33 citations · 200 across the 20 of their papers we have counts for
12 papers · 1 filter
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 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…
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…
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…