Analysis
Department of Mathematics and Statistics
Texas Tech University
We establish the precise asymptotic behavior, as time $t$ tends to infinity, for nontrivial, decaying solutions of genuinely nonlinear systems of ordinary differential equations. The lowest order term in these systems, when the spatial variables are small, is not linear, but rather positively homogeneous of a degree greater than one. We prove that the solution behaves like $\xi t^{-p}$, as $t\to\infty$, for a nonzero vector $\xi$ and an explicit number $p>0$.
This week's Analysis seminar may be attended at 4:00 PM CST (UT-6) in MA 115.
We will discuss some foundational aspects of three-dimensional incompressible fluid turbulence, including guiding experimental observations, Kolmogorov's 1941 theory on the structure of a turbulent flow, Onsager's 1949 conjecture on anomalous dissipation and weak Euler solutions, and Landau's Kazan remark concerning intermittency. Mathematical examples and constructions that exhibit features of turbulent behavior will be discussed.
This week's Analysis seminar may be attended at 4:00 PM CST (UT-6) via this Zoom link.
Meeting ID: 976 4978 7908
Passcode: 973073
The hyperbolic Navier-Stokes equations considered here are the incompressible Navier-Stokes equations with an extra double time derivative term. Whether or not classical solutions to the hyperbolic Navier-Stokes equations (even in the 2D case) can develop finite-time singularities remains a challenging open problem. The talk presents a global existence and stability result when the coefficient of the double time derivative term and the size of the initial data satisfy a suitable constraint.
This week's Analysis seminar may be attended in-person at 4:00 PM CST (UT-6)
This talk asks questions that arise in ecosystem models but have implications throughout math and engineering. We ask what are the solution properties that hold for "almost every" equation F(X)=C where X and C are finite dimensional. What does "almost every" F mean? "Almost every" is defined so as to be useful to the scientist. I think of this as pure math for the applied scientist. Systems of M equations in N unknowns are ubiquitous in mathematical modeling. These systems, often nonlinear, are used to identify equilibria of dynamical systems in ecology, genomics, control, and many other areas. Our goal is to describe the properties that will hold for "almost every" F that has some structure, and in particular the global properties of solutions for structured systems of smooth functions. As an application of these ideas I will show examples of Lotka-Volterra systems of differential equations. One example has 14 species. We show that 3 must die out exponentially fast. The technique is rather unique. We produce a "team" of 24 Lyapunov functions, each of which gives different information about which species will die out and all of which are essential.
References
[1] arXiv.org/abs/2203.01432 Extinction of multiple populations and a team of Die-out Lyapunov functions Akhavan & Yorke
[2] arXiv.org/abs/2203.00503 Robustness of solutions of almost every system of equations Jahedi, Sauer & Yorke
[3] arXiv/abs/2008.12140 Structured Systems of Nonlinear Equations Jahedi, Sauer & Yorke
This week's Analysis seminar may be attended at 4:00 PM CST (UT-6) via this Zoom link.
Meeting ID: 976 4978 7908
Passcode: 973073
We'll discuss a construction that produces non-unique entropy solutions of the isentropic compressible euler equations. The construction will be a Nash iterative convex integration scheme and will produce continuous velocity fields with smooth density. As a consequence, we'll demonstrate solutions that produce entropy without the formation of a shock. This is joint work with Hyunju Kwon.
We discuss several extremal problems in Complex Analysis and Potential Theory. Solutions to some of these problems were found by this speaker and his collaborators. But several other problems remain open. The main focus will be on open problems where it is reasonable to believe that the symmetrization methods and/or the theory of quadratic differentials can be applied.
The Ricci flow is instrumental to the resolution of Poincare's and other important conjectures. Gradient Ricci solitons are modeling singularities that might develop along the flow. Understanding them would significantly advance our capacity to apply the theory. In this talk, we'll report certain progress on this subject in the presence of an almost complex structure and a large symmetry group.
This week's Analysis seminar may be attended at 4:00 PM CDT (UT-5) via this Zoom link.
Meeting ID: 976 4978 7908
Passcode: 973073
We present a framework for simulating measure-preserving, ergodic dynamical systems with pure point spectrum by a finite-dimensional quantum system amenable to implementation on a quantum computer. The framework is based on a quantum feature map for representing classical states by density operators on a reproducing kernel Hilbert space, H, of functions on classical state space with Banach algebra structure. Simultaneously, a mapping is employed from classical observables into self-adjoint operators on H such that quantum mechanical expectation values are consistent with pointwise function evaluation. With this approach, quantum states and observables on H evolve under the action of a unitary group of Koopman operators in a consistent manner with classical dynamical evolution. Moreover, the state of the quantum system can be projected onto a finite-rank density operator on a 2^n-dimensional tensor product Hilbert space associated with n qubits, enabling efficient implementation in a quantum circuit. We illustrate our approach with quantum circuit simulations of low-dimensional dynamical systems, as well as actual experiments on the IBM Quantum System One.
This week's Analysis seminar may be attended at 4:00 PM CDT (UT-5) via this Zoom link.
Meeting ID: 976 4978 7908
Passcode: 973073
We will discuss the sharpness of the 2/3rds Conjecture, Brannan’s Conjecture , Smale’s Conjecture, the Sendov Conjecture, and a version of the Riemann Hypothesis. This is joint work with Kendall Richards.
Abstract pdf
In this talk, I will discuss the non-uniqueness of global weak solutions to the isentropic system of gas dynamics. In particular, I will show that for any initial data belonging to a dense subset of the energy space, there exists infinitely many global weak solutions to the isentropic Euler equations for any ...
1 < γ ≤ 1 + 2/n
The proof is based on a generalization of convex integration techniques and weak vanishing viscosity limit of the Navier-Stokes equations. This talk is based on the joint work with M. Chen and A. Vasseur.
This week's Analysis seminar may be attended at 4:00 PM CDT (UT-5) via this Zoom link.
Meeting ID: 976 4978 7908
Passcode: 973073