arXiv subjects

Rossana Marra

Publications and source records attributed to Rossana Marra.

16 recordsLinked to original sources

arXiv2023

Ghost Effect from Boltzmann Theory

Taking place naturally in a gas subject to a given wall temperature distribution [Maxwell1879], the ``ghost effect'' exhibits a rare kinetic effect beyond the prediction of classical fluid theory and Fourier law in such a classical problem in physics. As the Knudsen number $\varepsilon$ goes to zero, the finite variation of temperature in the bulk is determined by an $\varepsilon$ infinitesimal, ghost-like velocity field, created by a given finite variation of the tangential wall temperature as predicted by Maxwell's slip boundary condition. Mathematically, such a finite variation leads to the presence of a severe $\varepsilon^{-1}$ singularity and a Knudsen layer approximation in the fundamental energy estimate. Neither difficulty is within the reach of any existing PDE theory on the steady Boltzmann equation in a general 3D bounded domain. Consequently, in spite of the discovery of such a ghost effect from temperature variation in as early as 1960's, its mathematical validity has been a challenging and intriguing open question, causing confusion and suspicion. We settle this open question in affirmative if the temperature variation is small but finite, by developing a new $L^2-L^6-L^{\infty}$ framework with four major innovations: 1) a key $\mathscr{A}$-Hodge decomposition and its corresponding local $\mathscr{A}$-conservation law eliminate the severe $\varepsilon^{-1}$ bulk singularity, leading to a reduced energy estimate; 2) A surprising $\varepsilon^{\frac{1}{2}}$ gain in $L^2$ via momentum conservation and a dual Stokes solution; 3) the $\mathscr{A}$-conservation, energy conservation and a coupled dual Stokes-Poisson solution reduces to an $\varepsilon^{-\frac{1}{2}}$ boundary singularity; 4) a crucial construction of $\varepsilon$-cutoff boundary layer eliminates such boundary singularity via new Hardy and BV estimates.

math.AP↗

arXiv2023

Ghost Effect from Boltzmann Theory: Expansion with Remainder

Consider the limit $\varepsilon\rightarrow0$ of the steady Boltzmann problem \begin{align} v\cdot\nabla_x\mathfrak{F}=\varepsilon^{-1}Q[\mathfrak{F},\mathfrak{F}],\quad \mathfrak{F}\big|_{v\cdot n<0}=M_w\displaystyle\int_{v'\cdot n>0} \mathfrak{F}(v')|v'\cdot n|\mathrm{d}{v'}, \end{align} where $\displaystyle M_w(x_0,v):=\frac{1}{2π\big(T_w(x_0)\big)^2} \exp\bigg(-\frac{|v|^2}{2T_w(x_0)}\bigg)$ for $x_0\in\partialΩ$ is the wall Maxwellian in the diffuse-reflection boundary condition. In the natural case of $|\nabla T_w|=O(1)$, for any constant $P>0$, the Hilbert expansion leads to \begin{align}\label{expansion} \mathfrak{F}\approx μ+\varepsilon\bigg\{μ\bigg(ρ_1+u_1\cdot v+T_1\frac{|v|^2-3T}{2}\bigg)-μ^{\frac{1}{2}}\left(\mathscr{A}\cdot\frac{\nabla_xT}{2T^2}\right)\bigg\} \end{align} where $\displaystyleμ(x,v):=\frac{ρ(x)}{\big(2πT(x)\big)^{\frac{3}{2}}} \exp\bigg(-\frac{|v|^2}{2T(x)}\bigg)$, and $(ρ,u_1,T)$ is determined by a Navier-Stokes-Fourier system with "ghost" effect. The goal of this paper is to construct $\mathfrak{F}$ in the form of \begin{align}\label{aa 08} \mathfrak{F}(x,v)=&μ+μ^{\frac{1}{2}}\Big(\varepsilon f_1+\varepsilon^2f_2\Big)+μ_w^{\frac{1}{2}}\Big(\varepsilon f^B_1\Big)+\varepsilon^αμ^{\frac{1}{2}}R, \end{align} for interior solutions $f_1$, $f_2$ and boundary layer $f^B_1$, where $μ_w$ is $μ$ computed for $T=T_w$, and derive equation for the remainder $R$ with some constant $α\geq1$. To prove the validity of the expansion suitable bounds on $R$ are needed, which are provided in the companion paper [Esposito-Guo-Rossana-Wu2023].

