[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82723-en":3,"doc-seo-82723-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82723,4398048949847,"Eliana","https://ap-avatar.wpscdn.com/avatar/400002536579ef2da7f?_k=1778318612642679267",8,"Research & Report","An Onsager Variational Scheme for Pressure-Driven Tumor Growth and Hele–Shaw Limits","Pressure-driven tumor growth models couple cell proliferation to mechanical pressure and lead naturally to moving free boundary problems that are difficult to approximate numerically. Challenges arise from degenerate diffusion, pressure-dependent proliferation, and stiffness induced by the law p = n^γ for large γ. A structure-preserving finite difference method is introduced using an Onsager variational principle and a modified energy shifted by the homeostatic pressure. The resulting scheme yields nonnegativity, homeostatic upper bounds, discrete modified energy dissipation, and convergence toward the Hele–Shaw limit while capturing free boundary topology changes.","arXiv :2607 .03252v1 [math .NA] 3 Jul 2026  \nAn Onsager Variational Scheme for Pressure-Driven Tumor Growth and Hele–Shaw Limits  \nWeijie Huang∗ and Xinran Ruan†  \nAbstract  \nPressure-driven tumor growth models describe the coupling between cell proliferation and mechanical pressure and naturally lead to moving free boundary problems. Their numerical approximation is challenging due to degenerate diffusion, pressure-dependent proliferation, and the stiffness of the pressure law p = nγ for large γ . In this paper, we propose a structure-preserving finite difference method for this class of pressure-driven tumor growth models with pressure-dependent proliferation. The method is derived from the Onsager variational principle. The key idea is to introduce a modified energy shifted by the homeostatic pressure, so that the growth term can be written in a dissipative form and incorporated together with the transport part into a unified Rayleighian formulation. This formulation leads to a time-discrete constrained minimization problem and a fully discrete scheme with explicit mobilities and an implicit pressure update. We prove that the scheme preserves nonnegativity and the homeostatic upper bound, satisfies a discrete modified energy dissipation law, and admits a fixed-grid stiff-pressure limiting structure. Numerical experiments in one and two spatial dimensions demonstrate the accuracy of the method, its convergence toward the Hele–Shaw limit for large γ, and its ability to capture free boundary evolutions with topology changes.  \nKey words. Pressure-driven tumor growth, Onsager variational principle, structure-preserving scheme, energy stability, Hele–Shaw limit.  \nMSC 2020 . 65M06, 65M12, 35K65, 35R35, 92C50 .  \n1 Introduction  \nLiving tissues are active materials in which proliferation, mechanical pressure, and crowding effects are strongly coupled [1 , 11 , 25] . Continuum models for such mechanobiological processes must describe both bulk evolution and the motion of free boundaries separating occupied and empty regions, and, in the stiff-pressure regime, they naturally lead to singular limits in which strong mechanical resistance enforces an incompressibility constraint in densely packed tissues [22 , 23 , 24] . Tumor growth is a prototypical setting in which these mechanisms appear. For this reason, pressure-driven tumor growth models have become a useful class of problems for studying the connection between degenerate diffusion, free boundary motion, and reliable numerical simulation.  \nIn this paper, we consider the pressure-driven tumor growth model [23]  \n∂tn + ∇ · j = nG (p), j = −n∇p, p = nγ , (1.1)  \nposed in a bounded domain Ω ⊂ Rd. Here n = n (x, t) ≥ 0 denotes the tumor cell density, j = j (x, t) is the cell flux, p = p (x, t) is the mechanical pressure, and γ > 1 is the stiffness  \n∗ School of Mathematics and Statistics, Beijing Jiaotong University, Beijing 100044, People’s Republic of China; Beijing Key Laboratory of Biological Big Data and Topological Statistics, Beijing Jiaotong University, Beijing 100044, People’s Republic of China ([wjhuang@bjtu.edu.cn](wjhuang@bjtu.edu.cn) ).  \n†(Corresponding author) School of Mathematical Sciences, Capital Normal University, Beijing 100048, People’s Republic of China ([xinran.ruan@cnu.edu.cn](xinran.ruan@cnu.edu.cn) ).  \nparameter in the pressure law. The function G is the net proliferation rate. We assume pressureinhibited proliferation with a homeostatic pressure pH > 0 [1 , 22 , 23], namely  \nG (pH ) = 0, G′(p) ≤ 0, p ≥ 0. (1.2)  \nThis implies  \n(p − pH )G (p) ≤ 0, p ≥ 0 , (1.3)  \nwhich is the key structural condition behind the dissipative formulation used in this work.  \nFor fixed γ, (1.1) is a degenerate reaction-diffusion equation of porous-medium type. The degeneracy gives finite speed of propagation and allows the formation of sharp moving interfaces. As γ → ∞ , the pressure law enforces a congestion constraint and the pressure becomes a Lagrange multiplier for the saturated region","cbCaie8cYD0Vqq1Q","https://ap.wps.com/l/cbCaie8cYD0Vqq1Q","pdf",1355544,1,23,"English","en",105,"# Introduction\n## Problem setup and mechanobiological background\n## Pressure law stiff regime and Hele–Shaw limiting structure\n## Numerical challenges motivating a structure-preserving method\n## Contributions and main properties of the proposed scheme","[{\"question\":\"What model class is studied in the paper?\",\"answer\":\"The paper studies a pressure-driven tumor growth model with pressure-dependent net proliferation G(p) and a mechanical pressure law p = n^γ in a bounded domain.\"},{\"question\":\"Why does large γ create numerical difficulties?\",\"answer\":\"Large γ makes the pressure law stiff and enforces a congestion constraint, so numerical density must approach the limiting behavior without spurious overshoots while respecting structural bounds.\"},{\"question\":\"What are the main structural and stability properties of the proposed numerical scheme?\",\"answer\":\"The method preserves nonnegativity and the homeostatic upper bound, satisfies a discrete modified energy dissipation law, and admits a fixed-grid stiff-pressure limiting structure, with convergence toward the Hele–Shaw limit.\"}]",1784182500,58,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"an-onsager-variational-scheme-for-pressure-driven-tumor-growth-and-heleshaw-limits","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/an-onsager-variational-scheme-for-pressure-driven-tumor-growth-and-heleshaw-limits/82723/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What model class is studied in the paper?","Question",{"text":75,"@type":76},"The paper studies a pressure-driven tumor growth model with pressure-dependent net proliferation G(p) and a mechanical pressure law p = n^γ in a bounded domain.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why does large γ create numerical difficulties?",{"text":80,"@type":76},"Large γ makes the pressure law stiff and enforces a congestion constraint, so numerical density must approach the limiting behavior without spurious overshoots while respecting structural bounds.",{"name":82,"@type":73,"acceptedAnswer":83},"What are the main structural and stability properties of the proposed numerical scheme?",{"text":84,"@type":76},"The method preserves nonnegativity and the homeostatic upper bound, satisfies a discrete modified energy dissipation law, and admits a fixed-grid stiff-pressure limiting structure, with convergence toward the Hele–Shaw 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