Analysis
Department of Mathematics and Statistics
Texas Tech University
The long-time behavior of solutions of the three-dimensional Navier–Stokes equations in a periodic domain is studied. The time-dependent body force decays, as time t tends to infinity, in a coherent manner. In fact, it is assumed to have a general and complicated asymptotic expansion which involves complex powers of et, t, ln t, or other iterated logarithmic functions of t. We prove that all Leray–Hopf weak solutions admit an asymptotic expansion which is independent of the solutions and is uniquely determined by the asymptotic expansion of the body force. The proof makes use of the complexifications of the Gevrey–Sobolev spaces together with those of the Stokes operator and the bilinear form of the Navier–Stokes equations.
Recent developments of convex integration technique led to many non-uniqueness results of various PDEs. I will try to describe the mechanism of how this technique works for a simple case, specifically a construction of a solution to the 3D Euler equations that does not conserve its energy (although it is well-known that the classical approach yields local unique solution that conserves energy). Afterwards, if time permits, I will describe related open problems of my interest. This talk is intended to be accessible to graduate students.
The theory of kernel integral operators lies at the intersection of diverse fields such as Operator theory, Harmonic analysis, Geometric analysis, and even Dynamical systems theory. Kernels used in Analysis are usually equipped with a bandwidth parameter ϵ controlling the decay of tails. There are various results, theoretical and empirical - about such integral operators revealing the local geometry as ϵ→0, such as curvature, the Laplace and advection operators. However, little is known about the limiting behavior of these operators in non-smooth surfaces, such as fractals. I shall present to you an axiomatic framework which proves existing results on the asymptotic behavior of kernel integral operators. It also established spectral convergence, an elusive property in operator theory. This framework also provides directions in which one can generalize these results to non-smooth manifolds.
We will introduce the theory of circle packings on Riemann surfaces. This talk is designed to be accessible to graduate students.
We show the existence of determining wavenumber for the Navier-Stokes equation in both 3D and 2D. Estimates on the determining wavenumber are established in terms of the phenomenological Kolmogorov’s dissipation number (3D) and Kraichnan’s number (2D). The results justify the prediction of Kolmogorov’s dissipation wavenumber and Kraichnan’s number in a rigorous way.
This Analysis seminar may be attended Monday the 24th at 4:00 PM CDT (UT-5) via this Zoom link.
Meeting ID: 972 0263 8160
Passcode: 549916
The Mahler measure is defined as the geometric mean of a polynomial over the unit circle. This quantity is of importance in analysis and number theory. We show that it grows exponentially fast if we iterate a polynomial in the sense of complex dynamics. The exact base of that exponential growth is described by an integral over the invariant measure for the Julia set of the polynomial we iterate. We also provide sharp bounds for such integrals by using some results from complex function theory.
This Analysis seminar may be attended Monday the 7th at 4:00 PM CST (UT-6) via this Zoom link.
Meeting ID: 972 0263 8160
Passcode: 549916
PDF abstract
The Navier-Stokes and Euler equations are the fundamental models for describing viscous and inviscid fluids, respectively. Based on ideas which date back to Kolmogorov and Onsager, solutions to these equations are expected to dissipate energy, which in turn suggests that such solutions are somewhat rough and thus only weak solutions. At these low regularity levels, however, one may construct wild weak solutions using convex integration methods. In this talk, I will discuss the motivation and methodology behind joint work with Tristan Buckmaster, Nader Masmoudi, and Vlad Vicol in which we construct wild solutions to the Euler equations which deviate from the predictions of Kolmogorov's classical K41 phenomenological theory.
This Analysis seminar may be attended Monday the 28th at 4:00 PM CST (UT-6) via this Zoom link.
Meeting ID: 972 0263 8160
Passcode: 549916
We study collections of measures that are negligible in a sense of "modules". The idea is originated in complex analysis as "a conformal module of a family of curves" in looking for an invariant object under conformal transformations on the complex plane. Later the definition of the module was successfully applied to the nonlinear potential theory and quasiconformal analysis in a wider sense in Euclidean spaces. B. Fuglede, by studying the completion of functional spaces, generalized the notion of the module of a family of curves to the module of a family of measures. A collection of measures is exceptional if the corresponding module vanishes. We are interested in finding exceptional families of measures on Carnot groups, related to geometric objects such as "intrinsic graphs". It leads to the notion of a Grassmannian on specific Carnot groups.