heat reflection coefficient in the phonon transport equation

The reflection coefficient is an important thermal property of materials, especially at the nanoscale, and determining this property requires solving an inverse problem based on macroscopic temperature measurements. In this manuscript, we investigate the stability of this inverse problem to infer the reflection coefficient in the phonon transport equation. We show that the problem becomes ill-posed as the system transitions from the ballistic to the diffusive regime, characterized by the Knudsen number converging to zero. Such a stability estimate clarifies the discrepancy observed in previous studies on the well-posedness of this inverse problem. Furthermore, we quantify the rate at which the stability deteriorates with respect to the Knudsen number and confirm the theoretical result with numerical evidence.

math.AP↗

arXiv2026

Entropy-stable positivity-preserving DG schemes for Boltzmann-Poisson models of collisional electronic transport along energy bands

This work is related to the development of entropy-stable positivity-preserving DG methods as a computational scheme for Boltzmann-Poisson systems modeling the probability density of collisional electronic transport along energy bands in semiconductors. We pose, in momentum coordinates representing spherical / energy-angular variables, the respective Vlasov-Boltzmann eq. with a linear collision operator and a singular measure, modeling scatterings as functions of the bandstructure appropriately for hot electron nanoscale transport. We show stability results of semidiscrete DG schemes under an entropy norm for 1D, 2D position, using dissipative properties of the collisional operator given its entropy inequality. The latter depends on an exponential of the Hamiltonian rather than the Maxwellian associated with only kinetic energy. For the 1D problem, knowing the analytic solution to the Poisson equation and convergence to a constant current is crucial to obtaining full stability. For the 2D problem, specular reflection boundary conditions and periodicity are considered in estimating stability under an entropy norm. Regarding the positivity-preservation in the DG scheme for the 1D problem, we treat collisions as a source and find convex combinations of the transport and collision terms which guarantee positivity of the cell average of our numerical probability density at the next time. The positivity of the numerical solution to the probability density in the domain is guaranteed by applying well-known limiters that preserve the cell average modifying the slope of the piecewise linear solutions to make the function non-negative. The use of a spherical coordinate system whose radial component is the momentum magnitude is slightly different from choices in previous DG solvers for Boltzmann-Poisson, since the proposed DG formulation gives simpler integrals involving piecewise polynomials.

math.NA↗

arXiv2026

A fast scheme for the homogeneous Boltzmann equation based on lifting and tensor train approximation

We propose a fast deterministic scheme for the space-homogeneous Boltzmann equation that exploits the low-rank structure of the velocity distribution. This paper consists of two independent contributions. The first is a \emph{lifting-projection (LP) scheme}, inspired by the approach in the recent theoretical breakthroughs \cite{guillen2025landau, imbert2026monotonicity, guillen2025landau2} on the well-posedness of the Landau and Boltzmann equations. In particular, the approach lifts the nonlinear 3D Boltzmann equation to the 6D linear Kac master equation, advanced over a single time step, and projected back to its marginal in 3D. The second contribution is a \emph{low-rank tensor method} for evaluating the collision operator, in which the lifted solution is represented in tensor train (TT) format and computed via a TT cross approximation algorithm with interpolation, complemented by a TT-friendly conservation correction that enforces conservation of mass, momentum, and energy. When the solution is low-rank in velocity, the method scales linearly in $n$ when cubic interpolation is used (and quadratic in $n$ when spectral interpolation is used), where $n$ is the number of grid points in each velocity direction. Therefore, our methods offer significant computational savings over existing deterministic solvers in such cases. Numerical experiments on 2D and 3D benchmarks, including the BKW exact solution and anisotropic initial data, confirm the computational scaling, the expected order of accuracy and verify the effectiveness of the conservation correction.

math.NA↗

arXiv2025

A structure-preserving multiscale solver for particle-wave interaction in non-uniform magnetized plasmas

