arXiv subjects

Richard M. Höfer

Publications and source records attributed to Richard M. Höfer.

At least 19 recordsLinked to original sources

arXiv2026

Reactive Flow around Spherically Evolving Particles in Critical and Supercritical Dilute Regimes

We study a coupled Stokes-reaction-diffusion-advection system in a three dimensional domain perforated by a large number of evolving spherical inclusions with radii of order $\varepsilon^α$ for $α\in(1,3]$. Their centers are separated on the order of $\varepsilon$ but are not assumed to form a periodic lattice. The inclusions grow or shrink through an interfacial adsorption-desorption mechanism, so that the evolution of the geometry is coupled to the Stokes-reaction-diffusion-advection system. We first establish well-posedness on arbitrary large time intervals for sufficiently small $\varepsilon$. We then derive quantitative homogenization limits in terms of the limiting empirical measure describing the spatial distribution and size of the inclusions. Its evolution is governed by a continuity equation in the radius variable, and the convergence of the empirical measures is controlled in the $2$-Wasserstein distance. In the critical case, $α=3$, the Stokes system converges to a Brinkman system, while non-vanishing concentration boundary layers modify the microscopic exchange law for the reaction-diffusion-advection equation. For $α\in(1,3)$, the limit is of Darcy type and the original exchange law is retained. The homogenization results are quantitative and provide explicit error estimates.

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arXiv2026

Runaway avalanches in plasmas with external electric fields: spatially inhomogeneous case in a perturbation framework

We consider the Landau-Coulomb equation for a (hydrogen) plasma heated by an external electric field. In this setting, theoretical and experimental results in plasma physics show the emergence of so-called \emph{runaway electrons} which are linearly accelerating but only lead to a minimal increase of the plasma temperature. Runaway electrons are a major obstacle in nuclear fusion since they can overcome the confinement and damage the structure of the reactor. We rigorously prove the well-posedness of the underlying nonlinear \emph{open} Landau-Coulomb system in a perturbative setting and the conjectured growth bounds for the mean velocity and plasma temperature. We show that the mean velocity is linearly increasing in time, and capture the sharp logarithmic growth of the temperature. Furthermore, we prove that the electron distribution can be asymptotically described by a scattering-type Maxwellian. Due to the different nature of the electron-electron and electron-ion interactions, we recast the equation as a novel coupled system that allows us to isolate the dissipation structures of the two operators. For the coupled system, we perform a micro-macro decomposition to show convergence to the scattering-type Maxwellian.

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arXiv2026

Brinkman's term for the steady-state Navier-Stokes equations with prescribed flux rate or pressure drop in perforated pipes

The steady motion of a viscous incompressible fluid in a distorted pipe, containing several small particles of diameter $\eps^3$ and mutual distance $\eps$, is modeled through the Navier-Stokes equations with mixed boundary conditions. Apart from inhomogeneous Dirichlet boundary conditions on the particles, these involve the Bernoulli pressure and the tangential velocity on the inlet and outlet of the tube, while either the transversal flux rate or the pressure drop is prescribed along the pipe. Applying the energy method in homogenization theory, we study the asymptotic behavior of the solutions to these systems as $\eps \to 0$, without any restriction on the magnitude of the data, and show that the effective equations display an additional Brinkman term. An important feature of the present work concerns the required uniform bounds, which are achieved (in the case of the prescribed flux problem) by a contradiction argument based on Bernoulli's law for solutions of the stationary Euler equations.

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arXiv2026

Homogenization of the Navier-Stokes equations in a randomly perforated domain in the inviscid limit

We study the behaviour of the solution $u_\varepsilon$ to the Navier-Stokes equations with vanishing viscosity and a non-slip condition in a randomly perforated domain. We consider the space $\mathbb{R}^3$ where we remove $N$ holes that are i.i.d. distributed. The behaviour depends on the particle size $\varepsilon^α=N^{-α/3}$ and the viscosity $\varepsilon^γ=N^{-γ/3}$ of the fluid. We prove quantitative convergence results to a function $u$, provided that the local Reynolds number is small, in the subcritical ($α+γ>3$) and critical ($α+γ=3$) regime. In the first case, $u$ solves the Euler equations, whereas in the second case $u$ solves the Euler-Brinkman equations. This extends the results of https://doi.org/10.1088/1361-6544/acfe56 from the periodic to the random setting. We only treat the case $α>2$ so that the particles do not overlap with overwhelming probability.

