Section 4 seminar, Autumn 2022

The Section 4 seminar for the Autumn of 2022 will be held on Thursdays from 10:15–12:00 (see the schedule)

Upcoming talks

Speaker: Kenneth H. Karlsen
Title: Camassa-Holm equation with transport noise.
Abstract: 

We consider a complex and fascinating SPDE that is  known as the stochastic Camassa-Holm equation. There are many properties of this SPDE that are not well understood, and there is much work to be done in order to fully understand its behavior. In these talks, we will eventually prove existence of global-in-time weak solutions.

The stochastic Camassa-Holm equation is a nonlinear partial differential equation that describes the motion of water waves in a fluid with random disturbances. It has a number of mathematical properties that make it useful for modeling wave behavior in a variety of contexts. Some of the key properties of the equation include:

    Nonlinearity: The equation takes into account the nonlinear interactions between waves.

    Dispersion: The equation accounts for the dispersive effects of waves, which describes how the speed and wavelength of a wave can vary depending on its frequency.

    Randomness: The equation incorporates the effects of random disturbances on the motion of waves, which is important for capturing the unpredictable nature of wave behavior in a fluid.

The stochastic Camassa-Holm equation can exhibit complex behavior, such as the formation of non-smooth solutions. A non-smooth solution is a solution that is not differentiable, or in other words, has sharp or abrupt changes in its slope. This is related to the formation of wave breaking, which is the point at which the height of a water wave becomes so large that it starts to tumble over itself. This complicates the search for globally defined solutions and what we mean by a solution.

Past talks

Thursday 10 November
Speaker: Ulrik Skre Fjordholm
Title: Why is 1D easier than multi-D? Part II

Thursday 3 November
Speaker: Ulrik Skre Fjordholm
Title: Why is 1D easier than multi-D?
Abstract: I will review some theory on transport/continuity equations with irregular velocity fields. I will contrast the 1D case to the multi-D case, and focus on a technique which allows a rather complete well-posedness theory of generalized solutions to 1D transport equations. The technique fails in multi-D, and we look at a counterexample. The talk is mostly based on a work by Panov (2008).

Thursday 27 October
Speaker: Kenneth Karlsen
Title: A singular limit problem for stochastic conservation laws, part 4

Thursday 20 October
Speaker: Kenneth Karlsen
Title: A singular limit problem for stochastic conservation laws, part 3

Thursday 13 October
Speaker: Kenneth Karlsen
Title: A singular limit problem for stochastic conservation laws, part 3

Thursday 6 October
Speaker: Geir Dahl
Title: The Permutation and Alternating Sign Matrix Cones

Abstract: CMT deals with combinatorics related to matrix theory. We will look at two classes of matrices, the permutation matrices and the so-called alternating sign matrices (ASM). The goal of the talk is to give some basic facts on ASMs, mention a connection to physics, and to discuss how polyhedra (in particular, polyhedral cones) associated with permutation matrices and ASMs have some interesting properties. The talk will be non-technical, and focus on concepts and main results. The talk is based on work done in collaboration with Richard Brualdi (Univ. Wisconsin, Madison).

Thursday 29 September
Speaker: Kenneth Karlsen
Title: A singular limit problem for stochastic conservation laws, part 2

Thursday 22 September
Speaker: Kenneth Karlsen
Title: A singular limit problem for stochastic conservation laws
Abstract: We investigate a singular limit problem for stochastic conservation laws with discontinuous flux, perturbed by vanishing diffusion—dynamic capillarity terms. Our convergence arguments use kinetic formulations, H-measures and velocity averaging for stochastic transport equations, and a.s. representations of random variables in quasi-Polish spaces. This technical talk is based on joint work M. Kunzinger and D. Mitrovic.

Published Sep. 5, 2022 3:48 PM - Last modified Dec. 7, 2022 11:52 AM