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NPDE Lecture 2 Discussion

II. Weak Solutions and Uniqueness
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Weak Solutions

Let me remind you what I did in a previous lecture. I considered the problem
\begin{align}
u_{t}-\text{div}\big( u^{\sigma} \nabla u\big) &=0,~x \in\R^{N},~t>0; \ \sigma > 0,\label{eq:inst-pme-1}\\
\int_{\R^{N}}u(x,t)\,dx &=1,~t > 0\label{eq:inst-pme-2}\\
u(x,0) = \delta(x)\label{eq:inst-pme-3}
\end{align}
and constructed instantaneous point-source solution, also called the Zeldovich-Kompaneets-Barenblatt (ZKB) solution.
\begin{equation} u_*(x,t)=t^{-\frac{N}{2+N\sigma}}\left[\frac{\sigma}{2(2+N\sigma)} \left( \eta_{0}^{2}-\frac{|x|^{2}}{t^{\frac{2}{2+N\sigma}}}\right)_{+} \right]^{\frac{1}{\sigma}}\end{equation}
Note that the nonlinear diffusion equation is invariant under translation of time and space coordinate, and the solution to the problem
\[ \begin{cases}
u_{t}=\text{div}\big(u^{\sigma }u\big),~&(x,t) \in \R^{N}\times \R_{+}\\
u(x,0)=u_*(x,T),~&x \in \R^{N}
\end{cases}\]
is exactly $u_*(x,t+T)$, which has compact support for all $t\geq 0$.

However, several important questions are left open. First of all, ZKB solution is not a classical solution: it is not even differentiable on the boundary of the support. First important question which we need to answer is the following: In what sense is $u_*$ actually a solution of the problem? Hence, we need to define the notion of the weak solution. Second important question we need to answer is the following: May be there is a smooth solution of the same problem with different properties, and ZKB solution is just physically irrelevant mathematical example. Main goal of this lecture is to answer these important questions.
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NPDE Localization Effect in Diffusion-Absorption Equation

Reaction-diffusion equations can exhibit a phenomenon known as space localization: not only do they exhibit finite speed of propagation, but heat can be completely contained in a compact region for all time under certain conditions. This is a physically pertinent phenomena as well, having been observed in plasmas, but not encountered in solutions to the linear heat equation.

Consider the reaction-diffusion equation
\begin{equation} u_{t}=\text{div} \big( u^{\sigma}\nabla u\big)-\gamma u,\quad u \in \R^{N},~t>0\label{eq:pme-loc-1}\end{equation}
where $\sigma >0$, $\gamma>0$. That means nonlinear diffusion is accompanied with linear absorption.
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NPDE Lecture 1 Discussion

I. Nonlinear Diffusion Equation (Introduction)
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I am going to start a series of lectures on nonlinear partial differential equations. Advancements made in the theory of nonlinear PDEs is one of the main achievements in XX century mathematics. Let me first make a remark on linear PDEs. In some sense one can say that there is a complete theory of linear PDEs, and perhaps the best source would be the four volumes of “The Analysis of Linear Partial Differential Operators” by Lars Hörmander [3].

The importance of the nonlinear PDEs is associated with the fact that they reflect more precise nonlinear laws of the nature, while linear PDEs were derived from the linearized versions of the nonlinear laws. Despite the great advances in the theory of nonlinear PDEs it is far from being complete. It is remarkable that almost every major nonlinear PDE has its own personality and requires unique approach. However, looking to current state of art in the theory of nonlinear PDEs, one can observe a similarity with the classical linear theory in the following sense: there are some major individual nonlinear PDEs, and analysis and understanding of those PDEs is a key towards the general theory of a broad class of nonlinear PDEs. I am going to concentrate on one of those key nonlinear PDEs called the nonlinear diffusion equation, which is a generalization of the classical heat/diffusion equation.
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Nonlinear Partial Differential Equations

Partial differential equations (PDEs) are central to mathematics, whether pure or applied. They arise in mathematical models of real world problems, where dependent variables vary continuously as functions of several independent variables, usually space and time. PDEs arising in a majority of real world applications are nonlinear. Despite its complexity, in the theory of nonlinear PDEs one can
observe key equations or systems which are essential for the development of the theory for a particular class of equations. One of those key equations is the nonlinear reaction-diffusion-convection equation:

\[u_t=(u^m)_{xx}+bu^{\beta}+c(u^{\gamma})_x,\]

where \[u=u(x,t),x\in \mathbb{R}, t>0, b,c \in \mathbb{R}, m>0, \beta>0, \gamma>0\]
It arises, for instance, in the theory of gas flow through a porous medium, heat radiation in plasmas, spatial spread of biological populations, diffusion of particles in plasma etc.

In summer session of 2014 students will pursue research on the open problem of evolution of interfaces
of solutions to the reaction-diffusion-convection equations.

Full solution of this problem for the reaction-diffusion equation is given in the following two papers by using significantly the theory of reaction-diffusion equations in general non-smooth domains developed in my JDE paper.

Introductory lectures on this topic will be posted soon.

Welcome to Ugur Abdulla’s Mathematical Blog

Welcome to my Mathematical Blog .  The purpose is to share updates in my research,  discussion of mathematical ideas in the fields of my research expertise, discussion of open problems, updates and discussions on my new papers and posted lectures, and in general discussion of any math related topics.  It is open to everyone in the mathematical community worldwide.

In particular, this is the weblog for the Research Experience for Undergraduates (REU) Site on Partial Differential Equations and Dynamical Systems, which I am running for 2014-2016 with the grant from the US National Science Foundation.  For application, grant, and program information, please see the REU Main Page.

Each summer from 2014 to 2016, 9 undergraduate students will participate in an 8 week summer REU Site on Partial Differential Equations and Dynamical Systems held at the Florida Institute of Technology Mathematical Sciences Department.

The REU Site is designed to involve undergraduate students in innovative research in nonlinear partial differential equations, optimal control and inverse problems for systems with distributed parameters, and dynamical systems and chaos theory, while utilizing modern tools of mathematical and numerical analysis. Students will have a great opportunity to pursue hands-on, original research on the frontier of modern mathematics, which will include the evolution of interfaces for nonlinear reaction-diffusion-convection equations, inverse free boundary problems and optimal control of phase transition processes, and the fine classification of minimal periodic orbits of discrete dynamical systems with application in chaos theory.

For more information on the REU faculty and program, see Dr. Abdulla’s REU page. For an extended description and other related links, see the FIT Research Portal. I will post here periodically both before, during, and after the REU to communicate with other students and the community at large about our results and other interesting information.