Analysis
Department of Mathematics and Statistics
Texas Tech University
We examine how stationary solutions to Galerkin approximations of the Navier--Stokes equations behave in the limit as the Grashof number $G$ tends to $\infty$. An appropriate scaling is used to place the Grashof number as a new coefficient of the nonlinear term, while the body force is fixed. A new type of asymptotic expansion, as $G\to\infty$, for a family of solutions is introduced. Relations among the terms in the expansion are obtained by following a procedure that compares and totally orders positive sequences generated by the expansion. The same methodology applies to the case of perturbed body forces and similar results are obtained. We demonstrate with a class of forces and solutions that have convergent asymptotic expansions in $G$. All the results hold in both two and three dimensions, as well as for both no-slip and periodic boundary conditions.
This is the second part of the talks on the topic. We examine how stationary solutions to Galerkin approximations of the Navier--Stokes equations behave in the limit as the Grashof number $G$ tends to $\infty$. An appropriate scaling is used to place the Grashof number as a new coefficient of the nonlinear term, while the body force is fixed. A new type of asymptotic expansion, as $G\to\infty$, for a family of solutions is introduced. Relations among the terms in the expansion are obtained by following a procedure that compares and totally orders positive sequences generated by the expansion. The same methodology applies to the case of perturbed body forces and similar results are obtained. We demonstrate with a class of forces and solutions that have convergent asymptotic expansions in $G$. All the results hold in both two and three dimensions, as well as for both no-slip and periodic boundary conditions. This is joint work with Ciprian Foias and Michael S. Jolly (Indiana University).
Many branches of theoretical and applied mathematics require a quantifiable notion of complexity. One such circumstance is a topological dynamical system - which involves a continuous self-map on a metric space. There are many notions of complexity one can assign to the repeated iterations of the map. One of the foundational discoveries of dynamical systems theory is that these have a common limit, known as the topological entropy of the system. This talk presents a category-theoretic view of entropy. Category theory is a structural approach to mathematics, in which mathematical properties are derived as consequences of structural arrangements rather than internal properties. The category theoretic approach reveals how the common limit is a consequence of the structural assumptions on these notions. The various notions of complexity are expressed as functors, and natural transformations between these functors lead to their joint convergence to the common limit.
Preprint link
This is the second and final part of the talk. In this session, we shall discuss how the various notions of open covers, diameters, metric span and metric cover come together into a diagram of interweaving functors. The colimits of these functors are precisely the various definitions of entropy. We shall discuss a theorem about how in such setups, the functors have a common colimit, thus reproving a fundamental result from the theory of Dynamical systems.
Preprint link
Turbulence occurs in our daily lives. Kolmogorov's zeroth law of turbulence from 1941 was supported by numerical analysis under the name of "anomalous dissipation." Closely related is the famous Onsager's conjecture in 1949. The magnetohydrodynamics (MHD) system has applications in astrophysics, geophysics, and plasma physics, and has been studied since 1940s led by the pioneering works of Alfven. In the community of researchers on MHD turbulence, there is a long-standing belief, namely Taylor's conjecture in 1974, concerning the conservation of the magnetic helicity in the infinite conductivity limit. Via the technique of convex integration, we prove non-uniqueness in law of the 3D MHD system forced by random noise. The solution we construct has properties such that all of its total energy, magnetic helicity, and cross helicity grow twice faster than those constructed via the classical approach, namely the Galerkin approximation. Consequently, it represents a counterexample to Taylor's conjecture in the stochastic setting.
This week's Analysis seminar may be attended at 4:00 PM CDT (UT-5) via this Zoom link.
Meeting ID: 966 0483 8930
Passcode: 871299
I will discuss several extremal problems for the logarithmic capacity of constellations (collections) of disks in the plane. Solutions for some of these problems will be provided and a few conjectures for problems which remain open will be suggested. Similar problems for the hyperbolic capacity of constellations of disks in the hyperbolic plane also will be mentioned.