Journal ArticleDOI
Variatonal formulation for the lubrication approximation of the Hele-Shaw flow
Lorenzo Giacomelli,Felix Otto +1 more
Reads0
Chats0
TLDR
In this paper, it has been shown that the time-discretized lubrication approximation of the one-phase Hele-Shaw flow is the same as the constant limit of the continuous limit of timediscrete gradient flows of the corresponding surface energy functionals with respect to the Wasserstein metric.Abstract:
It has been recently discovered that both the surface tension driven one-phase Hele-Shaw flow and its lubrication approximation can be understood as (continuous limits of time-discretized) gradient flows of the corresponding surface energy functionals with respect to the Wasserstein metric. Here we complete the connection between the two problems, proving that the time-discretized lubrication approximation is the \(\Gamma\)-limit of suitably rescaled time-discretized Hele-Shaw flows in half space.read more
Citations
More filters
Book
Gradient Flows: In Metric Spaces and in the Space of Probability Measures
TL;DR: In this article, Gradient flows and curves of Maximal slopes of the Wasserstein distance along geodesics are used to measure the optimal transportation problem in the space of probability measures.
Journal ArticleDOI
A Family of Nonlinear Fourth Order Equations of Gradient Flow Type
TL;DR: In this article, global existence and long-time behavior of solutions to a family of nonlinear fourth order evolution equations on R d are studied, which constitute gradient flows for the perturbed information functionals with respect to the L 2-Wasserstein metric.
Posted Content
A new class of efficient and robust energy stable schemes for gradient flows
Jie Shen,Jie Xu,Jiang Yang +2 more
TL;DR: In this paper, a scalar auxiliary variable (SAV) approach is proposed to deal with nonlinear terms in gradient flows, which is not restricted to specific forms of the nonlinear part of the free energy and only requires to solve linear equations with constant coefficients.
Journal ArticleDOI
Dewetting films: bifurcations and concentrations
TL;DR: In this paper, the authors derived the global structure of the bifurcation diagram for steady-state solutions and studied the behavior of solutions in the limit that short-range repulsive forces are neglected.
Posted Content
A Family of Nonlinear Fourth Order Equations of Gradient Flow Type
TL;DR: In this paper, global existence and long-time behavior of solutions to a family of nonlinear fourth order evolution equations on $R^d$ were studied, which constitute gradient flows for the perturbed information functionals $F[u] = 1/(2α) \int | D (u^α) |^2 dx + \lambda/2 \int|x|^2 u dx.
References
More filters
Book
Measure theory and fine properties of functions
TL;DR: In this article, the authors define and define elementary properties of BV functions, including the following: Sobolev Inequalities Compactness Capacity Quasicontinuity Precise Representations of Soboleve Functions Differentiability on Lines BV Function Differentiability and Structure Theorem Approximation and Compactness Traces Extensions Coarea Formula for BV Functions isoperimetric inequalities The Reduced Boundary The Measure Theoretic Boundary Gauss-Green Theorem Pointwise Properties this article.
Journal ArticleDOI
Long-scale evolution of thin liquid films
TL;DR: In this article, a unified mathematical theory is presented that takes advantage of the disparity of the length scales and is based on the asymptotic procedure of reduction of the full set of governing equations and boundary conditions to a simplified, highly nonlinear, evolution equation or to a set of equations.
Journal ArticleDOI
The geometry of optimal transportation
TL;DR: In this paper, the existence and uniqueness of optimal maps are discussed. But the uniqueness of the optimal map is not discussed. And the role of the map in finding the optimal solution is left open.
Book
Probability Metrics and the Stability of Stochastic Models
TL;DR: General topics in the theory of probability metrics relations between compound, simple and primary distances applications of minimal p.
Journal ArticleDOI
Higher order nonlinear degenerate parabolic equations
Francisco Bernis,Avner Friedman +1 more
TL;DR: In this article, the authors considered nonlinear degenerate parabolic equations of the form ut + (−1)m − 1 D(f(u) D2m + 1u) = 0 with f(u)-n (n ⩾ 1) near u = 0 and D = ∂∂x.