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. Continue reading NPDE Lecture 2 Discussion

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. nike requin 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 $\sigma >0$, $\gamma>0$. nike air max 90 That means nonlinear diffusion is accompanied with linear absorption. adidas stan smith Continue reading NPDE Localization Effect in Diffusion-Absorption Equation