ALEX 2018 Workshop: Abstracts

Global existence and stability for dissipative processes coupled across volume and surface

Karoline Disser

TU Darmstadt, Fachbereich Mathematik (Germany) and

Weierstraß-Institut Berlin (Germany)

Alexander Mielke and Annegret Glitzky have systematically modelled the dynamics of processes coupled across volume and surface domains using gradient structures [1]. The aim of this talk is to analyze a typical dissipative class of these models and show global well-posedness [2]. The dynamics can be highly nonlinear but the structure of the coupling preserves $L^{\infty}$ -bounds. To show not only existence but also stability based on $L^{\infty}$ -bounds, we use a functional analytic framework based on the isomorphism property of second-order divergence-form operators in $W^{-1,q}$ for $q>d$ larger than the spatial dimension $d$ of the volume domain [3].

References

1 A. Glitzky and A. Mielke. A gradient structure for systems coupling reaction-diffusion effects in bulk and interfaces. Z. angew. Math. Physik (ZAMP), 64(1): 29-52, 2013.
2 K. Disser. Global existence, uniqueness and stability for nonlinear dissipative systems of bulk-interface interaction, 2016.
3 K. Disser, H.-C. Kaiser and J. Rehberg. ptimal Sobolev regularity for linear second-order divergence elliptic operators occurring in real-world problems. SIAM J. Math. Anal. 47, no. 3, 1719-1746, 2015.