Travelling wave solutions for gravity fingering in porous media flows

多孔介质中$A + B \rightarrow C$反应流的适定性与指进模式

Well-posedness and Fingering Patterns in $A + B \rightarrow C$ Reactive Porous Media Flow

本文研究多孔介质中$A+B→C$反应与密度驱动流耦合的对流-扩散-反应系统，通过Galerkin方法证明弱解适定性，结合有限元模拟揭示指进不稳定性，为相关应用提供理论基础。

Propagation of solutions of the Porous Medium Equation with reaction and their travelling wave behaviour

We consider reaction-diffusion equations of porous medium type, with different kind of reaction terms, and nonnegative bounded initial data. For all the reaction terms under consideration there are initial data for which the solution converges to 1 uniformly in compact sets for large times. We will characterize for which reaction terms this happens for all nontrivial nonnegative initial data, and for which ones there are also solutions converging uniformly to 0. Problems in this family have a unique (up to translations) travelling wave with a finite front and we will see how its speed gives th…

Interface dynamics of the porous medium equation with a bistable reaction term

We consider a degenerate partial differential equation arising in population dynamics, namely the porous medium equation with a bistable reaction term. We study its asymptotic behavior as a small parameter, related to the thickness of a diffuse interface, tends to zero. We prove the rapid formation of transition layers which then propagate. We prove the convergence to a sharp interface limit whose normal velocity, at each point, is that of the underlying degenerate travelling wave.

System of Porous Medium Equations

We investigate the evolution of population density vector, $\bold{u}=\left(u^1,\cdots,u^k\right)$, of $k$-species whose diffusion is controlled by its absolute value $\left|\bold{u}\right|$. More precisely we study the properties and asymptotic large time behaviour of solution $\bold{u}=\left(u^1,\cdots,u^k\right)$ of degenerate parabolic system \begin{equation*} \left(u^i\right)_t=\nabla\cdot\left(\left|\bold{u}\right|^{m-1}\nabla u^i\right) \qquad \mbox{for $m>1$ and $i=1,\cdots,k$}. \end{equation*} Under some regularity assumption, we prove that the function $u^i$ which describes the popula…

Flat Fronts and Stability for the Porous Medium Equation

This work is concerned with the equation $ \partial_t ρ= Δ_x ρ^m $, $ m > 1 $, known as the porous medium equation. It shows stability of the pressure of solutions close to flat travelling wave fronts in the homogeneous Lipschitz sense that is in a way optimal for the treatment of the equation. This is the first result of this type and implies global regularity estimates for any number of derivatives of the pressure. Consequences include smoothness, analyticity in temporal and tangential directions, and analyticity of the interface between empty and occupied regions. In the course of the argum…

Travelling-wave behaviour in doubly nonlinear reaction-diffusion equations

We study a family of reaction-diffusion equations that present a doubly nonlinear character given by a combination of the $p$-Laplacian and the porous medium operators. We consider the so-called slow diffusion regime, corresponding to a degenerate behaviour at the level 0, \normalcolor in which nonnegative solutions with compactly supported initial data have a compact support for any later time. For some results we will also require $p\ge2$ to avoid the possibility of a singular behaviour away from 0. Problems in this family have a unique (up to translations) travelling wave with a finite fron…