2013/11/14 by Gonzalo Galiano, Galiano, Gonzalo, Sergey Shmarev +3
Computer Science · Mathematics · #35K55 #35K57 #35K65 #35R35 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1311.3454
openalex publication_date 2013/11/14 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We study the the Dirichlet problem for the cross-diffusion system ∂tui=div(aiui∇ (u1+u2))+fi(u1,u2), i=1,2, ai=constgt;0, in the cylinder Q=Ω× (0,T]. The functions fi are assumed to satisfy the conditions f1(0,r)=0, f2(s,0)=0, f1(0,r), f2(s,0) are locally Lipschitz-continuous. It is proved that for suitable initial data u0, v0 the system admits segregated solutions (u1,u2) such that ui∈ L∞(Q), u1+u2∈ C0(Q), u1+u2>0 and u1⋅ u2=0 everywhere in Q. We show that the segregated solution is not unique and derive the equation of motion of the surface Γ which separates the parts of Q where u1>0, or u2>0. The equation of motion of Γ is a modification of the Darcy law in filtration theory. Results of numerical simulation are presented.