2013/10/08 by Leonid Berlyand, Berlyand, Leonid, Volodymyr Rybalko +3
Computer Science · Materials Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Nonlinear Dynamics and Pattern Formation #Solidification and crystal growth phenomena
paper · pdf · doi:10.48550/arxiv.1310.2285
openalex publication_date 2013/10/08 · openalex created_date 2022/08/30 · openalex updated_date 2026/07/28
We consider a system of two PDEs arising in modeling of motility of\neukariotic cells on substrates. This system consists of the Allen-Cahn equation\nfor the scalar phase field function coupled with another vectorial parabolic\nequation for the orientation of the actin filament network. The two key\nproperties of this system are (i) presence of gradients in the coupling terms\n(gradient coupling) and (ii) mass (volume) preservation constraints. We first\nprove that the sharp interface property of initial conditions is preserved in\ntime. Next we formally derive the equation of the motion of the interface,\nwhich is the mean curvature motion perturbed by a nonlinear term that appears\ndue to the properties (i)-(ii). This novel term leads to surprising features of\nthe the motion of the interface. Because of these properties maximum principle\nand classical comparison techniques do not apply to this system. Furthermore,\nthe system can not be written in a form of gradient flow, which is why recently\ndeveloped Gamma-convergence techniques also can not be used for the\njustification of the formal derivation. Such justification is presented in a\none-dimensional model problem and it leads to a stability result in a class of\n'sharp interface' initial data.\n