# American Institute of Mathematical Sciences

July  2016, 21(5): 1455-1468. doi: 10.3934/dcdsb.2016006

## Free boundary problems for systems of Stokes equations

 1 The Ohio State University, Department of Mathematics, Columbus, OH 43210, United States

Received  May 2015 Revised  November 2015 Published  April 2016

Recent years have seen a dramatic increase in the number and variety of new mathematical models describing biological processes. Some of these models are formulated as free boundary problems for systems of PDEs. Relevant biological questions give rise to interesting mathematical questions regarding properties of the solutions. In this review we focus on models whose formulation includes Stokes equations. They arise in describing the evolution of tumors, both at the macroscopic and molecular levels, in wound healing of cutaneous wounds, and in biofilms. We state recent results and formulate some open problems.
Citation: Avner Friedman. Free boundary problems for systems of Stokes equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1455-1468. doi: 10.3934/dcdsb.2016006
##### References:
 [1] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.  Google Scholar [2] H. Flemming and J. Wingender, The biofilm matrix, Nat. Rev. Microbiol., 8 (2010), 623-633. doi: 10.1038/nrmicro2415.  Google Scholar [3] S. J. H. Franks, H. M. Byrne, J. P. King, J. C. E. Underwood and C. E. Lewis, Modelling the early growth of ductal carcinoma in situ of the breast, J. Math. Biol., 47 (2003), 424-452. doi: 10.1007/s00285-003-0214-x.  Google Scholar [4] S. J. H. Franks, H. M. Byrne, J. C. E. Underwood and C. E. Lewis, Biological inferences from a mathematical model of comedo ductal carcinoma in situ of the breast, J. Theoret. Biol., 232 (2005), 523-543. doi: 10.1016/j.jtbi.2004.08.032.  Google Scholar [5] S. J. H. Franks, H. M. Byrne, J. P. King, J. C. E. Underwood and C. E. Lewis, Modelling the growth of ductal carcinoma in situ, Math. Med. Biol., 20 (2003), 277-308. Google Scholar [6] S. J. H. Franks and J. P. King, Interaction between a uniformly proliferating tumor and its surroundings: Uniform material properties, Math. Med. Biol., 20 (2003), 47-89. Google Scholar [7] A. Friedman, A free boundary problem for a coupled system of elliptic, hyperbolic, and Stokes equations modeling tumor growth, Interfaces& Free Bound., 8 (2006), 247-261. doi: 10.4171/IFB/142.  Google Scholar [8] A. Friedman, A multiscale tumor model, Interfaces & Free Bound., 10 (2008), 245-262. doi: 10.4171/IFB/188.  Google Scholar [9] A. Friedman, Free boundary value problems associated with multiscale tumor models, Mathematical Modeling of Natural Phenomena, 4 (2009), 134-155. doi: 10.1051/mmnp/20094306.  Google Scholar [10] A. Friedman and B. Hu, Bifurcation from stability to instability for a free boundary problem arising in tumor model, Arch. Rat. Mech. Anal., 180 (2006), 293-330. doi: 10.1007/s00205-005-0408-z.  Google Scholar [11] A. Friedman and B. Hu, Asymptotic stability for a free boundary problem arising in a tumor model, J. Diff. Eqs., 227 (2006), 598-639. doi: 10.1016/j.jde.2005.09.008.  Google Scholar [12] A. Friedman and B. Hu, Bifurcation from stability to instability for a free boundary problem modeling tumor growth by Stokes equation, Math. Anal & Appl., 327 (2007), 643-664. doi: 10.1016/j.jmaa.2006.04.034.  Google Scholar [13] A. Friedman and B. Hu, Bifurcation for a free boundary problem modeling tumor growth by Stokes equation, SIAM J. Math. Anal., 39 (2007), 174-194. doi: 10.1137/060656292.  Google Scholar [14] A. Friedman, B. Hu and C-Y Kao, Cell cycle control at the first restriction point and its effect on tissue growth, J. Math. Biol., 60 (2010), 881-907. doi: 10.1007/s00285-009-0290-7.  Google Scholar [15] A. Friedman, B. Hu and C. Xue, A three dimensional model of wound healing: Analysis and computation, Disc. Cont. Dynam. Syst., 17 (2012), 2691-2712. doi: 10.3934/dcdsb.2012.17.2691.  Google Scholar [16] A. Friedman, B. Hu and C. Xue, On multiphase multicomponent model of biofilm growth, Archive Rat. Mech. Anal., 211 (2014), 257-300. doi: 10.1007/s00205-013-0665-1.  Google Scholar [17] A. Friedman and C. Xue, A mathematical model for chronic wounds, Mathematical Biosciences and Engineering, 8 (2011), 253-261. doi: 10.3934/mbe.2011.8.253.  Google Scholar [18] I. Klapper and J. Dockery, Mathematical description of microbial biofilms, SIAM Rev., 52 (2010), 221-265. doi: 10.1137/080739720.  Google Scholar [19] L. Ma, M. Conover, H. Lu, M. R. Parsek, K. Bayles and D. J. Wozniak, Assembly and development of the Pseudomonas aeruginosa biofilm matrix, PLoS Pathog., 5 (2009), e1000, 354. doi: doi:10.1371/journal.ppat.1000354.  Google Scholar [20] V. Solonnikov, On quasistationary approximation in the problem of motion of a capillary drop, Progress in Nonlinear Differential Equations and Their Applications, 35 (1999), 643-671.  Google Scholar [21] V. Solonnikov, On the quasistationary approximation in the problem of evolution of an isolated liquid mass, Proceedings of International Conference on: Free Boundary Problems, Theory and Applications, 13 (2000), 327-342. Gakkōtosho.  Google Scholar [22] Q. Wang and T. Zhang, Review of mathematical models for biofilms, Solid State Commun., 150 (2010), 1009-1022. doi: 10.1016/j.ssc.2010.01.021.  Google Scholar [23] C. Xue, A. Friedman and C. K. Sen, A mathematical model of ischemic cutaneous wounds, PNAS, 106 (2009), 16782-16787. doi: 10.1073/pnas.0909115106.  Google Scholar