math.AP↗

arXiv2023

On the derivation of new non-classical hydrodynamic equations for Hamiltonian particle systems

We consider a Hamiltonian system of particles, interacting through of a smooth pair potential. We look at the system on a space scale of order ε^1, times of order ε^2, and mean velocities of order ε, with ε a scale parameter, under initial conditions where the system is in a local Gibbs state with parameters corresponding to density and temperature with gradients of order 1. Assuming that the phase space density of the particles is given by a suitable series in ε the behavior of the system under this rescaling is described, to the lowest order in ε, by new non-classical hydrodynamic equations that cannot be derived from the compressible Navier-Stokes equations in the small Mac number limit. The analogous equations in kinetic theory are called ghost effect equations.

math-ph↗

arXiv2022

Hydrodynamic Limit of a Kinetic Gas Flow Past an Obstacle

Given an obstacle in $\mathbb{R}^3$ and a non-zero velocity with small amplitude at the infinity, we construct the unique steady Boltzmann solution flowing around such an obstacle with the prescribed velocity as $|x|\to \infty$, which approaches the corresponding Navier-Stokes steady flow, as the mean-free path goes to zero. Furthermore, we establish the error estimate between the Boltzmann solution and its Navier-Stokes approximation. Our method consists of new $L^6$ and $L^3$ estimates in the unbounded exterior domain, as well as an iterative scheme preserving the positivity of the distribution function.

math-ph↗

arXiv2019

Diffusive limit for a Boltzmann-like equation with non-conserved momentum

We consider a kinetic model whose evolution is described by a Boltzmann-like equation for the one-particle phase space distribution $f(x,v,t)$. There are hard-sphere collisions between the particles as well as collisions with randomly fixed scatterers. As a result, this evolution does not conserve momentum but only mass and energy. We prove that the diffusively rescaled $f^\varepsilon(x,v,t)=f(\varepsilon^{-1}x,v,\varepsilon^{-2}t)$, as $\varepsilon\to 0$ tends to a Maxwellian $M_{ρ, 0, T}=\fracρ{(2πT)^{3/2}}\exp[{-\frac{|v|^2}{2T}}]$, where $ρ$ and $T$ are solutions of coupled diffusion equations and estimate the error in $L^2_{x,v}$.

math-ph↗

arXiv2019

Uniqueness of the Non-Equilibrium Steady State for a $1$d BGK model in kinetic theory

We continue our investigation of kinetic models of a one-dimensional gas in contact with homogeneous thermal reservoirs at different temperatures. Nonlinear collisional interactions between particles are modeled by a so-called BGK dynamics which conserves local energy and particle density. Weighting the nonlinear BGK term with a parameter $α\in [0,1]$, and the linearinteraction with the reservoirs by $(1-α)$, we prove that for all $α$ close enough to zero, the explicit spatially uniform non-equilibrium stable state (NESS) is \emph{unique}, and there are no spatially non-uniform NESS with a spatial density $ρ$ belonging to $L^p$ for any $p>1$. We also show that for all $α\in [0,1]$, the spatially uniform NESS is dynamically stable, with small perturbation converging to zero exponentially fast.

math-ph↗

arXiv2017

Approach to the steady state in kinetic models with thermal reservoirs at different temperatures