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arXiv2025

Qualitative/quantitative homogenization of some non-Newtonian flows in perforated domains

In this paper, we consider the homogenization of stationary and evolutionary incompressible viscous non-Newtonian flows of Carreau-Yasuda type in domains perforated with a large number of periodically distributed small holes in $\mathbb{R}^{3}$, where the mutual distance between the holes is measured by a small parameter $\varepsilon>0$ and the size of the holes is $\varepsilon^α$ with $α\in (1, 3)$. The Darcy's law is recovered in the limit, thus generalizing the results from https://doi.org/10.1016/0362-546X(94)00285-P and [https://doi.org/10.1016/j.jde.2024.08.021] for $α=1$. Instead of using their restriction operator to derive the estimates of the pressure extension by duality, we use the Bogovski\uı type operator in perforated domains (constructed in [https://doi.org/10.1051/cocv/2016016]) to deduce the uniform estimates of the pressure directly. Moreover, quantitative convergence rates are given.

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arXiv2025

Derivation of the Monokinetic Vlasov-Stokes equations

We consider a microscopic model of spherical particles with inertia in a Stokes flow. As the particle number grows to infinity and their size goes to zero we derive the monokinetic Vlasov-Stokes equations as mean-field limit. We do this under the assumption that the particles have initial velocities given by a Lipschitz velocity profile and prove the mean-field limit for times of the order of the inverse Lipschitz constant. Notably this is not a perturbative result. In particular, we do not require the inertia of the particles to vanish in the limit. Thereby the result improves upon the perturbative derivation in [HS23] in the case of a monokinetic limit density.

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arXiv2025

Propagation of chaos for first-order mean-field systems with non-attractive moderately singular interaction

We consider particle systems that evolve by inertialess binary interaction through general non-attractive kernels of singularity $|x|^{-α}$ with $α<d-1$. We prove a quantitative mean-field limit in terms of Wasserstein distances under certain conditions on the initial configuration while maintaining control of the particle configuration in the form of the minimal distance and certain singular sums of the particle distances. As a corollary, we show propagation of chaos for $α<\frac{d-1}{2}$ for $d\ge 3$ and $α<\frac 13=\frac{2d-3}{3}$ for $d=2$. This extends the results of Hauray (https://doi.org/10.1142/S0218202509003814), which yield propagation of chaos for $α< \frac{d-2}{2}$ without an assumption on the sign of the interaction. The main novel ingredient is that due to the non-attraction property it is enough to control the distance to the next-to-nearest neighbour particle.

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arXiv2025

Correction to Doi type models for suspensions

Starting from microscopic $N$ particle systems, we study the derivation of Doi type models for suspensions of non-spherical particles in Stokes flows. While Doi models accurately describe the effective evolution of the spatial particle density to the first order in the particle volume fraction, this accuracy fails regarding the evolution of the particle orientations. We rigorously attribute this failure to the singular interaction of the particles via a $-3$-homogeneous kernel. In the situation that the particles are initially distributed according to a stationary ergodic point process, we identify the limit of this singular interaction term. It consists of two parts. The first corresponds to a classical term in Doi type models. The second new term depends on the (microscopic) $2$-point correlation of the point process. By including this term, we provide a modification of the Doi model that is accurate to first order in the particle volume fraction.

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arXiv2025

Hydrodynamic limit of multiscale viscoelastic models for rigid particle suspensions

We study the multiscale viscoelastic Doi model for suspensions of Brownian rigid rod-like particles, as well as its generalization by Saintillan and Shelley for self-propelled particles. We consider the regime of a small Weissenberg number, which corresponds to a fast rotational diffusion compared to the fluid velocity gradient, and we analyze the resulting hydrodynamic approximation. More precisely, we show the asymptotic validity of macroscopic nonlinear viscoelastic models, in form of so-called ordered fluid models, as an expansion in the Weissenberg number. The result holds for zero Reynolds number in 3D and for arbitrary Reynolds number in 2D. Along the way, we establish several new well-posedness and regularity results for nonlinear fluid models, which may be of independent interest.