Particle-wave interaction is of fundamental interest in plasma physics, especially in the study of runaway electrons in magnetic confinement fusion. Analogous to the concept of photons and phonons, wave packets in plasma can also be treated as quasi-particles, called plasmons. To model the ``mixture" of electrons and plasmons in plasma, a set of ``collisional" kinetic equations has been derived, based on weak turbulence limit and the Wentzel-Kramers-Brillouin (WKB) approximation. There are two main challenges in solving the electron-plasmon kinetic system numerically. Firstly, non-uniform plasma density and magnetic field results in high dimensionality and the presence of multiple time scales. Secondly, a physically reliable numerical solution requires a structure-preserving scheme that enforces the conservation of mass, momentum, and energy. In this paper, we propose a struture-preserving multiscale solver for particle-wave interaction in non-uniform magnetized plasmas. The solver combines a conservative local discontinuous Galerkin (LDG) scheme for the interaction part with a trajectory averaging method for the plasmon Hamiltonian flow part. Numerical examples for a non-uniform magnetized plasma in an infinitely long symmetric cylinder are presented. It is verified that the LDG scheme rigorously preserves all the conservation laws, and the trajectory averaging method significantly reduces the computational cost.

math.NA↗

arXiv2025

A structure-preserving local discontinuous Galerkin method for the Fokker-Planck-Landau equation

In this work, we introduce a structure-preserving local discontinuous Galerkin (LDG) method \cite{cockburn1998local} for solving the non-local non-linear Fokker-Planck-Landau (FPL) equations. We rephrase the structure-preserving strategy of Shiroto and Sentoku\cite{shiroto2019structure} in the language of numerical analysis, and extend it to the LDG framework. We propose a method that is not only conservative, but also stabilized through upwind flux. The apparent contradiction between conservation laws and numerical stabilization is elegantly resolved by leveraging the properties of the jump terms inherent to the LDG framework. In the numerical experiments, our scheme is tested with benchmark examples.

math.NA↗

arXiv2025

Reconstruction of heat relaxation index in phonon transport equation

For nano-materials, heat conductivity is an ill-defined concept. This classical concept assumes the validity of Fourier's law, which states the heat flux is proportional to temperature gradient, with heat conductivity used to denote this ratio. However, this macroscopic constitutive relation breaks down at nano-scales. Instead, heat is propagated using phonon transport equation, an ab initio model derived from the first principle. In this equation, a material's thermal property is coded in a coefficient termed the relaxation time ($τ$). We study an inverse problem in this paper, by using material's temperature response upon heat injection to infer the relaxation time. This inverse problem is formulated in a PDE-constrained optimization, and numerically solved by Stochastic Gradient Descent (SGD) method and its variants. In the execution of SGD, Fréchet derivative is computed and Lipschitz continuity is proved. This approach, in comparison to the earlier studies, honors the nano-structure of of heat conductivity in a nano-material, and we numerically verify the break down of the Fourier's law.

math.NA↗

arXiv2024

Weak solutions for weak turbulence models in electrostatic plasmas

The weak turbulence model, also known as the quasilinear theory in plasma physics, has been a cornerstone in modeling resonant particle-wave interactions in plasmas. This reduced model stems from the Vlasov-Poisson/Maxwell system under the weak turbulence assumption, incorporating the random phase approximation and ergodicity. The interaction between particles and waves (plasmons) can be treated as a stochastic process, whose transition probability bridges the momentum space and the spectral space. Therefore, the operators on the right hand side resemble collision forms, such as those in Boltzmann and Landau interacting models. For them, there have been results on well-posedness and regularity of solutions. However, as far as we know, there is no such preceding work for the quasilinear theory addressed in this manuscript. In this paper, we establish the existence of global weak solutions for the system modeling electrostatic plasmas in one dimension. Our key contribution consists of associating the original integral-differential system to a degenerate inhomogeneous porous medium equation(PME) with nonlinear source terms, and leveraging advanced techniques from the PME literature. This approach opens a novel pathway for analyzing weak turbulence models in plasma physics. Moreover, our work offers new tools for tackling related problems in the broader context of nonlinear nonlocal PDEs.

math.AP↗

arXiv2024

Exponentially-tailed regularity and time asymptotic for the homogeneous Boltzmann equation

We present in this document the Lebesgue and Sobolev propagation of exponential tails for solutions of the homogeneous Boltzmann equation for hard and Maxwell interactions. In addition, we show the $L^{p}$-integrability creation of such tails in the case of hard interactions. The document also presents a result on exponentially-fast convergence to thermodynamical equilibrium and propagation of singularities and regularization of such solutions. All these results are valid under the mere Grad's cut-off condition for the angular scattering kernel. Highlights of this contribution include: (1) full range of $L^{p}$-norms with $p\in[1,\infty]$, (2) analysis for the critical case of Maxwell interactions, (3) propagation of fractional Sobolev exponential tails using pointwise conmutators, and (4) time asymptotic and propagation of regularity and singularities under general physical data. In many ways, this work is an improvement and an extension of several classical works in the area; we use known techniques and introduce new and flexible ideas that achieve the proofs in an elementary manner.

