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September  2016, 15(5): 1941-1974. doi: 10.3934/cpaa.2016023

The Hele-Shaw problem with surface tension in the case of subdiffusion

Nataliya Vasylyeva 1, and  Vitalii Overko 1,

1. 

Institute of Applied Mathematics and Mechanics of NAS of Ukraine, Dobrovolskogo, 1, Slov'iansk, 84100, Ukraine, Ukraine

Received  July 2015 Revised  February 2016 Published  July 2016

In this paper we analyze anomalous diffusion version of the multidimensional Hele-Shaw problem with a nonzero surface tension of a free boundary. We prove the one-valued solvability of this moving boundary problem in the Hölder classes. In the two-dimensional case some numerical solutions are constructed.
Keywords: regularizer, anomalous diffusion, coercive estimates., Quasistationary Stefan problem, Caputo derivative.
Mathematics Subject Classification: Primary: 35R35, 35C15; Secondary: 35B65, 35R1.
Citation: Nataliya Vasylyeva, Vitalii Overko. The Hele-Shaw problem with surface tension in the case of subdiffusion. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1941-1974. doi: 10.3934/cpaa.2016023
References:
[1]

C. Atkinson, Moving boundary problems for time fractional and composition dependent diffusion,, \emph{Frac. Calcul. and Appl. Analysis}, 15 (2012), 207.  doi: 10.2478/s13540-012-0015-2.  Google Scholar

[2]

B. V. Bazaliy, Stefan problem for the Laplace equation with regard for the curvature of the free boundary,, \emph{Ukr. Math. J.}, 40 (1997), 1465.   Google Scholar

[3]

B. V. Bazaliy and A. Friedman, The Hele-Shaw problem with surface tension in a half-plane: a model problem,, \emph{J. Diff. Equs.}, 216 (2005), 387.  doi: 10.1016/j.jde.2005.03.007.  Google Scholar

[4]

B. V. Bazaliy and A. Friedman, The Hele-Shaw problem with surface tension in a half-plane,, \emph{J. Diff. Equs.}, 216 (2005), 439.  doi: 10.1016/j.jde.2005.03.017.  Google Scholar

[5]

B. V. Bazaliy and N. Vasylyeva, The two-phase Hele-Shaw problem with a nonregular initial interface and without surface tension,, \emph{J. Math. Phys., 10 (2014), 3.   Google Scholar

[6]

J.-P. Bouchard and A. Georges, Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications,, \emph{Phys. Rep.}, 195 (1990), 127.  doi: 10.1016/0370-1573(90)90099-N.  Google Scholar

[7]

A. Cartea and D. del Castillo-Negrete, Fractional diffusion models of option prices in markets with jumps,, \emph{Physica A}, 374 (2007), 749.   Google Scholar

[8]

X. Chen, The Hele-Shaw problem and area preserving curve shortening motions,, \emph{Arch. Rat. Mech. Anal.}, 123 (1993), 117.  doi: 10.1007/BF00695274.  Google Scholar

[9]

S. Chen, B. Merriman, S. Osher and P. Smereka, A simple level set method for solving Stefan problems,, \emph{J. Comput. Phys.}, 135 (1997), 8.  doi: 10.1006/jcph.1997.5721.  Google Scholar

[10]

X. Chen and F. Reitich, Local existence and uniqueness of solutions of the Stefan problem with surface tension and kinetic indercooling,, \emph{J. Math. Anal. Appl.}, 164 (1992), 350.  doi: 10.1016/0022-247X(92)90119-X.  Google Scholar

[11]

K. Diethelm, N. J. Ford and A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations,, \emph{Nonlinear Dynam.}, 29 (2002), 3.  doi: 10.1023/A:1016592219341.  Google Scholar

[12]

G. Drazer and D. H. Zanette, Experimental evidence of power-law trapping-time distributions in porous media,, \emph{Phys. Rev. E}, 60 (1999), 5858.   Google Scholar

[13]

A. Erdélyi, Higher Transcendental Functions v.3,, Mc Graw-Hill, (1955).   Google Scholar

[14]

J. Escher and G. Simonett, Classical solutions multidimensional Hele-Shaw models,, \emph{SIAM J. Math. Anal.}, 28 (1997), 1028.  doi: 10.1137/S0036141095291919.  Google Scholar

[15]

