2017/06/25 by Pierluigi Colli, Colli, Pierluigi, Michele Colturato +1
Engineering · Materials Science · Mathematics · #35D30 #35K20 #35K61 #80A22 #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Solidification and crystal growth phenomena #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1706.08108
openalex publication_date 2017/06/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the present contribution we consider a singular phase field system located\nin a smooth and bounded three-dimensional domain. The entropy balance equation\nis perturbed by a logarithmic nonlinearity and by the presence of an additional\nterm involving a possibly nonlocal maximal monotone operator and arising from a\nclass of sliding mode control problems. The second equation of the system\naccounts for the phase dynamics, and it is deduced from a balance law for the\nmicroscopic forces that are responsible for the phase transition process. The\nresulting system is highly nonlinear; the main difficulties lie in the\ncontemporary presence of two nonlinearities, one of which under time\nderivative, in the entropy balance equation. Consequently, we are able to prove\nonly the existence of solutions. To this aim, we will introduce a backward\nfinite differences scheme and argue on this by proving uniform estimates and\npassing to the limit on the time step.\n