show all references

##### References:
 [1] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.  Google Scholar [2] H. Flemming and J. Wingender, The biofilm matrix, Nat. Rev. Microbiol., 8 (2010), 623-633. doi: 10.1038/nrmicro2415.  Google Scholar [3] S. J. H. Franks, H. M. Byrne, J. P. King, J. C. E. Underwood and C. E. Lewis, Modelling the early growth of ductal carcinoma in situ of the breast, J. Math. Biol., 47 (2003), 424-452. doi: 10.1007/s00285-003-0214-x.  Google Scholar [4] S. J. H. Franks, H. M. Byrne, J. C. E. Underwood and C. E. Lewis, Biological inferences from a mathematical model of comedo ductal carcinoma in situ of the breast, J. Theoret. Biol., 232 (2005), 523-543. doi: 10.1016/j.jtbi.2004.08.032.  Google Scholar [5] S. J. H. Franks, H. M. Byrne, J. P. King, J. C. E. Underwood and C. E. Lewis, Modelling the growth of ductal carcinoma in situ, Math. Med. Biol., 20 (2003), 277-308. Google Scholar [6] S. J. H. Franks and J. P. King, Interaction between a uniformly proliferating tumor and its surroundings: Uniform material properties, Math. Med. Biol., 20 (2003), 47-89. Google Scholar [7] A. Friedman, A free boundary problem for a coupled system of elliptic, hyperbolic, and Stokes equations modeling tumor growth, Interfaces& Free Bound., 8 (2006), 247-261. doi: 10.4171/IFB/142.  Google Scholar [8] A. Friedman, A multiscale tumor model, Interfaces & Free Bound., 10 (2008), 245-262. doi: 10.4171/IFB/188.  Google Scholar [9] A. Friedman, Free boundary value problems associated with multiscale tumor models, Mathematical Modeling of Natural Phenomena, 4 (2009), 134-155. doi: 10.1051/mmnp/20094306.  Google Scholar [10] A. Friedman and B. Hu, Bifurcation from stability to instability for a free boundary problem arising in tumor model, Arch. Rat. Mech. Anal., 180 (2006), 293-330. doi: 10.1007/s00205-005-0408-z.  Google Scholar [11] A. Friedman and B. Hu, Asymptotic stability for a free boundary problem arising in a tumor model, J. Diff. Eqs., 227 (2006), 598-639. doi: 10.1016/j.jde.2005.09.008.  Google Scholar [12] A. Friedman and B. Hu, Bifurcation from stability to instability for a free boundary problem modeling tumor growth by Stokes equation, Math. Anal & Appl., 327 (2007), 643-664. doi: 10.1016/j.jmaa.2006.04.034.  Google Scholar [13] A. Friedman and B. Hu, Bifurcation for a free boundary problem modeling tumor growth by Stokes equation, SIAM J. Math. Anal., 39 (2007), 174-194. doi: 10.1137/060656292.  Google Scholar [14] A. Friedman, B. Hu and C-Y Kao, Cell cycle control at the first restriction point and its effect on tissue growth, J. Math. Biol., 60 (2010), 881-907. doi: 10.1007/s00285-009-0290-7.  Google Scholar [15] A. Friedman, B. Hu and C. Xue, A three dimensional model of wound healing: Analysis and computation, Disc. Cont. Dynam. Syst., 17 (2012), 2691-2712. doi: 10.3934/dcdsb.2012.17.2691.  Google Scholar [16] A. Friedman, B. Hu and C. Xue, On multiphase multicomponent model of biofilm growth, Archive Rat. Mech. Anal., 211 (2014), 257-300. doi: 10.1007/s00205-013-0665-1.  Google Scholar [17] A. Friedman and C. Xue, A mathematical model for chronic wounds, Mathematical Biosciences and Engineering, 8 (2011), 253-261. doi: 10.3934/mbe.2011.8.253.  Google Scholar [18] I. Klapper and J. Dockery, Mathematical description of microbial biofilms, SIAM Rev., 52 (2010), 221-265. doi: 10.1137/080739720.  Google Scholar [19] L. Ma, M. Conover, H. Lu, M. R. Parsek, K. Bayles and D. J. Wozniak, Assembly and development of the Pseudomonas aeruginosa biofilm matrix, PLoS Pathog., 5 (2009), e1000, 354. doi: doi:10.1371/journal.ppat.1000354.  Google Scholar [20] V. Solonnikov, On quasistationary approximation in the problem of motion of a capillary drop, Progress in Nonlinear Differential Equations and Their Applications, 35 (1999), 643-671.  Google Scholar [21] V. Solonnikov, On the quasistationary approximation in the problem of evolution of an isolated liquid mass, Proceedings of International Conference on: Free Boundary Problems, Theory and Applications, 13 (2000), 327-342. Gakkōtosho.  Google Scholar [22] Q. Wang and T. Zhang, Review of mathematical models for biofilms, Solid State Commun., 150 (2010), 1009-1022. doi: 10.1016/j.ssc.2010.01.021.  Google Scholar [23] C. Xue, A. Friedman and C. K. Sen, A mathematical model of ischemic cutaneous wounds, PNAS, 106 (2009), 16782-16787. doi: 10.1073/pnas.0909115106.  Google Scholar
