Page 498 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
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HER-ORDER EVOLUTION EQUATIONS (MS-43)
Thin-film problems with dynamic contact angle
Dirk Peschka, peschka@wias-berlin.de
Weierstraß-Institut für Angewandte Analysis und Stochastik, Germany
The evolution of a thin viscous fluid over a solid surface is often described using the thin-film
equations
∂th − ∇ · m|h|α∇ δE = 0, E = 1 |∇h|2 + W (x, h) dx,
δh ω2
which are fourth-order degenerate parabolic equations for the height h of the fluid layer with
the support ω(t) = {x : h(t, x) > 0} . The motion of the liquid layer is driven by an energy E,
which in addition to a surface energy contains other sources of internal energy in W . By com-
plementing this PDE with suitable boundary conditions on ∂ω(t), this becomes a free boundary
problem with a moving contact line.
In this talk I will introduce the gradient structure underlying the thin-film problem, detail
the variational structure of the bulk-interface coupling that leads to dynamic contact angles, and
investigate different limiting cases of low and high viscosities, i.e., m → 0 and m → ∞ for
0 < α < 3.
Higher-Order Total Directional Variation
Carola Bibiane Schönlieb, cbs31@cam.ac.uk
University of Cambridge, United Kingdom
Coauthors: Simone Parisotto, Simon Masnou, Jan Lellmann
In this talk we discuss a new higher-order and anisotropic total variation model for image pro-
cessing. This new model combines higher-order total variation regularisation with possibly
inhomogeneous, smooth elliptic anisotropies.
We prove some properties of this total variation model and of the associated spaces of ten-
sors with finite variations. We show the existence of solutions to a related regularity-fidelity op-
timization problem and prove a decomposition formula which we will use to develop a primal-
dual hybrid gradient approach for its numerical approximation.
This choice of total variation regularisation allows to preserve and enhance intrinsic anisotro-
pic features in images. We illustrate this on various examples from different imaging applica-
tions: image denoising, wavelet-based image zooming, and reconstruction of surfaces from
scattered height measurements.
This talk is based on the two papers:
References
[1] Parisotto, Simone; Lellmann, Jan; Masnou, Simon; Schönlieb, Carola-Bibiane: Higher-
order total directional variation: Imaging Applications. In: SIAM Journal on Imaging
Sciences, 13 (4), pp. 2063-2104, 2020.
[2] Parisotto, Simone; Masnou, Simon; Schönlieb, Carola-Bibiane: Higher-order Total Direc-
tional Variation: Analysis. In: SIAM Journal on Imaging Sciences, 13 (1), pp. 474-496,
2020.
496
Thin-film problems with dynamic contact angle
Dirk Peschka, peschka@wias-berlin.de
Weierstraß-Institut für Angewandte Analysis und Stochastik, Germany
The evolution of a thin viscous fluid over a solid surface is often described using the thin-film
equations
∂th − ∇ · m|h|α∇ δE = 0, E = 1 |∇h|2 + W (x, h) dx,
δh ω2
which are fourth-order degenerate parabolic equations for the height h of the fluid layer with
the support ω(t) = {x : h(t, x) > 0} . The motion of the liquid layer is driven by an energy E,
which in addition to a surface energy contains other sources of internal energy in W . By com-
plementing this PDE with suitable boundary conditions on ∂ω(t), this becomes a free boundary
problem with a moving contact line.
In this talk I will introduce the gradient structure underlying the thin-film problem, detail
the variational structure of the bulk-interface coupling that leads to dynamic contact angles, and
investigate different limiting cases of low and high viscosities, i.e., m → 0 and m → ∞ for
0 < α < 3.
Higher-Order Total Directional Variation
Carola Bibiane Schönlieb, cbs31@cam.ac.uk
University of Cambridge, United Kingdom
Coauthors: Simone Parisotto, Simon Masnou, Jan Lellmann
In this talk we discuss a new higher-order and anisotropic total variation model for image pro-
cessing. This new model combines higher-order total variation regularisation with possibly
inhomogeneous, smooth elliptic anisotropies.
We prove some properties of this total variation model and of the associated spaces of ten-
sors with finite variations. We show the existence of solutions to a related regularity-fidelity op-
timization problem and prove a decomposition formula which we will use to develop a primal-
dual hybrid gradient approach for its numerical approximation.
This choice of total variation regularisation allows to preserve and enhance intrinsic anisotro-
pic features in images. We illustrate this on various examples from different imaging applica-
tions: image denoising, wavelet-based image zooming, and reconstruction of surfaces from
scattered height measurements.
This talk is based on the two papers:
References
[1] Parisotto, Simone; Lellmann, Jan; Masnou, Simon; Schönlieb, Carola-Bibiane: Higher-
order total directional variation: Imaging Applications. In: SIAM Journal on Imaging
Sciences, 13 (4), pp. 2063-2104, 2020.
[2] Parisotto, Simone; Masnou, Simon; Schönlieb, Carola-Bibiane: Higher-order Total Direc-
tional Variation: Analysis. In: SIAM Journal on Imaging Sciences, 13 (1), pp. 474-496,
2020.
496
