Analysis
Department of Mathematics and Statistics
Texas Tech University
A theorem of Lehto (1955) states that for every measurable function $\psi$ on the unit circle $\mathbb T,$ there is a
function $f$ holomorphic in the unit disc $\mathbb{D}$, having $\psi$ as radial limit a.e. on $\mathbb{T}.$ In this talk,
an analogous boundary value problem for holomorphic functions in the polydisc in $\mathbb{C}^n,$ will be presented.
Lehto's proof was based on an approximation theorem of Bagemihl and Seidel (1954). They showed that for a continuous
function $\psi$ on $\mathbb{D},$ and an $F_\sigma$ set $E\subset \mathbb{T}$ of the first category, there exists a
holomorphic function $h$ in $\mathbb{D}$ such that $\psi-h$ has vanishing radial limits as we move to the boundary
via radii ending in $E$. We generalized similar approximating theorems for harmonic functions on Riemannian manifolds
(with completely different proofs); that is, for Riemannian manifolds $M$ and $N,$ and a (harmonic) line bundle
$(M, N, \pi, \mathbb{R}),$ continuous functions on $M$ can be approximated by harmonic ones such that the limit on some
special subset of $M$ (roughly fibres of an $F_\sigma$ polar set in $N$) tends to zero as we move to the ideal
boundary of $M.$
Joint work with Paul. M. Gauthier, Université de Montréal, Montréal, Québec, Canada.
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In this talk, we will investigate the propagation of Lipschitz regularity by solutions to various nonlinear, nonlocal parabolic equations. We will locally analyze models such as the Michelson-Sivashinsky equation, incompressible Navier-Stokes system, and advection diffusion problems that include the dissipative SQG equation. Depending on the model, we will either show global well-posedness, derive new regularity criteria, or provide different proofs to and generalize previously obtained results. In particular, and if time allows, we will show that for abstract drift-diffusion problems, it is possible to break certain, supercritical Holder-type barriers and get regularity, which is rather surprising.
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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.
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The symmetric decreasing rearrangement (symmetrization) of a function $f$ in
$n$ variables is the unique radially decreasing function $Sf$ equimeasurable
with $f$. Classical inequalities tell us that symmetrization reduces the overall
oscillation of functions; for instance, it shrinks $L^p$-distances and decreases
gradient norms. But how does symmetrization behave with respect to mean oscillation?
In this talk, I will describe some recent results with Almut Burchard and Galia Dafni.
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Since Hausdorff dimension was first introduced in 1918, many different notions of
dimension have been defined and used throughout many areas of Mathematics. An interesting
topic has always been the distortion of said dimensions of a given set under a specific
class of mappings. More specifically, Gehring and Väisälä proved in 1973 a theorem
concerning the distortion of Hausdorff dimension under quasiconformal maps, while Kaufman
in 2000 proved the analogous result for Box-counting dimension. In this talk, an introduction
to the different types of dimensions will be presented, along with the results of of Gehring,
Väisälä and Kaufman. We will then proceed to discuss analogous theorems we proved for
the Assouad dimension and spectrum, which describe how K-quasiconformal maps change these
notions of a given subset of $\mathbb{R}^n$. We will conclude the talk by demonstrating how
said theorems can be applied to fully classify polynomial spirals up to quasiconformal equivalence.
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In the past twenty years, the method of convex integration has attracted considerable attention as a means to construct weak solutions with "wild" energy profiles to prominent equations in fluid dynamics. This led to the resolution of the Onsager conjecture (2018) and a proof of non-uniqueness for weak solutions of the 3D Navier-Stokes equations (2018).
I will give an introduction to the theory of convex integration, tracing the analytic and geometric developments back to its roots, namely the celebrated 1954 work of John F. Nash Jr. on $C^1$ embeddings. I will provide an overview of the embedding problem and Nash's solution as well as some comments on the modern convex integration techniques in the context of fluid mechanics.
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We study the generalized Forchheimer flows of slightly compressible fluids in rotating
porous media. In the problem's model, the varying density in the Coriolis force is fully
accounted for without any simplifications. It results in a doubly nonlinear parabolic
equation for the density. We derive a priori estimates for the solutions in terms of the
initial, boundary data and physical parameters, emphasizing on the case of unbounded data.
Weighted Poincaré-Sobolev inequalities suitable to the equation's nonlinearity, adapted
Moser's iteration and maximum principle are used and combined to obtain different types
of estimates. This is joint work with Emine Celik and Thinh Kieu.
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In this talk, based on a joint paper by Prof. Dimitrios Betsakos
from the Aristotle University of Thessaloniki in Greece,
Prof. Matti Vuorinen from the University of Turku in Finland and the
presenter, we will discuss problems concerning the conformal capacity of ``hedgehogs'', which are compact sets $E$ in the unit
disk $\mathbb{D}=\{z:\,|z|<1\}$ consisting of a central body $E_0$
that is typically a smaller disk
$\overline{\mathbb{D}}_r=\{z:\,|z|\le r\}$, $r\in(0,1)$, and several
spikes $E_k$ that are compact sets lying on radial intervals
$I(\alpha_k)=\{te^{i\alpha_k}:\,0\le t<1\}$. The main questions we
are concerned with are the following: (1) How does the conformal
capacity ${\rm cap}(E)$ of $E=\cup_{k=0}^n E_k$ behave when the
spikes $E_k$, $k=1,\ldots,n$, move along the intervals
$I(\alpha_k)$ toward the central body if their hyperbolic lengths
are preserved during the motion? (2) How does the
capacity ${\rm cap}(E)$ depend on the distribution of
angles between the spikes $E_k$? We prove several results related
to these questions and discuss methods of applying symmetrization
type transformations to study the capacity of hedgehogs. Several
open problems, including problems on the
capacity of hedgehogs in the three-dimensional
hyperbolic space, also will be suggested.
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