Bidisperse filtration problem with non-monotonic retention profiles (Q2088051)

The authors consider the evolution system \(\frac{\partial c_{i}}{\partial t}+\frac{ \partial c_{i}}{\partial x}+\frac{\partial s_{i}}{\partial t}=0\), \(\frac{ \partial s_{i}}{\partial t}=(1-b)\lambda _{i}c_{i}\), \(i=1,2\), with \( b=B_{1}c_{1}^{0}s_{1}+B_{2}c_{2}^{0}s_{2}\), posed in the quadrant \(\Omega =\{x\geq 0,t\geq 0\}\).\ Here \(c_{i},s_{i}\), \(i=1,2\), are the suspended and retained particles concentrations, respectively, \(b\) is the concentration of occupied sites, \(B_{i}>0\) is the individual area that an attached particle occupies at the rock surface, \(c_{i}^{0}>0\) is the particle concentration in the injected suspension with \(c_{1}^{0}+c_{2}^{0}=1\), and \(\lambda _{1}>\lambda _{2}>0\). Looking for solutions to this system in the form of travelling waves: \(c_{i}=c_{i}(x-ut)\), \(s_{i}=s_{i}(x-ut)\), \(i=1,2\), where \(u \) is the unknown constant velocity of the traveling wave, the authors obtain the system of ordinary differential equations \((1-u)c_{i}^{\prime }-us_{i}^{\prime }=0\), \(-us_{i}^{\prime }=(1-b)\lambda _{i}c_{i}\), \(i=1,2\), with the conditions at infinity: \(w\rightarrow +\infty \): \(c_{i}\rightarrow 0 \), \(s_{i}\rightarrow 0\), \(w\rightarrow -\infty \): \(c_{i}\rightarrow c_{i}^{0} \), \(i=1,2\). The authors solve this system and finally obtain an expression of travelling wave velocity, and they establish properties of the solution. They then consider the initial-boundary conditions at \(x=0\): \(c_{i}=c_{i}^{0}\), and at \(t=0\): \(c_{i}=0\), \(s_{i}=0\), \(i=1,2\). They here observe that the solutions \(c_{1}(x,t),c_{2}(x,t)\) have a discontinuity on the characteristic line \(t=x\). Coming back to the evolution system, they define a weak solution with the preceding initial-boundary conditions as a set of functions \( c_{i},s_{i}\), \(i=1,2\), such that the functions \(c_{1},c_{2}\) are piecewise differentiable in \(\Omega \), the functions \(s_{1},s_{2}\) are piecewise differentiable and continuous in \(\Omega \), the equations of the evolution system are satisfied in a weak sense (through a variational formulation), and the initial-boundary conditions are satisfied in the strong (pointwise) sense. The first main result of the paper proves the existence of a weak unique solution of the evolution system satisfying the initial-boundary conditions. The further main results prove properties of this weak solution in the subdomains \(\Omega _{0}=\{x>0,0<t<x\}\) and \(\Omega _{1}=\{x>0,t>x\}\). The paper ends with some examples for which the authors also present and discuss results of numerical simulations.