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Abstract: TBA

Abstract: TBA

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. 

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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. 

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Abstract: Last time we have derived the Korteweg-de Vries (KdV) equation. This talk is structured into two parts. First we will derive an explicit formula of a solitary-wave solution for the KdV equation. In the second part we will discuss shortly the idea to show existence of solitary waves by variational methods. 

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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. 

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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. 

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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. 

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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. 

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Abstract: Last time we have derived an equation for the dynamic behaviour of a capillary-driven thin film. 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. 

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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. We discuss how global classical solutions to this problem (with regularised initial data) can be found. 

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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 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 is greater than 4 , positive initial data yield positive weak solutions. Finally, we are going to introduce a more sophisticated regularisation scheme yielding positive approximate solutions. 

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Abstract: We discuss traveling-wave solutions for a thin-film equation with general mobility coefficient, where is the wave speed and the function satisfies the ODE with suitable regularity and initial conditions. We will discuss existence and properties of such traveling waves. 

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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. 

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