Asymptotic Models in Fluid Dynamics

Joint Research Seminar Lund-Stuttgart

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!

List of Talks
Season 2: Water waves and the KdV equation (tentative schedule)
08/03/2023 Erik Wahlén (Lund) The water-wave problem (and its many shapes)
Abstract: In this talk I will introduce the classical water wave problem, modelled by the incompressible Euler equations with a free boundary (the water surface). The focus will be on the two-dimensional irrotational problem. The problem has a very long history, going back to the 18th century, but I will try to give a summary of the type of questions that have been studied by mathematicians and some important open questions that still remain.
15/03/2023 Bastian Hilder (Lund) Derivation of the KdV equation
Abstract: The Korteweg-de Vries (KdV) equation appears as an asymptotic model to describe the dynamics of small, long-wavelength water waves. In this talk, I will formally derive the KdV equation as an amplitude equation for the water wave problem. The derivation is based on a multiple-scaling ansatz for solutions of the water wave problem in Eulerian coordinates.
22/03/2023 Sarah Hofbauer (Stuttgart) Travelling-wave solutions for the KdV equation
Abstract: Last time we have derived the Korteweg-de Vries (KdV) equation $$\partial_t A = \mu \partial_x^3 A + \nu A \partial_x A.$$ This talk is structured into two parts. First we will derive an explicit formula of a solitary-wave solution for the KdV equation, which is given by \(A(x,t) = v(x-ct)\), where \(c \in \mathbb{R}\) is the wave speed and v satisfies the ODE \(-cv' + v''' - 6vv' = 0\) for \(\mu = -1\) and \(\nu = 6\). In the second part we will discuss shortly the idea to show existence of solitary waves by variational methods.
17/04/2023 Jörg Weber (Lund) Stability of solitary waves for the KdV equation
Abstract: Having constructed solitary waves for KdV in the last session, the natural question to ask is whether or not they are stable. Here, due to translational invariance of the equation, one has to be careful when speaking about “stability” – the correct notion being orbital stability. In this talk, I show how our situation fits into the classical, seminal theory of Grillakis, Shatah and Strauss, and thus answer the question on stability, while keeping an eye on the great, more widely applicable power of their theory.
24/04/2023 Wolf-Patrick Düll (Stuttgart) Validity of the KdV equation
Abstract: In this talk, we explain how to jusitify rigorously the KdV approximation of the water wave problem with and without surface tension by proving error estimates over the physically relevant timespan. The error estimates are derived in the arc length formulation of the water wave problem, which yields bounds that are uniform with respect to the strength of the surface tension, as the height of the wave packet and the surface tension go to zero.
02/05/2023 Arnaud Eychenne (University of Bergen) Existence of N-solitary waves for the fractional Korteweg-de Vries equation
Abstract: The fractional Korteweg-de Vries equation (fKdV) equation is a dispersion generalized of the KdV equation. As the KdV equation, the fKdV equation enjoys the existence of solitary waves and N-solitary waves. Due to the singularity of the non-local dispersion, the technics involved in proving the existence and derived properties of solitary waves and N-solitary waves are different from the one of the KdV equation (local context). In the first part of the talk, we will give the results known of the solitary waves of the fKdV equation and the differences in the techniques between the local and non-local contexts. In the second part, we will present the existence of the N-solitary waves, with a short explanation of the Martel-Merle method to prove the existence of such objects and the main difficulties due to the non-locality of the dispersive operator.
Previous Seasons
Season 1: Introduction to Thin-Film Equations
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.