We continue the investigation of kinetic models of a system in contact via stochastic interactions with several spatially homogeneous thermal reservoirs at different temperatures. Considering models different from those investigated in earlier work, we explicitly compute the unique spatially uniform non-equilibrium steady state (NESS) and prove that it is approached exponentially fast from any uniform initial state. This leaves open the question of whether there exist NESS that are not spatially uniform. Making a further simplification of our models, we then prove non-existence of such NESS and exponential approach to the unique spatially uniform NESS (with a computably boundable rate). The method of proof relies on refined Doeblin estimates and other probabilisitic techniques, and is quite different form the analysis in earlier work that was based on contraction mapping methods.

math-ph↗

arXiv2016

Stationary solutions to the Boltzmann equation in the Hydrodynamic limit

Despite its conceptual and practical importance, the rigorous derivation of the steady incompressible Navier-Stokes-Fourier system from the Boltzmann theory has been {an} outstanding {open problem} for general domains in 3D. We settle this open question in {the} affirmative, in the presence of a small external field and a small boundary temperature variation for the diffuse boundary condition. We employ a recent quantitative $L^{2}-L^{\infty }$ approach with new $L^{6}$ estimates for the hydrodynamic part $\mathbf{P}f$ of the distribution function. Our results also imply the validity of Fourier law in the hydrodynamical limit, and our method {leads to {asymptotical} stability of steady Boltzmann solutions as well as the derivation of the {unsteady} Navier-Stokes Fourier system}.

math.AP↗

arXiv2012

Froth-like minimizers of a non local free energy functional with competing interactions

We investigate the ground and low energy states of a one dimensional non local free energy functional describing at a mean field level a spin system with both ferromagnetic and antiferromagnetic interactions. In particular, the antiferromagnetic interaction is assumed to have a range much larger than the ferromagnetic one. The competition between these two effects is expected to lead to the spontaneous emergence of a regular alternation of long intervals on which the spin profile is magnetized either up or down, with an oscillation scale intermediate between the range of the ferromagnetic and that of the antiferromagnetic interaction. In this sense, the optimal or quasi-optimal profiles are "froth-like": if seen on the scale of the antiferromagnetic potential they look neutral, but if seen at the microscope they actually consist of big bubbles of two different phases alternating among each other. In this paper we prove the validity of this picture, we compute the oscillation scale of the quasi-optimal profiles and we quantify their distance in norm from a reference periodic profile. The proof consists of two main steps: we first coarse grain the system on a scale intermediate between the range of the ferromagnetic potential and the expected optimal oscillation scale; in this way we reduce the original functional to an effective "sharp interface" one. Next, we study the latter by reflection positivity methods, which require as a key ingredient the exact locality of the short range term. Our proof has the conceptual interest of combining coarse graining with reflection positivity methods, an idea that is presumably useful in much more general contexts than the one studied here.

math-ph↗

arXiv2012

Ghost effect by curvature in planar Couette flow

We study a rarefied gas, described by the Boltzmann equation, between two coaxial rotating cylinders in the small Knudsen number regime. When the radius of the inner cylinder is suitably sent to infinity, the limiting evolution is expected to converge to a modified Couette flow which keeps memory of the vanishing curvature of the cylinders (ghost effect). In the 1-d stationary case we prove the existence of a positive isolated L_2-solution to the Boltzmann equation and its convergence. This is obtained by means of a truncated bulk-boundary layer expansion which requires the study of a new Milne problem, and an estimate of the remainder based on a generalized spectral inequality.

