Asymptotic analyses and error estimates for a Cahn-Hilliard type phase field system modelling tumor growth

TL;DR

This paper analyzes a Cahn-Hilliard type phase field system modeling tumor growth, focusing on asymptotic behavior and error estimates as viscosity coefficients tend to zero, extending previous work on well-posedness and convergence.

Contribution

It provides new convergence results, solution uniqueness, and error estimates for the system when only one viscosity coefficient approaches zero, advancing understanding of the model's asymptotic limits.

Findings

Proved convergence of solutions as one viscosity coefficient tends to zero.

Established uniqueness of solutions for the limit problems.

Derived error estimates quantifying the approximation accuracy.

Abstract

This paper is concerned with a phase field system of Cahn-Hilliard type that is related to a tumor growth model and consists of three equations in terms of the variables order parameter, chemical potential and nutrient concentration. This system has been investigated in the recent contributions arXiv:1401.5943 [math.AP] and arXiv:1501.07057 [math.AP] from the viewpoint of well-posedness, long time behavior and asymptotic convergence as two positive viscosity coefficients tend to zero at the same time. Here, we continue the analysis performed in arXiv:1501.07057 [math.AP] by showing two independent sets of results as just one of the coefficients tends to zero, the other remaining fixed. We prove convergence results, uniqueness of solutions to the two resulting limit problems, and suitable error estimates.