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A multiphase hele-shaw flow with solidification
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
(English)Manuscript (preprint) (Other academic)
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-223549OAI: oai:DiVA.org:kth-223549DiVA, id: diva2:1184770
Note

QC 20180223

Available from: 2018-02-22 Created: 2018-02-22 Last updated: 2018-02-23Bibliographically approved
In thesis
1. Two Problems in non-linear PDE’s with Phase Transitions
Open this publication in new window or tab >>Two Problems in non-linear PDE’s with Phase Transitions
2018 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis is in the field of non-linear partial differential equations (PDE), focusing on problems which show some type of phase-transition. A single phase Hele-Shaw flow models a Newtoninan fluid which is being injected in the space between two narrowly separated parallel planes. The time evolution of the space that the fluid occupies can be modelled by a semi-linear PDE. This is a problem within the field of free boundary problems. In the multi-phase problem we consider the time-evolution of a system of phases which interact according to the principle that the joint boundary which emerges when two phases meet is fixed for all future times. The problem is handled by introducing a parameterized equation which is regularized and penalized. The penalization is non-local in time and tracks the history of the system, penalizing the joint support of two different phases in space-time. The main result in the first paper is the existence theory of a weak solution to the parameterized equations in a Bochner space using the implicit function theorem. The family of solutions to the parameterized problem is uniformly bounded allowing us to extract a weakly convergent subsequence for the case when the penalization tends to infinity.

The second problem deals with a parameterized highly oscillatory quasi-linear elliptic equation in divergence form. As the regularization parameter tends to zero the equation gets a jump in the conductivity which occur at the level set of a locally periodic function, the obstacle. As the oscillations in the problem data increases the solution to the equation experiences high frequency jumps in the conductivity, resulting in the corresponding solutions showing an effective global behaviour. The global behavior is related to the so called homogenized solution. We show that the parameterized equation has a weak solution in a Sobolev space and derive bounds on the solutions used in the analysis for the case when the regularization is lost. Surprisingly, the limiting problem in this case includes an extra term describing the interaction between the solution and the obstacle, not appearing in the case when obstacle is the zero level-set. The oscillatory nature of the problem makes standard numerical algorithms computationally expensive, since the global domain needs to be resolved on the micro scale. We develop a multi scale method for this problem based on the heterogeneous multiscale method (HMM) framework and using a finite element (FE) approach to capture the macroscopic variations of the solutions at a significantly lower cost. We numerically investigate the effect of the obstacle on the homogenized solution, finding empirical proof that certain choices of obstacles make the limiting problem have a form structurally different from that of the parameterized problem.

Place, publisher, year, edition, pages
KTH Royal Institute of Technology, 2018. p. 35
Series
TRITA-SCI-FOU ; 2018:04
Keywords
Conductivity jump, Free boundary problem, Quasilinear elliptic equation, Mathematical techniques, Nonlinear analysis, Free boundary, Free-boundary problems, Quasi-linear elliptic, Quasilinear elliptic equations, Hele-Shaw flow, non-local, implicit function theorem, Heterogeneous Multi Scale, HMM, Finite Element, FEM, FE
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-223562 (URN)978-91-7729-682-9 (ISBN)
Presentation
2018-02-23, F11, Lindstedtsvägen 22, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

QC 20180222

Available from: 2018-02-22 Created: 2018-02-22 Last updated: 2018-02-22Bibliographically approved

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