Events
Department of Mathematics and Statistics
Texas Tech University
Weak solutions of the incompressible Navier-Stokes equations are unique in the so-called Ladyzhenskaya-Prodi-Serrin regime. A scaling analysis suggests that classical uniqueness results are sharp, but previous nonuniqueness constructions of convex integration are far below the critical threshold. In this talk, I will show sharp nonuniqueness results on two end-points of the Ladyzhenskaya-Prodi-Serrin regime.
Joint work with Alexey Cheskidov.
To join the talk on Zoom please click
here.
 | Wednesday
Feb. 9
|
| Algebra and Number Theory
No Seminar
|
In realistic flow problems described by partial differential equations (PDEs), where the dynamics are not known, or in which the variables are changing rapidly, the robust, adaptive time-stepping is central to accurately and efficiently predict the long-term behavior of the solution. This is especially important in the coupled flow problems, such as the fluid-structure interaction (FSI), which often exhibit complex dynamic behavior. While the adaptive spatial mesh refinement techniques are well established and widely used, less attention has been given to the adaptive time-stepping methods for PDEs. We will discuss novel, adaptive, partitioned numerical methods for FSI problems with thick and thin structures. The time integration in the proposed methods is based on the refactorized Cauchy's one-legged 'theta-like' method, which consists of a backward Euler method, where the fluid and structure sub-problems are sub-iterated until convergence, followed by a forward Euler method. The bulk of the computation is done by the backward Euler method, as the forward Euler step is equivalent to (and implemented as) a linear extrapolation. We will present the numerical analysis of the proposed methods showing linear convergence of the sub-iterative process and unconditional stability. The time adaptation strategies will be discussed. The properties of the methods, as well as the selection of the parameters used in the adaptive process, will be explored in numerical examples.
Please virtually attend the Applied Math seminar via this Zoom link Wednesday the 9th at 4 PM -- meeting ID: 937 2431 1192
In our study, the exposure to equity markets is measured by a convex risk measure,
and the stock returns follow normal variance-mean mixture distributions.
We convert the high-dimensional problem into a two-dimensional one by using a scheme motivated by Shi and Kim (2001).
We further prove the dimension reduction scheme can preserve the convexity of the problem.
An empirical study will be presented for illustration.