There will be a poster session for graduate students on Saturday, April 13. We encourage all participants to attend and provide feedback and advice to the students.
One poster per individual is allowed. Display boards and pins for the posters will be provided. The organizers will dispose of any posters left behind after the conclusion of the conference. To be considered for presenting a poster at this conference, please submit a title and abstract for your poster on the registration form by March 15, 2024.
Geophysicists have studied 3D Quasi-Geostrophic systems extensively. These systems describe stratified flows in the atmosphere on a large time scale and are widely used for forecasting atmospheric circulation. They couple an inviscid transport equation in $\mathbb{R}_{+} \times \Omega$ with an equation on the boundary satisfied by the trace, where $\Omega$ is either $2 D$ torus or a bounded convex domain in $\mathbb{R}^2$. In this talk, we will show the existence and some regularity results of global in time weak solutions to a family of singular 3D quasi-geostrophic systems with Ekman pumping, where the background density profile degenerates at the boundary. The main difficulty is handling the degeneration of the background density profile at the boundary.
In this paper, we rigorously prove the existence of self-similar converging shock wave solutions for the non-isentropic Euler equations for $\gamma\in (1,3]$. These solutions are analytic away from the shock interface before collapse, and the shock wave reaches the origin at the time of collapse. The region behind the shock undergoes a sonic degeneracy, which causes numerous difficulties for smoothness of the flow and the analytic construction of the solution. The proof is based on continuity arguments, nonlinear invariances, and barrier functions.
A rigorous derivation of point vortex systems from kinetic equations has been a challenging open problem, due to singular layers in the inviscid limit, giving a large velocity gradient in the Boltzmann equations. In this paper, we derive the Helmholtz-Kirchhoff point-vortex system from the hydrodynamic limits of the Boltzmann equations. We construct Boltzmann solutions by the Hilbert-type expansion associated to the point vortices solutions of the 2D Navier-Stokes equations. We give a precise pointwise estimate for the solution of the Boltzmann equations with small Strouhal number and Knudsen number.
We establish the theory of p-modulus of a family of infinite paths on an infinite rooted tree, then explore its interpretation and properties. One key result is the formulation of p-modulus in the infinite tree as a limit of p-modulus on truncated trees, with a formula given in terms of a series. Another key result is the existence of a critical p-value for radially symmetric infinite binary trees that assigns a kind of dimension to the boundaries of these trees.
Abstract: We prove error estimates for a certain class numerical schemes
approximating nonlocal conservation laws, modeling traffic flow and crowd dynamics.
Numerical simulations are presented to illustrate this result.