scholarly journals Well‐posedness and stability for a mixed order system arising in thin film equations with surfactant

2020 ◽  
Vol 293 (5) ◽  
pp. 879-892
Author(s):  
Gabriele Bruell
Keyword(s):  
Thin Film ◽  
Order System ◽  
Well Posedness ◽  
Thin Film Equations ◽  
Mixed Order

scholarly journals Non-negative Martingale Solutions to the Stochastic Thin-Film Equation with Nonlinear Gradient Noise

2021 ◽  
Author(s):  
Konstantinos Dareiotis ◽  
Benjamin Gess ◽  
Manuel V. Gnann ◽  
Günther Grün
Keyword(s):  
Thin Film ◽  
Energy Estimates ◽  
Approximation Procedure ◽  
Thin Film Flow ◽  
Thin Film Equations ◽  
Scalar Conservation Law ◽  
Thin Film Equation ◽  
Degenerate Parabolic ◽  
Scalar Conservation ◽  
Martingale Solutions

AbstractWe prove the existence of non-negative martingale solutions to a class of stochastic degenerate-parabolic fourth-order PDEs arising in surface-tension driven thin-film flow influenced by thermal noise. The construction applies to a range of mobilites including the cubic one which occurs under the assumption of a no-slip condition at the liquid-solid interface. Since their introduction more than 15 years ago, by Davidovitch, Moro, and Stone and by Grün, Mecke, and Rauscher, the existence of solutions to stochastic thin-film equations for cubic mobilities has been an open problem, even in the case of sufficiently regular noise. Our proof of global-in-time solutions relies on a careful combination of entropy and energy estimates in conjunction with a tailor-made approximation procedure to control the formation of shocks caused by the nonlinear stochastic scalar conservation law structure of the noise.

scholarly journals Further results on a space-time FOSLS formulation of parabolic PDEs

2020 ◽  
Author(s):  
Gregor Gantner ◽  
Rob Stevenson
Keyword(s):  
Finite Element Method ◽  
Least Squares ◽  
Parabolic Equations ◽  
Space Time ◽  
Adaptive Finite Element Method ◽  
Order System ◽  
Well Posedness ◽  
Adaptive Finite Element ◽  
Parabolic Pdes ◽  
First Order

In [2019, Space-time least-squares finite elements for parabolic equations, arXiv:1911.01942] by Führer&Karkulik, well-posedness of a space-time First-Order System Least-Squares formulation of the heat equation was proven.  In the present work, this result is generalized to general second order parabolic PDEs with possibly inhomogenoeus boundary conditions, and plain convergence of a standard adaptive finite element method driven by the least-squares estimator is demonstrated.  The proof of the latter easily extends to a large class of least-squares formulations.

scholarly journals Space-Time Estimates of Mild Solutions of a Class of Higher-Order Semilinear Parabolic Equations in Lp

2014 ◽  
Vol 1 (1) ◽  
Author(s):  
Albert N. Sandjo ◽  
Célestin Wafo Soh
Keyword(s):  
Thin Film ◽  
Epitaxial Growth ◽  
Parabolic Equations ◽  
Mild Solutions ◽  
Film Theory ◽  
Well Posedness ◽  
Semilinear Parabolic Equations ◽  
Unified Framework ◽  
Time Estimates ◽  
Semilinear Parabolic

AbstractWe establish the well-posedness of boundary value problems for a family of nonlinear higherorder parabolic equations which comprises some models of epitaxial growth and thin film theory. In order to achieve this result, we provide a unified framework for constructing local mild solutions in C

scholarly journals Patterns formed in a thin film with spatially homogeneous and non-homogeneous Derjaguin disjoining pressure

2021 ◽  
pp. 1-25
Author(s):  
ABDULWAHED S. ALSHAIKHI ◽  
MICHAEL GRINFELD ◽  
STEPHEN K. WILSON
Keyword(s):  
Thin Film ◽  
Steady State ◽  
Bifurcation Theory ◽  
Disjoining Pressure ◽  
Thin Film Equations ◽  
Planar Substrate ◽  
Spatially Homogeneous ◽  
The One ◽  
Steady State Solutions ◽  
Average Film Thickness

We consider patterns formed in a two-dimensional thin film on a planar substrate with a Derjaguin disjoining pressure and periodic wettability stripes. We rigorously clarify some of the results obtained numerically by Honisch et al. [Langmuir 31: 10618–10631, 2015] and embed them in the general theory of thin-film equations. For the case of constant wettability, we elucidate the change in the global structure of branches of steady-state solutions as the average film thickness and the surface tension are varied. Specifically we find, by using methods of local bifurcation theory and the continuation software package AUTO, both nucleation and metastable regimes. We discuss admissible forms of spatially non-homogeneous disjoining pressure, arguing for a form that differs from the one used by Honisch et al., and study the dependence of the steady-state solutions on the wettability contrast in that case.

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