arXiv subjects

Stefanie Schindler

Publications and source records attributed to Stefanie Schindler.

5 recordsLinked to original sources

arXiv2025

Time-asymptotic self-similarity of the damped compressible Euler equations in parabolic scaling variables

We study the long-time behavior of solutions to the compressible Euler equations with frictional damping in the whole space, where we prescribe direction-dependent values for the density at spatial infinity. To this end, we transform the system into parabolic scaling variables and derive a relative entropy inequality, which allows to conclude the convergence of the density towards a self-similar solution to the porous medium equation while the associated limit momentum is governed by Darcy's law. Moreover, we obtain convergence rates that explicitly depend on the flatness of the limit profile. While we focus on weak solutions in the one-dimensional case, we extend our results to energy-variational solutions in the multi-dimensional setting.

math.AP↗

arXiv2023

Existence of similarity profiles for diffusion equations and systems

We study the existence of similarity profiles for diffusion equations and reaction diffusion systems on the real line, where the different nontrivial limits are imposed for $ x \to -\infty$ and $x \to +\infty$. Theses similarity profiles solve a coupled system of nonlinear ODEs that can be treated by monotone operator theory.

math.AP↗

arXiv2023

Convergence to self-similar profiles in reaction-diffusion systems

We study a reaction-diffusion system on the real line, where the reactions of the species are given by one reversible reaction according to the mass-action law. We describe different positive limits at both sides of infinity and investigate the long-time behavior. Rescaling space and time according to the parabolic scaling, we show that solutions converge exponentially to a constant profile. In the original variables these profiles correspond to asymptotically self-similar behavior describing the diffusive mixing or equilibration of the different states at infinity. Our method provides global exponential convergence for all initial states with finite relative entropy.

math.AP↗

arXiv2023

Self-Similar Pattern in Coupled Parabolic Systems as Non-Equilibrium Steady States

We consider reaction-diffusion systems and other related dissipative systems on unbounded domains which would have a Liapunov function (and gradient structure) when posed on a finite domain. In this situation, the system may reach local equilibrium on a rather fast time scale but the infinite amount of mass or energy leads to persistent mass or energy flow for all times. In suitably rescaled variables the system converges to a steady state that corresponds to asymptotically self-similar behavior in the original system.

math.AP↗

arXiv2021

Gradient flows for bounded linear evolution equations

We study linear evolution equations in separable Hilbert spaces defined by a bounded linear operator. We answer the question which of these equations can be written as a gradient flow, namely those for which the operator is real diagonalisable. The proof is constructive, from which we also derive geodesic lambda-convexity.

math.AP↗