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arXiv2025

Derivation of a Multiscale Ferrofluid Model: Superparamagnetic Behavior due to Fast Spin Flip

We consider a microscopic model of $N$ magnetic nanoparticles in a Stokes flow. We assume that the temperature is above the critical Néel temperature such that the particles' magnetizations undergo random flip with rate $1/\varepsilon$. The microscopic system is the modeled through a piecewise deterministic Markov jump process. We show that for large $N$, small particle volume fraction and small $\varepsilon$, the system can be effectively described by a multiscale model.

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arXiv2025

Derivation of the viscoelastic stress in Stokes flows induced by non-spherical Brownian rigid particles through homogenization

We consider a microscopic model of $n$ identical axis-symmetric rigid Brownian particles suspended in a Stokes flow. We rigorously derive in the homogenization limit of many small particles a classical formula for the viscoelastic stress that appears in so-called Doi models which couple a Fokker-Planck equation to the Stokes equations. We consider both Deborah numbers of order $1$ and very small Deborah numbers. Our microscopic model contains several simplifications, most importantly, we neglect the time evolution of the particle centers as well as hydrodynamic interaction for the evolution of the particle orientations. The microscopic fluid velocity is modeled by the Stokes equations with given torques at the particles in terms of Stratonovitch noise. We give a meaning to this PDE in terms of an infinite dimensional Stratonovitch integral. This requires the analysis of the shape derivatives of the Stokes equations in perforated domains, which we accomplish by the method of reflections.

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arXiv2025

Sedimentation of particles with very small inertia I: Convergence to the transport-Stokes equation

We consider the sedimentation of $N$ spherical particles with identical radii $R$ in a Stokes flow in $\mathbb R^3$. The particles satisfy a no-slip boundary condition and are subject to constant gravity. The dynamics of the particles is modeled by Newton's law but with very small particle inertia as $N$ tends to infinity and $R$ to $0$. In a mean-field scaling, we show that the particle evolution is well approximated by the transport-Stokes system which has been derived previously as the mean-field limit of inertialess particles. In particular this justifies to neglect the particle inertia in the microscopic system, which is a typical modelling assumption in this and related contexts. The proof is based on a relative energy argument that exploits the coercivity of the particle forces with respect to the particle velocities in a Stokes flow. We combine this with an adaptation of Hauray's method for mean-field limits to $2$-Wasserstein distances. Moreover, in order to control the minimal distance between particles, we prove a representation of the particle forces. This representation makes the heuristic \enquote{Stokes law} rigorous that the force on each particle is proportional to the difference of the velocity of the individual particle and the mean-field fluid velocity generated by the other particles.

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arXiv2024

Homogenization of the Navier-Stokes equations in perforated domains in the inviscid limit

We study the solution $u_\varepsilon$ to the Navier-Stokes equations in $\mathbb R^3$ perforated by small particles centered at $(\varepsilon \mathbb Z)^3$ with no-slip boundary conditions at the particles. We study the behavior of $u_\varepsilon$ for small $\varepsilon$, depending on the diameter $\varepsilon^α$, $α> 1$, of the particles and the viscosity $\varepsilon^γ$, $γ> 0$, of the fluid. We prove quantitative convergence results for $u_\varepsilon$ in all regimes when the local Reynolds number at the particles is negligible. Then, the particles approximately exert a linear friction force on the fluid. The obtained effective macroscopic equations depend on the order of magnitude of the collective friction. We obtain a) the Euler-Brinkman equations in the critical regime, b) the Euler equations in the subcritical regime and c) Darcy's law in the supercritical regime.

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arXiv2024

Quantitative homogenization of the compressible Navier-Stokes equations towards Darcy's law

We consider the solutions $ρ_\varepsilon, \mathbf{u}_\varepsilon$ to the compressible Navier-Stokes equations (NSE) in a domain periodically perforated by holes of diameter $\varepsilon>0$. We focus on the case where the diameter of the holes is of the same order as the distance between neighboring holes. This is the same setting investigated in the paper by Masmoudi [\url{http://www.numdam.org/article/COCV_2002__8__885_0.pdf}], where convergence $ρ_\varepsilon, \mathbf{u}_\varepsilon$ of the system to the porous medium equation has been shown. We prove a quantitative version of this convergence result provided that the solution of the limiting system is sufficiently regular. The proof builds on the relative energy inequality satisfied by the compressible NSE.

