Analysis
Department of Mathematics and Statistics
Texas Tech University
 | Monday
Jan. 20
4 PM
MATH 12
|
| MLK Day
No Talk
Department of Mathematics and Statistics, Texas Tech University
|
N/A
In this talk, we discuss the steady-state distribution of heat on
long pipes in $\mathbb{R}^3$ heated along some regions of their
surfaces. In particular, we prove
that, if the pipe $P=\{(x,y,z):x^2+y^2<1\}$ is heated along its
surface belt
$$B(a)=\{(x,y,z):\,x^2+y^2=1,z\in(-a,a)\}, \qquad a>0,$$
then
the temperature in its cross-sections $D_c=\{(x,y,z)\in P:\,
z=c\}$ is increasing in the radial direction for all $c$ in the
interval $[-a, a]$.
This is a joint work with Prof. Dimitrios Betsakos
from the Aristotle University of Thessaloniki, Greece.
In this talk, we discuss the steady-state distribution of heat on
long pipes in $\mathbb{R}^3$ heated along some regions of their
surfaces. In particular, we prove
that, if the pipe $P=\{(x,y,z):x^2+y^2<1\}$ is heated along its
surface belt
$$B(a)=\{(x,y,z):\,x^2+y^2=1,z\in(-a,a)\}, \qquad a>0,$$
then
the temperature in its cross-sections $D_c=\{(x,y,z)\in P:\,
z=c\}$ is increasing in the radial direction for all $c$ in the
interval $[-a, a]$.
This is a joint work with Prof. Dimitrios Betsakos
from the Aristotle University of Thessaloniki, Greece.
Motivated by recent lecture of Professor Alex Solynin I want review a very powerful and influential Lemma of Growth and its application to behavior of generalized harmonic measure in unbounded domain. Evgeni Mihailovich Landis was Professor at Mechanical and Mathematical Department of Moscow Sate University. From view of many Soviet-Russian mathematician he was underestimate scientist of second part of 20 century. His method of lemma of Growth and machinery of real analysis lie behind proof of the 16 Hilbert problem for solution of elliptic equation in non-divergent. He was very gifted scientist and also contributed in the area which now called deep learning. His AVL algorithm considered by many expert in Google and other software researcher as one of the best fast searching algorithm. He has many followers now days well recognized world class scientists. I was one of his pupil and learned from him not only science but also method of thinking.
We study the long time behavior of the solutions of the Navier-Stokes-Boussinesq equations in geophysical fluid dynamics. We obtain the asymptotic expansions of Foias-Saut type, as time tends to infinity, for both of the velocity and temperature. We show that these expansions can be easily determined thanks to the ``asymptotic decoupling" property of the system.
This is a joint work with Animikh Biswas (University of Maryland Baltimore County).
We study the long time behavior of the solutions of the Navier-Stokes-Boussinesq equations in geophysical fluid dynamics. We obtain the asymptotic expansions of Foias-Saut type, as time tends to infinity, for both of the velocity and temperature. We show that these expansions can be easily determined thanks to the ``asymptotic decoupling" property of the system.
This is a joint work with Animikh Biswas (University of Maryland Baltimore County).
We study the long-time asymptotics of prototypical non-linear diffusion equations. Specifically, we consider the case of a non-degenerate diffusivity
function that is a (non-negative) polynomial of the dependent variable of the
problem. We motivate these types of equations using Einstein’s random walk
paradigm, leading to a partial differential equation in non-divergence form. On
the other hand, using conservation principles leads to a partial differential equation in divergence form. A transformation is derived to handle both cases. Then,
a maximum principle (on both an unbounded and a bounded domain) is proved,
in order to obtain bounds above and below for the time-evolution of the solution to the non-linear diffusion problem. Specifically, these bounds are based on
the fundamental solution of the linear problem (the so-called Aranson’s Green
function). Having thus sandwiched the long-time asymptotics of solutions to
the non-linear problems between two fundamental solutions of the linear problem, we prove that, unlike the case of degenerate diffusion, a non-degenerate
diffusion equation’s solution converges onto the linear diffusion solution at long
times. Select numerical examples support the mathematical theorems and illustrate the convergence process. Our results have implications on how to interpret
asymptotic scalings of potentially anomalous diffusion processes (such as in the
flow of particulate materials) that have been discussed in the applied physics
literature.This will be an expository talk about the direction of research about which the speaker is currently learning. A lot of effort has been devoted by many mathematicians to prove the ``positive'' answer to the Navier-Stokes problem, specifically that for every sufficiently smooth initial data with uniformly bounded kinetic energy, there exists a global smooth solution that preserves the same uniform bound on the energy. Smoothness immediately deduces uniqueness; thus, by contrapositive, the lack of uniqueness leads to the lack of smoothness. Hence, the ``negative'' answer to the Navier-Stokes problem requires finding an initial data that is smooth, has uniformly bounded kinetic energy, and the solution emanating from it eventually loses uniqueness. This is a very difficult direction of research; I end this abstract with the following quote by Terence Tao in 2007: ``Unfortunately, even though the Navier-Stokes equation is known to be very unstable, it is not clear at all how to pass from this to a rigorous demonstration of a blowup solution.''