 [1] Shihe Xu. Analysis of a delayed free boundary problem for tumor growth. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 293-308. doi: 10.3934/dcdsb.2011.15.293 [2] Zejia Wang, Suzhen Xu, Huijuan Song. Stationary solutions of a free boundary problem modeling growth of angiogenesis tumor with inhibitor. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2593-2605. doi: 10.3934/dcdsb.2018129 [3] Shihe Xu, Yinhui Chen, Meng Bai. Analysis of a free boundary problem for avascular tumor growth with a periodic supply of nutrients. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 997-1008. doi: 10.3934/dcdsb.2016.21.997 [4] Avner Friedman, Bei Hu, Chuan Xue. A three dimensional model of wound healing: Analysis and computation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (8) : 2691-2712. doi: 10.3934/dcdsb.2012.17.2691 [5] Haiyan Wang, Shiliang Wu. Spatial dynamics for a model of epidermal wound healing. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1215-1227. doi: 10.3934/mbe.2014.11.1215 [6] Sophia A. Maggelakis. Modeling the role of angiogenesis in epidermal wound healing. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 267-273. doi: 10.3934/dcdsb.2004.4.267 [7] Junde Wu, Shangbin Cui. Asymptotic behavior of solutions for parabolic differential equations with invariance and applications to a free boundary problem modeling tumor growth. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 737-765. doi: 10.3934/dcds.2010.26.737 [8] Jian-Guo Liu, Min Tang, Li Wang, Zhennan Zhou. Analysis and computation of some tumor growth models with nutrient: From cell density models to free boundary dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3011-3035. doi: 10.3934/dcdsb.2018297 [9] Shihe Xu, Meng Bai, Fangwei Zhang. Analysis of a free boundary problem for tumor growth with Gibbs-Thomson relation and time delays. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3535-3551. doi: 10.3934/dcdsb.2017213 [10] John A. Adam. Inside mathematical modeling: building models in the context of wound healing in bone. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 1-24. doi: 10.3934/dcdsb.2004.4.1 [11] Junde Wu, Shangbin Cui. Asymptotic behavior of solutions of a free boundary problem modelling the growth of tumors with Stokes equations. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 625-651. doi: 10.3934/dcds.2009.24.625 [12] Avner Friedman. Free boundary problems arising in biology. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 193-202. doi: 10.3934/dcdsb.2018013 [13] Yaodan Huang, Zhengce Zhang, Bei Hu. Bifurcation from stability to instability for a free boundary tumor model with angiogenesis. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2473-2510. doi: 10.3934/dcds.2019105 [14] Serena Dipierro, Enrico Valdinoci. (Non)local and (non)linear free boundary problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 465-476. doi: 10.3934/dcdss.2018025 [15] Noriaki Yamazaki. Almost periodicity of solutions to free boundary problems. Conference Publications, 2001, 2001 (Special) : 386-397. doi: 10.3934/proc.2001.2001.386 [16] Jiayue Zheng, Shangbin Cui. Bifurcation analysis of a tumor-model free boundary problem with a nonlinear boundary condition. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4397-4410. doi: 10.3934/dcdsb.2020103 [17] Junde Wu. Bifurcation for a free boundary problem modeling the growth of necrotic multilayered tumors. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3399-3411. doi: 10.3934/dcds.2019140 [18] Weiqing Xie. A free boundary problem arising from the process of Czochralski crystal growth. Conference Publications, 2001, 2001 (Special) : 380-385. doi: 10.3934/proc.2001.2001.380 [19] Hantaek Bae. Solvability of the free boundary value problem of the Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 769-801. doi: 10.3934/dcds.2011.29.769 [20] Boris Muha, Zvonimir Tutek. Note on evolutionary free piston problem for Stokes equations with slip boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1629-1639. doi: 10.3934/cpaa.2014.13.1629

2020 Impact Factor: 1.327

## Metrics

• PDF downloads (194)
• HTML views (0)
• Cited by (5)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]