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 and 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.

[2]

H. Flemming and J. Wingender, The biofilm matrix, Nat. Rev. Microbiol., 8 (2010), 623-633. doi: 10.1038/nrmicro2415.

[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.

[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.

[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.

[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.

[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.

[8]

A. Friedman, A multiscale tumor model, Interfaces & Free Bound., 10 (2008), 245-262. doi: 10.4171/IFB/188.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

I. Klapper and J. Dockery, Mathematical description of microbial biofilms, SIAM Rev., 52 (2010), 221-265. doi: 10.1137/080739720.

[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.

[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.

[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.

[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.

[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.

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.

[2]

H. Flemming and J. Wingender, The biofilm matrix, Nat. Rev. Microbiol., 8 (2010), 623-633. doi: 10.1038/nrmicro2415.

[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.

[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.

[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.

[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.

[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.

[8]

A. Friedman, A multiscale tumor model, Interfaces & Free Bound., 10 (2008), 245-262. doi: 10.4171/IFB/188.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

I. Klapper and J. Dockery, Mathematical description of microbial biofilms, SIAM Rev., 52 (2010), 221-265. doi: 10.1137/080739720.

[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.

[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.

[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.

[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.

[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.

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