math-ph↗

arXiv2023

A conservative Galerkin solver for the quasilinear diffusion model in magnetized plasmas

The quasilinear theory describes the resonant interaction between particles and waves with two coupled equations: one for the evolution of the particle probability density function(\textit{pdf}), the other for the wave spectral energy density(\textit{sed}). In this paper, we propose a conservative Galerkin scheme for the quasilinear model in three-dimensional momentum space and three-dimensional spectral space, with cylindrical symmetry. We construct an unconditionally conservative weak form, and propose a discretization that preserves the unconditional conservation property, by "unconditional" we mean that conservation is independent of the singular transition probability. The discrete operators, combined with a consistent quadrature rule, will preserve all the conservation laws rigorously. The technique we propose is quite general: it works for both relativistic and non-relativistic systems, for both magnetized and unmagnetized plasmas, and even for problems with time-dependent dispersion relations. We represent the particle \textit{pdf} by continuous basis functions, and use discontinuous basis functions for the wave \textit{sed}, thus enabling the application of a positivity-preserving technique. The marching simplex algorithm, which was initially designed for computer graphics, is adopted for numerical integration on the resonance manifold. We introduce a semi-implicit time discretization, and discuss the stability condition. In addition, we present numerical examples with a "bump on tail" initial configuration, showing that the particle-wave interaction results in a strong anisotropic diffusion effect on the particle \textit{pdf}.

physics.comp-ph↗

arXiv2023

The Cauchy problem for Boltzmann bi-linear systems: The mixing of monatomic and polyatomic gases

From a unified vision of vector valued solutions in weighted Banach spaces, this manuscript establishes the existence and uniqueness for space homogeneous Boltzmann bi-linear systems with conservative collisional forms arising in complex gas dynamical structures. This broader vision is directly applied to dilute multi-component gas mixtures composed of both monatomic and polyatomic gases. Such models can be viewed as extensions of scalar Boltzmann binary elastic flows, as much as monatomic gas mixtures with disparate masses and single polyatomic gases, providing a unify approach for vector valued solutions in weighted vector Banach spaces. Novel aspects of this work include developing the extension of a general ODE theory in vector valued weighted Banach spaces, precise lower bounds for the collision frequency in terms of the weighted Banach norm, energy identities and the consequently angular or compact manifold averaging lemmas which provide coerciveness resulting into global in time stability, a new combinatorics estimate for $p$-binomial forms producing sharper estimates for the $k$-moments of bi-linear collisional forms. These techniques enable the Cauchy problem improvement that resolves the model with initial data corresponding to strictly positive and bounded initial vector valued mass and total energy, in addition to only $2^+$ moment determined by the hard potential rates discrepancy, a result comparable in generality to the classical Cauchy theory of the scalar homogeneous Boltzmann equation.

math-ph↗

arXiv2023

The Boltzmann equation for hard potentials with integrable angular transition: Coerciveness, exponential tails rates, and Lebesgue integrability

This manuscript focus on an extensive survey with new techniques on the problem of solving the Boltzmann flow by bringing a unified approach to the Cauchy problem to homogeneous kinetic equations with Boltzmann-like collision operators under integrability assumption of the scattering profile in the particle-particle interaction mechanism. The work focuses on the relevant hard potential case where the solution properties are studied with a modern take. While many of the discussed results can be found the literature spread over several papers along the years, we bring a complete program that includes a new approach to the existence and uniqueness theorem securing the well-posedness theory, to moments estimates, and integrability propagation for the homogeneous Boltzmann flow. In particular, a detailed calculation of classical polynomial moments upper bounds as function of the coerciveness is described, which characterized the rate of exponential moments obtained by summability of the polynomial ones. In addition, a proof of uniform propagation of $L^\infty$ regularity under general integrable scattering kernels is performed. Along the way, constants appearing in estimates are carefully calculated, improving most of previous exiting results in the literature. For the non expert reader we also include a general discussion of the basic elements of the Boltzmann model and important key results for the understanding of the mathematical discussion of the equation and include an extensive set of references that enrich and motivate further discussions on Boltzmann flows for broader gas modeling configuration such as gas mixtures systems, polyatomic gases, multilinear collisional forms such as, ternary or quartic, to those derived from symmetry braking quantum mean field theories or weak turbulence models from spectral energy waves in classical fluid.

