thesis
Curvature driven interface evolution: Uniqueness properties of weak solution concepts
ISTA Thesis
published
Sebastian
Hensel
author 4D23B7DA-F248-11E8-B48F-1D18A9856A870000-0001-7252-8072
Julian L
Fischer
supervisor
GradSch
department
JuFi
department
International IST Doctoral Program
project
Bridging Scales in Random Materials
project
The present thesis is concerned with the derivation of weak-strong uniqueness principles for curvature driven interface evolution problems not satisfying a comparison principle. The specific examples being treated are two-phase Navier-Stokes flow with surface tension, modeling the evolution of two incompressible, viscous and immiscible fluids separated by a sharp interface, and multiphase mean curvature flow, which serves as an idealized model for the motion of grain boundaries in an annealing polycrystalline material. Our main results - obtained in joint works with Julian Fischer, Tim Laux and Theresa M. Simon - state that prior to the formation of geometric singularities due to topology changes, the weak solution concept of Abels (Interfaces Free Bound. 9, 2007) to two-phase Navier-Stokes flow with surface tension and the weak solution concept of Laux and Otto (Calc. Var. Partial Differential Equations 55, 2016) to multiphase mean curvature flow (for networks in R^2 or double bubbles in R^3) represents the unique solution to these interface evolution problems within the class of classical solutions, respectively. To the best of the author's knowledge, for interface evolution problems not admitting a geometric comparison principle the derivation of a weak-strong uniqueness principle represented an open problem, so that the works contained in the present thesis constitute the first positive results in this direction. The key ingredient of our approach consists of the introduction of a novel concept of relative entropies for a class of curvature driven interface evolution problems, for which the associated energy contains an interfacial contribution being proportional to the surface area of the evolving (network of) interface(s). The interfacial part of the relative entropy gives sufficient control on the interface error between a weak and a classical solution, and its time evolution can be computed, at least in principle, for any energy dissipating weak solution concept. A resulting stability estimate for the relative entropy essentially entails the above mentioned weak-strong uniqueness principles. The present thesis contains a detailed introduction to our relative entropy approach, which in particular highlights potential applications to other problems in curvature driven interface evolution not treated in this thesis.
https://research-explorer.ista.ac.at/download/10007/10008/thesis_final_Hensel.zip
application/x-zip-compressed
https://research-explorer.ista.ac.at/download/10007/10014/thesis_final_Hensel.pdf
application/pdfno
Institute of Science and Technology Austria2021
eng
2663-337X10.15479/at:ista:10007
300
https://research-explorer.ista.ac.at/record/10012 https://research-explorer.ista.ac.at/record/10013 https://research-explorer.ista.ac.at/record/7489
Hensel, Sebastian. <i>Curvature Driven Interface Evolution: Uniqueness Properties of Weak Solution Concepts</i>. Institute of Science and Technology Austria, 2021, doi:<a href="https://doi.org/10.15479/at:ista:10007">10.15479/at:ista:10007</a>.
S. Hensel, “Curvature driven interface evolution: Uniqueness properties of weak solution concepts,” Institute of Science and Technology Austria, 2021.
Hensel S. Curvature driven interface evolution: Uniqueness properties of weak solution concepts. 2021. doi:<a href="https://doi.org/10.15479/at:ista:10007">10.15479/at:ista:10007</a>
Hensel S. 2021. Curvature driven interface evolution: Uniqueness properties of weak solution concepts. Institute of Science and Technology Austria.
Hensel, Sebastian. “Curvature Driven Interface Evolution: Uniqueness Properties of Weak Solution Concepts.” Institute of Science and Technology Austria, 2021. <a href="https://doi.org/10.15479/at:ista:10007">https://doi.org/10.15479/at:ista:10007</a>.
Hensel, S. (2021). <i>Curvature driven interface evolution: Uniqueness properties of weak solution concepts</i>. Institute of Science and Technology Austria. <a href="https://doi.org/10.15479/at:ista:10007">https://doi.org/10.15479/at:ista:10007</a>
S. Hensel, Curvature Driven Interface Evolution: Uniqueness Properties of Weak Solution Concepts, Institute of Science and Technology Austria, 2021.
100072021-09-13T11:12:34Z2023-09-07T13:30:45Z