**Organisers:** Gabriele Brüll, Jonas Jansen (both Lund), Christina Lienstromberg, and Guido Schneider (both Stuttgart)

The seminar takes place roughly biweekly, Wednesdays at 11:00 (sharp) via Zoom. There will be a 45-60 minute talk plus discussion.

To follow the seminar, please contact me via mail.

The seminar will resume in October 2023. Have a great summer!

Season 1: Introduction to Thin-Film Equations | |||
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23/11/2022 | Espen Xylander (Stuttgart) | Modelling of Newtonian thin-film equations with mobility/slip-conditions | |

Abstract: The talk is structured into two parts: First, we introduce the initial topic of our seminar by looking at examples of thin films in nature and by briefly showcasing a couple of equations that govern such situations. Afterwards, in the second part, the goal is to derive a specific instance of these thin-film equations. We will discuss the modelling parameters of the underlying Navier-Stokes system before we simplify the problem by assuming the liquid's average height is proportionally small compared to the horizontal length scale. This treatment is part of the so-called lubrication approximation and allows us to reduce the system to a single evolution equation for the fluid height. You can find an excerpt from Espen's Master's thesis with a detailed overview of the modelling here. | |||

07/12/2022 | Jonas Jansen (Lund) | Short-time existence and standard parabolic theory | |

Abstract: Last time we have derived an equation for the dynamic behaviour of a capillary-driven thin film of the form $$ \partial_t h + \partial_x(h^n \partial_x^3 h) = 0.$$ This equation is quasilinear degenerate-parabolic and of fourth order. In this talk, we will recall semigroup methods / standard parabolic theory and show how these methods can be applied to obtain local-in-time, smooth, positive solutions. | |||

14/12/2022 | Juri Joussen (Stuttgart) | Naive global non-negative weak solutions | |

Abstract: One of the main difficulties in the study of thin-film equations is their degeneracy: As the solution approaches zero, the parabolicity of the problem breaks down. To overcome this issue, we consider the most natural regularisation of the thin-film equation: $$ \partial_t h_\varepsilon+\partial_x\left(\left(\left\vert h_\varepsilon\right\vert^n+\varepsilon\right)\partial_x^3 h_\varepsilon\right)=0. $$ We discuss how global classical solutions to to this problem (with regularised initial data) can be found. As \(\varepsilon\to 0\), we show that there is a limit \(h_\varepsilon\to h\) that satisfies the thin-film equation in a weak sense. | |||

18/01/2023 | Stefano Böhmer (soon Lund) | Entropy methods, non-negativity, and non-naive regularisation | |

Abstract: Last time, we have established global existence of weak solutions to the thin-film equation which turned out to be Hölder-continuous in space and time. We have removed the degeneracy of the thin-film equation by adding a small parameter to the mobility term \(|h|^n\) and we have employed an energy-dissipation inequality yielding uniform bounds. In this talk, we are going to introduce another integral estimate, the so-called entropy-dissipation inequality, allowing us to study non-negativity of weak solutions. Most importantly, weak solutions obtained from this regularisation scheme and emerging from non-negative initial data remain non-negative. If the slippage parameter satisfies \(n>4\), positive initial data yield positive weak solutions. Finally, we are going to introduce a more sophisticated regularisation scheme yielding positive approximate solutions. | |||

25/01/2023 | Gabriele Brüll (Lund) | Travelling-wave solutions for thin-film equations | |

Abstract: A traveling-wave solutions for the thin-film equation with general mobility coefficient \(n\) is given by \(h(t,x)=H(x-ct)\), where \(c\in \mathbb{R}\) is the wave speed and the function \(H\) satisfies the ODE $$cH^\prime = \left(H^nH^{\prime\prime\prime}\right)^\prime $$ with suitable regularity and initial conditions. We will discuss existence and properties of such traveling waves. | |||

08/02/2023 | Christina Lienstromberg (Stuttgart) | Summary of the first season, further properties of solutions, and outlook | |

Abstract: In this talk, we summarise the content of the first season on thin-film equations and provide an overview of possible further research directions. |