math-ph↗

arXiv2022

Moment estimates and well-posedness of the binary-ternary Boltzmann equation

In this paper, we show generation and propagation of polynomial and exponential moments, as well as global well-posedness of the homogeneous binary-ternary Boltzmann equation. We also show that the co-existence of binary and ternary collisions yields better generation properties and time decay, than when only binary or ternary collisions are considered. To address these questions, we develop for the first time angular averaging estimates for ternary interactions. This is the first paper which discusses this type of questions for the binary-ternary Boltzmann equation and opens the door for studying moments properties of gases with higher collisional density.

math.AP↗

arXiv2021

Global well-posedness of a binary-ternary Boltzmann equation

In this paper we show global well-posedness near vacuum for the binary-ternary Boltzmann equation. The binary-ternary Boltzmann equation provides a correction term to the classical Boltzmann equation, taking into account both binary and ternary interactions of particles, and may serve as a more accurate description model for denser gases in non-equilibrium. Well-posedness of the classical Boltzmann equation and, independently, the purely ternary Boltzmann equation follow as special cases. To prove global well-posedness, we use a Kaniel-Shinbrot iteration and related work to approximate the solution of the nonlinear equation by monotone sequences of supersolutions and subsolutions. This analysis required establishing new convolution type estimates to control the contribution of the ternary collisional operator to the model. We show that the ternary operator allows consideration of softer potentials than the one binary operator, consequently our solution to the ternary correction of the Boltzmann equation preserves all the properties of the binary interactions solution. These results are novel for collisional operators of monoatomic gases with either hard or soft potentials that model both binary and ternary interactions.

math.AP↗

arXiv2021

Error estimate of a bi-fidelity method for kinetic equations with random parameters and multiple scales

In this paper, we conduct uniform error estimates of the bi-fidelity method for multi-scale kinetic equations. We take the Boltzmann and the linear transport equations as important examples. The main analytic tool is the hypocoercivity analysis for kinetic equations, considering solutions in a perturbative setting close to the global equilibrium. This allows us to obtain the error estimates in both kinetic and hydrodynamic regimes.

math.NA↗

arXiv2020

Entropy Decay Rates for Conservative Spectral Schemes Modeling Fokker-Planck-Landau Type Flows in the Mean Field Limit

The focus of this work is to create benchmark simulations of decay rates to statistical equilibrium in transport plasma models for Coulomb particle interactions given by a coupled Vlasov-Poisson Fokker-Planck-Landau equation, as well as with Maxwell type and hard sphere interactions. The qualitative decay to the equilibrium Maxwell-Boltzmann distribution through relative entropy is studied in detail for all three types of particle interactions by means of a conservative hybrid spectral and discontinuous Galerkin scheme adapted from previous work. More precisely, the Coulomb case shows that there is a degenerate spectrum, with a decay rate close to the law of two thirds predicted by upper estimates in a work of Strain and Guo in 2006, while the Maxwell type and hard sphere examples both exhibit a spectral gap as predicted by Desvillettes and Villani in 2000. Such decay rate behavior indicates that the analytical estimates for the Coulomb case is sharp while, still to this date, there is no analytical proof of sharp degenerate spectral behaviour for the Fokker-Planck-Landau operator. Simulations are presented, both for the space-homogeneous case of just particle potential interactions and the space-inhomogeneous case for the mean field coupling through the Poisson equation for total charges in periodic domains. New explicit derivations of spectral collisional weights are presented in the case of Maxwell type and hard sphere interactions and the stability of all three scenarios, including Coulomb interactions, is investigated.

cond-mat.stat-mech↗

arXiv2020

Convergence and Error Estimates for the Conservative Spectral Method for Fokker-Planck-Landau Equations

Error estimates are rigorously derived for a semi-discrete version of a conservative spectral method for approximating the space-homogeneous Fokker-Planck-Landau (FPL) equation associated to hard potentials. The analysis included shows that the semi-discrete problem has a unique solution with bounded moments. In addition, the derivatives of such a solution up to any order also remain bounded in $L^2$ spaces globally time, under certain conditions. These estimates, combined with control of the spectral projection, are enough to obtain error estimates to the analytical solution and convergence to equilibrium states. It should be noted that this is the first time that an error estimate has been produced for any numerical method which approximates FPL equations associated to any range of potentials.

math.NA↗