Abstract
We extend previous weak well-posedness results obtained in Frigeri et al. (2017, Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs, Vol. 22, Springer, Cham, pp. 217–254) concerning a non-local variant of a diffuse interface tumour model proposed by Hawkins-Daarud et al. (2012, Int. J. Numer. Method Biomed. Engng.28, 3–24). The model consists of a non-local Cahn–Hilliard equation with degenerate mobility and singular potential for the phase field variable, coupled to a reaction–diffusion equation for the concentration of a nutrient. We prove the existence of strong solutions to the model and establish some high-order continuous dependence estimates, even in the presence of concentration-dependent mobilities for the nutrient variable in two spatial dimensions. Then, we apply the new regularity results to study an inverse problem identifying the initial tumour distribution from measurements at the terminal time. Formulating the Tikhonov regularised inverse problem as a constrained minimisation problem, we establish the existence of minimisers and derive first-order necessary optimality conditions.
Highlights
We extend previous weak well-posedness results obtained in Frigeri et al (2017, Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs, Vol 22, Springer, Cham, pp. 217–254) concerning a non-local variant of a diffuse interface tumour model proposed by Hawkins-Daarud et al
We prove the existence of strong solutions to the model and establish some high-order continuous dependence estimates, even in the presence of concentration-dependent mobilities for the nutrient variable in two spatial dimensions
Mathematical modelling for tumour growth dynamics has undergone a swift development in the last decades
Summary
Mathematical modelling for tumour growth dynamics has undergone a swift development in the last decades (see for instance pioneering works such as [16, 17, 69]). Since the weak well-posedness to (1.6)–(1.8), which we collectively call (P), is a direct consequence of the main results of [41], the focus of this work is to show the existence of strong solutions using techniques inspired by [33] for the non-local Cahn–Hilliard–Navier–Stokes system In our setting, this involves a bootstrapping argument in which we first improve the regularity of φ by fixing σ and employing a time discretisation of (1.6a) and we improve the regularity of σ with the help of new regularities for φ. Thanks to the new solution regularities to (P), practitioners interested in solving the inverse identification problem (1.9) that involve the non-local tumour model (1.6) with degenerate mobility and singular potentials can first obtain numerical approximations of {φα0 }α>0 by solving the optimality conditions, and sending α → 0 in an appropriate way to deduce a solution to (1.9).
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