Seminars

Title : On the derivation and justification of a Whitham-type model with bathymetry

Abstract:

The water wave equations are deduced from the Euler equations under the assumption of irrotational flow and are widely used to model the propagation of waves in a fluid. However, there are serval difficulties related to the complexity of these equations, both from an analytical perspective and from the perspective of applications. To overcome some of these issues, one typically considers simplified models characterized by dimensionless parameters that describe the main mechanisms involved. 

In this talk, I will explain the steps involved in deriving Whitham-type models with bathymetry from the water waves equations. The goal is to understand the difficulties related to having large amplitude variations of the bottom, where we derive asymptotic models that capture both the 'weakly nonlinear regime' and the 'shallow water regime'. 

This is based on joint work with Louis Emerald.

Chair : Kristoffer Varholm

Internal doubly periodic gravity-capillary waves with vorticity

Chair: Dag Nilsson

Title : One sided Hölder regularity of global weak solutions of negative order dispersive equations

Abstract:

We prove global existence, uniqueness and stability of entropy solutions with L2∩L∞ initial data for a general family of negative order dispersive equations. It is further demonstrated that this solution concept extends in a unique continuous manner to all L2 initial data. These weak solutions are found to satisfy one sided Hölder conditions whose coefficients decay in time. The latter result controls the height of solutions and further provides a way to bound the maximal lifespan of classical solutions from their initial data. 

The talk is based on a joint work with June Xue (NTNU). 

Chair : Didier Pilod

Title:  Guaranteed lift-off in non-Newtonian thin-film equations

Abstract:

The dynamics of (very) thin films of non-Newtonian fluids on a solid bottom constricted by lateral boundaries is described by a fourth-order doubly degenerate-parabolic PDE of the form

Here, m(h)=h^n corresponds to ascribing a slip condition at the fluid-solid interface and \alpha > 0 describes the fluid’s rheology. In this talk, I will give an overview about the most important questions arising in the theory of Non-Newtonian thin-film equations: for example the problem of existence and uniqueness of solutions as well as non-negativity and qualitative properties of solutions. Specifically, I will explain how to prove that solutions with low-energy initial data lift off after finite time and converge to the constant equilibrium with explicit rates. This is joint work with Peter Gladbach and Christina Lienstromberg.

Chair : Frédéric valet

Title: Periodic Hölder waves in a class of negative-order fKdV-like equations

Chair : Jörg Weber

Title: Fine properties of steady water waves 

Chair: Mats Ehrnström

Title: Gradient blow-up for dispersive and dissipative perturbations of the Burgers equation

Abstract: In this talk, I will describe a singularity formation scenario for a general class of dispersive and dissipative perturbations of the classical Burgers equation. This class includes the Whitham equation in water waves, the fractional KdV equation with dispersive term of order $\alpha \in [0,1)$, and the fractal Burgers equation with dissipative term of order $\beta \in [0,1)$.

We show the existence of solutions whose gradient blows up in finite time, starting from smooth initial data ("wave breaking"). Our theorem appears to be the first construction of gradient blow-up for fractional KdV in the range $\alpha \in [2/3,1)$. We follow a self-similar approach, treating the dispersive term perturbatively. We show that the blow-up is stable when $\alpha < 2/3$. On the other hand, for $\alpha \geq 2/3$, the solution is constructed by perturbing an underlying unstable self-similar Burgers profile. In the final part of the talk, I will indicate how these observations can be used to address certain blow-up problems in higher space dimensions.

This is joint work with Sung-Jin Oh (UC Berkeley).

Chair: Erik Wahlén

Title : Decay of solitary waves of fKdV-type equations

Abstract: In this talk, I will explain a study of the solitary waves of fractional Korteweg-de Vries type equations, that are related to the dimension one semi-linear fractional equations:

\begin{align*}

    \vert D \vert^\alpha u + u -f(u)=0,

\end{align*}

with $\alpha\in (0,2)$, a prescribed coefficient $p^*(\alpha)$, and a non-linearity $f(u)=\vert u \vert^{p-1}u$ for $p\in(1,p^*(\alpha))$, or $f(u)=u^p$ with an integer $p\in[2;p^*(\alpha))$. Asymptotic developments of order one at infinity of solutions are given, as well as second order developments for positive solutions, in terms of the coefficient of dispersion $\alpha$ and of the non-linearity $p$. The main tools are the kernel formulation introduced by Bona and Li, and an accurate description of the kernel by complex analysis theory. The talk is based on a collaboration with Arnaud Eychenne.

Chair : Gabriele Brüll 

Short presentation by the people in the three research groups.

Chair : Mats Ehrnström