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.  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.  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.  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.  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.   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.   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.  doi: 10.4171/IFB/142.  Google Scholar

[8]

A. Friedman, A multiscale tumor model,, Interfaces & Free Bound., 10 (2008), 245.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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).  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.   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, 13 (2000), 327.   Google Scholar

[22]

Q. Wang and T. Zhang, Review of mathematical models for biofilms,, Solid State Commun., 150 (2010), 1009.  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.  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.  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.  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.  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.  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.   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.   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.  doi: 10.4171/IFB/142.  Google Scholar

[8]

A. Friedman, A multiscale tumor model,, Interfaces & Free Bound., 10 (2008), 245.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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).  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.   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, 13 (2000), 327.   Google Scholar

[22]

Q. Wang and T. Zhang, Review of mathematical models for biofilms,, Solid State Commun., 150 (2010), 1009.  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.  doi: 10.1073/pnas.0909115106.  Google Scholar

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