{"publication_status":"published","publisher":"World Scientific Pub Co Pte Ltd","publication_identifier":{"issn":["0218-2025","1793-6314"]},"abstract":[{"text":"<jats:p> We introduce a new phase field model for tumor growth where viscoelastic effects are taken into account. The model is derived from basic thermodynamical principles and consists of a convected Cahn–Hilliard equation with source terms for the tumor cells and a convected reaction–diffusion equation with boundary supply for the nutrient. Chemotactic terms, which are essential for the invasive behavior of tumors, are taken into account. The model is completed by a viscoelastic system consisting of the Navier–Stokes equation for the hydrodynamic quantities, and a general constitutive equation with stress relaxation for the left Cauchy–Green tensor associated with the elastic part of the total mechanical response of the viscoelastic material. For a specific choice of the elastic energy density and with an additional dissipative term accounting for stress diffusion, we prove existence of global-in-time weak solutions of the viscoelastic model for tumor growth in two space dimensions [Formula: see text] by the passage to the limit in a fully-discrete finite element scheme where a CFL condition, i.e. [Formula: see text], is required. </jats:p><jats:p> Moreover, in arbitrary dimensions [Formula: see text], we show stability and existence of solutions for the fully-discrete finite element scheme, where positive definiteness of the discrete Cauchy–Green tensor is proved with a regularization technique that was first introduced by Barrett and Boyaval [Existence and approximation of a (regularized) Oldroyd-B model, Math. Models Methods Appl. Sci. 21 (2011) 1783–1837]. After that, we improve the regularity results in arbitrary dimensions [Formula: see text] and in two dimensions [Formula: see text], where a CFL condition is required. Then, in two dimensions [Formula: see text], we pass to the limit in the discretization parameters and show that subsequences of discrete solutions converge to a global-in-time weak solution. Finally, we present numerical results in two dimensions [Formula: see text]. </jats:p>","lang":"eng"}],"title":"Viscoelastic Cahn–Hilliard models for tumor growth","status":"public","doi":"10.1142/s0218202522500634","_id":"45970","intvolume":"        32","year":"2022","page":"2673-2758","type":"journal_article","issue":"13","publication":"Mathematical Models and Methods in Applied Sciences","department":[{"_id":"841"}],"author":[{"first_name":"Harald","full_name":"Garcke, Harald","last_name":"Garcke"},{"first_name":"Balázs","last_name":"Kovács","id":"100441","orcid":"0000-0001-9872-3474","full_name":"Kovács, Balázs"},{"last_name":"Trautwein","full_name":"Trautwein, Dennis","first_name":"Dennis"}],"volume":32,"keyword":["Applied Mathematics","Modeling and Simulation"],"citation":{"mla":"Garcke, Harald, et al. “Viscoelastic Cahn–Hilliard Models for Tumor Growth.” <i>Mathematical Models and Methods in Applied Sciences</i>, vol. 32, no. 13, World Scientific Pub Co Pte Ltd, 2022, pp. 2673–758, doi:<a href=\"https://doi.org/10.1142/s0218202522500634\">10.1142/s0218202522500634</a>.","apa":"Garcke, H., Kovács, B., &#38; Trautwein, D. (2022). Viscoelastic Cahn–Hilliard models for tumor growth. <i>Mathematical Models and Methods in Applied Sciences</i>, <i>32</i>(13), 2673–2758. <a href=\"https://doi.org/10.1142/s0218202522500634\">https://doi.org/10.1142/s0218202522500634</a>","short":"H. Garcke, B. Kovács, D. Trautwein, Mathematical Models and Methods in Applied Sciences 32 (2022) 2673–2758.","chicago":"Garcke, Harald, Balázs Kovács, and Dennis Trautwein. “Viscoelastic Cahn–Hilliard Models for Tumor Growth.” <i>Mathematical Models and Methods in Applied Sciences</i> 32, no. 13 (2022): 2673–2758. <a href=\"https://doi.org/10.1142/s0218202522500634\">https://doi.org/10.1142/s0218202522500634</a>.","ieee":"H. Garcke, B. Kovács, and D. Trautwein, “Viscoelastic Cahn–Hilliard models for tumor growth,” <i>Mathematical Models and Methods in Applied Sciences</i>, vol. 32, no. 13, pp. 2673–2758, 2022, doi: <a href=\"https://doi.org/10.1142/s0218202522500634\">10.1142/s0218202522500634</a>.","bibtex":"@article{Garcke_Kovács_Trautwein_2022, title={Viscoelastic Cahn–Hilliard models for tumor growth}, volume={32}, DOI={<a href=\"https://doi.org/10.1142/s0218202522500634\">10.1142/s0218202522500634</a>}, number={13}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World Scientific Pub Co Pte Ltd}, author={Garcke, Harald and Kovács, Balázs and Trautwein, Dennis}, year={2022}, pages={2673–2758} }","ama":"Garcke H, Kovács B, Trautwein D. Viscoelastic Cahn–Hilliard models for tumor growth. <i>Mathematical Models and Methods in Applied Sciences</i>. 2022;32(13):2673-2758. doi:<a href=\"https://doi.org/10.1142/s0218202522500634\">10.1142/s0218202522500634</a>"},"date_created":"2023-07-10T11:47:27Z","language":[{"iso":"eng"}],"user_id":"100441","date_updated":"2024-04-03T09:15:35Z"}