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arXiv2024

Convergence rates and fluctuations for the Stokes-Brinkman equations as homogenization limit in perforated domains

We study the homogenization of the Dirichlet problem for the Stokes equations in $\mathbb{R}^3$ perforated by $m$ spherical particles. We assume the positions and velocities of the particles to be identically and independently distributed random variables. In the critical regime, when the radii of the particles are of order $m^{-1}$, the homogenization limit $u$ is given as the solution to the Brinkman equations. We provide optimal rates for the convergence $u_m \to u$ in $L^2$, namely $m^{-β}$ for all $β< 1/2$. Moreover, we consider the fluctuations. In the central limit scaling, we show that these converge to a Gaussian field, locally in $L^2(\mathbb{R}^3)$, with an explicit covariance. Our analysis is based on explicit approximations for the solutions $u_m$ in terms of $u$ as well as the particle positions and their velocities. These are shown to be accurate in $\dot H^1(\mathbb{R}^3)$ to order $m^{-β}$ for all $β< 1$. Our results also apply to the analogous problem regarding the homogenization of the Poisson equations.

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arXiv2023

Convergence of the method of reflections for particle suspensions in Stokes flows

We study the convergence of the method of reflections for the Stokes equations in domains perforated by countably many spherical particles with boundary conditions typical for the suspension of rigid particles. We prove that a relaxed version of the method is always convergent in $\dot H^1$ under a mild separation condition on the particles. Moreover, we prove optimal convergence rates of the method in $\dot W^{1,q}$, $1 < q < \infty$ and in $L^\infty$ in terms of the particle volume fraction under a stronger separation condition of the particles.

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arXiv2023

A fast point charge interacting with the screened Vlasov-Poisson system

We consider the long-time behavior of a fast, charged particle interacting with an initially spatially homogeneous background plasma. The background is modeled by the screened Vlasov-Poisson equations, whereas the interaction potential of the point charge is assumed to be smooth. We rigorously prove the validity of the \emph{stopping power theory} in physics, which predicts a decrease of the velocity $V(t)$ of the point charge given by $\dot{V} \sim -|V|^{-3} V$, a formula that goes back to Bohr (1915). Our result holds for all initial velocities larger than a threshold value that is larger than the velocity of all background particles and remains valid until (i) the particle slows down to the threshold velocity, or (ii) the time is exponentially long compared to the velocity of the point charge. The long-time behavior of this coupled system is related to the question of Landau damping which has remained open in this setting so far. Contrary to other results in nonlinear Landau damping, the long-time behavior of the system is driven by the non-trivial electric field of the plasma, and the damping only occurs in regions that the point charge has already passed.

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arXiv2023

Sedimentation of particles with very small inertia II: Derivation, Cauchy problem and hydrodynamic limit of the Vlasov-Stokes equation

We consider the sedimentation of $N$ spherical particles with identical radii $R$ in a Stokes flow in $\mathbb R^3$. The particles satisfy a no-slip boundary condition and are subject to constant gravity. The dynamics of the particles is modeled by Newton's law but with very small particle inertia as $N$ tends to infinity and $R$ to $0$. In a mean-field scaling we show that the evolution of the $N$-particle system is well approximated by the Vlasov-Stokes equation. In contrast to the transport-Stokes equation considered in the first part of this series, \cite{HoferSchubert23}, the Vlasov-Stokes equation takes into account the (small) inertia. Therefore we obtain improved error estimates. We also improve previous results on the Cauchy problem for the Vlasov-Stokes equation and on its convergence to the transport-Stokes equation in the limit of vanishing inertia. The proofs are based on relative energy estimates. In particular, we show new stability estimates for the Vlasov-Stokes equation in the $2$-Wasserstein distance. By combining a Lagrangian approach with a study of the energy dissipation, we obtain uniform stability estimates for arbitrary small particle inertia. We show that a corresponding stability estimate continues to hold for the empirical particle density which formally solves the Vlasov-Stokes equation up to an error. To this end we exploit certain uniform control on the particle configuration thanks to results in the first part \cite{HoferSchubert23}.

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