math-ph↗

arXiv2012

Fourier Law and Non-Isothermal Boundary in the Boltzmann Theory

In the study of the heat transfer in the Boltzmann theory, the basic problem is to construct solutions to the steady problem for the Boltzmann equation in a general bounded domain with diffuse reflection boundary conditions corresponding to a non isothermal temperature of the wall. Denoted by δthe size of the temperature oscillations on the boundary, we develop a theory to characterize such a solution mathematically. We construct a unique solution F_s to the Boltzmann equation, which is dynamically asymptotically stable with exponential decay rate. Moreover, if the domain is convex and the temperature of the wall is continuous we show that F_s is continuous away from the grazing set. If the domain is non-convex, discontinuities can form and then propagate along the forward characteristics. We show that they actually form for a suitable smooth temperature profile. We remark that this solution differs from a local equilibrium Maxwellian, hence it is a genuine non equilibrium stationary solution. Our analysis is based on recent studies of the boundary value problems for the Boltzmann equation but with new constructive coercivity estimates for both steady and dynamic cases. A natural question in this setup is to determine if the general Fourier law, stating that the heat conduction vector q is proportional to the temperature gradient, is valid. As an application of our result we establish an expansion in δfor F_s whose first order term F_1 satisfies a linear, parameter free equation. Consequently, we discover that if the Fourier law were valid for F_s, then the temperature of F_1 must be linear in a slab. Such a necessary condition contradicts available numerical simulations, leading to the prediction of break-down of the Fourier law in the kinetic regime.

math-ph↗

arXiv2008

Displacement convexity and minimal fronts at phase boundaries

We show that certain free energy functionals that are not convex with respect to the usual convex structure on their domain of definition, are strictly convex in the sense of displacement convexity under a natural change of variables. We use this to show that in certain cases, the only critical points of these functionals are minimizers. This approach based on displacement convexity permits us to treat multicomponent systems as well as single component systems. The developments produce new examples of displacement convex functionals, and, in the multi-component setting, jointly displacement convex functionals.

math.FA↗

arXiv2007

Stability of the Front under a Vlasov-Fokker-Planck Dynamics

We consider a kinetic model for a system of two species of particles interacting through a longrange repulsive potential and a reservoir at given temperature. The model is described by a set of two coupled Vlasov-Fokker-Plank equations. The important front solution, which represents the phase boundary, is a one-dimensional stationary solution on the real line with given asymptotic values at infinity. We prove the asymptotic stability of the front for small symmetric perturbations.

math-ph↗

arXiv2005

Phase segregation and interface dynamics in kinetic systems

We consider a kinetic model of two species of particles interacting with a reservoir at fixed temperature, described by two coupled Vlasov-Fokker-Plank equations. We prove that in the diffusive limit the evolution is described by a macroscopic equation in the form of the gradient flux of the macroscopic free energy functional. Moreover, we study the sharp interface limit and find by formal Hilbert expansions that the interface motion is given in terms of a quasi stationary problem for the chemical potentials. The velocity of the interface is the sum of two contributions: the velocity of the Mullins-Sekerka motion for the difference of the chemical potentials and the velocity of a Hele-Shaw motion for a linear combination of the two potentials. These equations are identical to the ones in Otto-E modelling the motion of a sharp interface for a polymer blend.

cond-mat.stat-mech↗

arXiv2002

Hydrodynamics of binary fluid phase segregation

Starting with the Vlasov-Boltzmann equation for a binary fluid mixture, we derive an equation for the velocity field $\bm{u}$ when the system is segregated into two phases (at low temperatures) with a sharp interface between them. $\bm{u}$ satisfies the incompressible Navier-Stokes equations together with a jump boundary condition for the pressure across the interface which, in turn, moves with a velocity given by the normal component of $\bm{u} $. Numerical simulations of the Vlasov-Boltzmann equations for shear flows parallel and perpendicular to the interface in a phase segregated mixture support this analysis. We expect similar behavior in real fluid mixtures.

cond-mat.stat-mech↗

arXiv1997

Solutions to the Boltzmann equation in the Boussinesq regime

We consider a gas in a horizontal slab, in which the top and bottom walls are kept at different temperatures. The system is described by the Boltzmann equation (BE) with Maxwellian boundary conditions specifying the wall temperatures. We study the behavior of the system when the Knudsen number $ε$ is small and the temperature difference between the walls as well as the velocity field is of order $ε$, while the gravitational force is of order $ε^2$. We prove that there exists a solution to the BE for which is near a global Maxwellian, and whose moments are close, up to order $ε^2$ to the density, velocity and temperature obtained from the smooth solution of the Oberbeck-Boussinesq equations, up to the time this one stays regular.

cond-mat.stat-mech↗