II. Inverse Stefan Problem – Part 2
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Tag Archives: REU 2014
NPDE Lecture 4 Discussion
IV. Nonlinear Diffusion in Nonsmooth Domains
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ISP Lecture 1 Discussion
Optimal Control and Inverse Problems for PDEs
Development and investigation of the nonlinear models of the mathematical physics is one of the most important problems of the modern science. This project will shed light into modeling and control of an important class of nonlinear processes with phase transition, so called free boundary problems, arising in thermophysics and mechanics of continuous media, bioengineering and materials science. Free boundary problems involve the solution of the partial differential equations in domains that are unknown apriori: not only the solutions of the equations but also the domain of definition of the equations must be determined.
The main goal of this project is to gain insight into inverse free boundary problems. These problems arise when not only the solution of the PDEs and the free boundaries are unknown, but also some other characteristics of the media are not available. For example, temperature or heat flux on the fixed boundaries, intensity of the source term or phase transition temperature may not be available. Another important motivation for the analysis of the inverse Stefan problem arises in optimal control of processes with phase transitions.
In summer REU session of 2014 students will pursue research on the optimal control of the free boundary problem for the heat equation motivated by the new variational formulation suggested in a recent paper:

U. G. Abdulla, On the Optimal Control of the Free Boundary Problems for the Second Order Parabolic Equations.
I.Wellposedness and Convergence of the Method of Lines,
Inverse Problems and Imaging, Volume 7, Number 2(2013), 307340.
NPDE Lecture 3 Discussion
III. Energy Estimates and Existence
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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:instpme1}\\
\int_{\R^{N}}u(x,t)\,dx &=1,~t > 0\label{eq:instpme2}\\
u(x,0) = \delta(x)\label{eq:instpme3}
\end{align}
and constructed instantaneous pointsource solution, also called the ZeldovichKompaneetsBarenblatt (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 DiffusionAbsorption Equation
Reactiondiffusion 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 reactiondiffusion equation
\begin{equation} u_{t}=\text{div} \big( u^{\sigma}\nabla u\big)\gamma u,\quad u \in \R^{N},~t>0\label{eq:pmeloc1}\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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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 reactiondiffusionconvection 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 reactiondiffusionconvection equations.
Full solution of this problem for the reactiondiffusion equation is given in the following two papers by using significantly the theory of reactiondiffusion equations in general nonsmooth domains developed in my JDE paper.
 U. G. Abdulla and J. R. King, Interface development and local solutions to reactiondiffusion equations, SIAM J. Math. Anal., 32, 2, 2000, 235260.
 U. G. Abdulla, Evolution of interfaces and explicit asymptotics at infinity for the fast diffusion equation with absorption, Nonlinear Analysis, 50, 4, 2002, 541560.
 U. G. Abdulla, Reactiondiffusion in irregular domains, J. Differential Equations, 164, 2000, 321354.
Introductory lectures on this topic will be posted soon.