33 citations · 200 across the 20 of their papers we have counts for
20 papers
Long- and short-time behavior of hypocoercive evolution equations via modal decompositions
F. Achleitner, A. Arnold, V. Mehrmann +1
The long- and short-time behavior of solutions to dissipative evolution equations is studied by applying the concept of hypocoercivity. Aiming at partial differential equations tha…
Fractional Dissipative PDEs
Franz Achleitner, Goro Akagi, Christian Kuehn +4
In this chapter we provide an introduction to fractional dissipative partial differential equations (PDEs) with a focus on trying to understand their dynamics. The class of PDEs we…
Hypocoercivity for Linear ODEs and Strong Stability for Runge--Kutta Methods
Franz Achleitner, Anton Arnold, Ansgar Jüngel
In this note, we connect two different topics from linear algebra and numerical analysis: hypocoercivity of semi-dissipative matrices and strong stability for explicit Runge--Kutta…
Hypocoercivity in Hilbert spaces
Franz Achleitner, Anton Arnold, Volker Mehrmann +1
The concept of hypocoercivity for linear evolution equations with dissipation is discussed and equivalent characterizations that were developed for the finite-dimensional case are…
Hypocoercivity in algebraically constrained partial differential equations with application to Oseen equations
Franz Achleitner, Anton Arnold, Volker Mehrmann
The long-time behavior of solutions to different versions of Oseen equations of fluid flow on the 2D torus is analyzed using the concept of hypocoercivity. The considered models ar…
Hypocoercivity and hypocontractivity concepts for linear dynamical systems
Franz Achleitner, Anton Arnold, Volker Mehrmann
For linear dynamical systems (in continuous-time and discrete-time) we revisit and extend the concepts of hypocoercivity and hypocontractivity and give a detailed analysis of the r…
The hypocoercivity index for the short time behavior of linear time-invariant ODE systems
Franz Achleitner, Anton Arnold, Eric A. Carlen
We consider the class of conservative-dissipative ODE systems, which is a subclass of Lyapunov stable, linear time-invariant ODE systems. We characterize asymptotically stable, con…
Hypocoercivity and controllability in linear semi-dissipative Hamiltonian ODEs and DAEs
Franz Achleitner, Anton Arnold, Volker Mehrmann
For the classes of finite dimensional linear time-invariant semi-dissipative Hamiltonian ordinary differential equations and differential-algebraic equations, stability and hypocoe…
Metastable Speeds in the Fractional Allen-Cahn Equation
Franz Achleitner, Christian Kuehn, Jens Markus Melenk +1
We study numerically the one-dimensional Allen-Cahn equation with the spectral fractional Laplacian on intervals with homogeneous Neumann boundary conditions. In parti…
Large-time asymptotics of a fractional drift-diffusion-Poisson system via the entropy method
Franz Achleitner, Ansgar Jüngel, Masakazu Yamamoto
The self-similar asymptotics for solutions to the drift-diffusion equation with fractional dissipation, coupled to the Poisson equation, is analyzed in the whole space. It is shown…
On optimal decay estimates for ODEs and PDEs with modal decomposition
Franz Achleitner, Anton Arnold, Beatrice Signorello
We consider the Goldstein-Taylor model, which is a 2-velocity BGK model, and construct the "optimal" Lyapunov functional to quantify the convergence to the unique normalized steady…
Asymptotic stability of traveling wave solutions for nonlocal viscous conservation laws with explicit decay rates
Franz Achleitner, Yoshihiro Ueda
We consider scalar conservation laws with nonlocal diffusion of Riesz-Feller type such as the fractal Burgers equation. The existence of traveling wave solutions with monotone decr…
On multi-dimensional hypocoercive BGK models
Franz Achleitner, Anton Arnold, Eric A. Carlen
We study hypocoercivity for a class of linearized BGK models for continuous phase spaces. We develop methods for constructing entropy functionals that enable us to prove exponentia…
Two classes of nonlocal Evolution Equations related by a shared Traveling Wave Problem
Franz Achleitner
We consider reaction-diffusion equations and Korteweg-de Vries-Burgers (KdVB) equations, i.e. scalar conservation laws with diffusive-dispersive regularization. We review the exist…
Addendum to "Travelling waves for a non-local Korteweg-de Vries-Burgers equation" [J. Differential Equations 257 (2014), no. 3, 720--758]
Carlota M. Cuesta, Franz Achleitner
We add a theorem to [J. Differential Equations 257 (2014), no. 3, 720--758] by F. Achleitner, C.M. Cuesta and S. Hittmeir. In that paper we studied travelling wave solutions of a K…
On linear hypocoercive BGK models
Franz Achleitner, Anton Arnold, Eric A. Carlen
We study hypocoercivity for a class of linear and linearized BGK models for discrete and continuous phase spaces. We develop methods for constructing entropy functionals that prove…
Large-time behavior in non-symmetric Fokker-Planck equations
Franz Achleitner, Anton Arnold, Dominik Stürzer
We consider three classes of linear non-symmetric Fokker-Planck equations having a unique steady state and establish exponential convergence of solutions towards the steady state w…
Analysis and numerics of traveling waves for asymmetric fractional reaction-diffusion equations
Franz Achleitner, Christian Kuehn
We consider a scalar reaction-diffusion equation in one spatial dimension with bistable nonlinearity and a nonlocal space-fractional diffusion operator of Riesz-Feller type. We pre…
Traveling waves for a bistable equation with nonlocal-diffusion
Franz Achleitner, Christian Kuehn
We consider a single component reaction-diffusion equation in one dimension with bistable nonlinearity and a nonlocal space-fractional diffusion operator of Riesz-Feller type. Our…
On Bounded Positive Stationary Solutions for a Nonlocal Fisher-KPP Equation
Franz Achleitner, Christian Kuehn
We study the existence of stationary solutions for a nonlocal version of the Fisher-Kolmogorov-Petrovskii-Piscounov (FKPP) equation. The main motivation is a recent study by Berest…