A. Friedman and F. Reitich, Analysis of a mathematical model for the growth of tumors,, \emph{J. Math. Bio.}, 38 (1999), 262.  doi: 10.1007/s002850050149.  Google Scholar

[16]

D. Gilbarg and D. Ttrudinger, Elliptic Partial Differential Equations of Second Order,, Springer-Verlag Berlin Heidelberg, (1989).   Google Scholar

[17]

E. I. Hanzava, Classical solution of the Stefan problem,, \emph{Tohoku Math. J.}, 33 (1981), 297.   Google Scholar

[18]

H. S. Hele-Shaw, The flow of water,, \emph{Nature}, 58 (1898), 34.   Google Scholar

[19]

R. Hilfer, Applications of Fractional Calculus in Physics,, World Scientific, (2000).  doi: 10.1142/9789812817747.  Google Scholar

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S. D. Howison, Bibliography of free an moving boundary problems in Hele-Shaw and Stokes flow,, http://www.maths.ox.ac.uk/howison/Hele-Shaw, (2006).   Google Scholar

[21]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations,, North-Holland Mathematics Studies, (2006).   Google Scholar

[22]

A. N. Kochubei, Fractional-parabolic systems,, \emph{Potential Analysis}, 37 (2012), 1.  doi: 10.1007/s11118-011-9243-z.  Google Scholar

[23]

M. Krasnoschok and N. Vasylyeva, Existence and uniqueness of the solutions for some initial-boundary value problems with the fractional dynamic boundary condition,, \emph{Inter. J. PDE}, 2013 (2013).   Google Scholar

[24]

O. A. Ladyzhenskaia, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasilinear Parabolic Equations,, Academic Press, (1968).   Google Scholar

[25]

O. A. Ladyzhenskaia and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations,, Academic Press, (1968).   Google Scholar

[26]

T. A. M. Langlands and B. I. Henry, Fractional chemotaxis diffusion equation,, \emph{Physical Review E}, 81 (2010).  doi: 10.1103/PhysRevE.81.051102.  Google Scholar

[27]

B.-T. Liu and J.-P. Hsu, Theoretical analysis on diffusional release from ellipsoidal drug delivery devices,, \emph{Chem. Eng. Sci.}, 61 (2006), 1748.   Google Scholar

[28]

J. Y. Liu and M. Xu, An exact solution to the moving boundary problem with fractional anomalous diffusion in drug release devices,, \emph{Z. Angew. Math. Mech.}, 84 (2004), 22.  doi: 10.1002/zamm.200410074.  Google Scholar

[29]

J. Y. Liu and M. Xu, Some exact solutions to Stefan problems with fractional differential equations,, \emph{J. Math. Anal. Appl.}, 351 (2009), 536.  doi: 10.1016/j.jmaa.2008.10.042.  Google Scholar

[30]

A. Logg, K. A. Mardal and G. Wells eds., Automated Solution of Differential Equations by the Finite Element Method: The Fenics Book,, \textbf{84}, 84 (2012).  doi: 10.1007/978-3-642-23099-8.  Google Scholar

[31]

A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems,, \emph{Progress in Nonlinear Differential Equations and their Applications 16}, (1995).  doi: 10.1007/978-3-0348-9234-6.  Google Scholar

[32]

F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics,, in \emph{Fractals and Fractional Calculus in Continuum Mechanics} (A. Garpinteri and F. Mainardi Eds.), (1997), 291.  doi: 10.1007/978-3-7091-2664-6_7.  Google Scholar

[33]

G. Prokert, Existence results for Hele-Shaw problem driven by surface tension,, \emph{European J. Appl. Math.}, 9 (1998), 195.  doi: 10.1017/S0956792597003276.  Google Scholar

[34]

A. V. Pskhu, Partial Differential Equations of the Fractional Order,, Nauka, (2005).   Google Scholar

[35]

A. V. Pskhu, The fundamental solution of a diffusion-wave equation of fractional order,, \emph{Izvestia RAN}, 73 (2009), 141.  doi: 10.1070/IM2009v073n02ABEH002450.  Google Scholar

[36]

S. Z. Rida, A. M. A. El-Sayed and A. A. M. Arafa, Effect of bacterial memory dependent growth by using fractional derivatives reaction-diffusion chemotactic model,, \emph{J. Statistical Physics}, 140 (2010), 797.  doi: 10.1007/s10955-010-0007-8.  Google Scholar

[37]

W. Tan, C. Fu, C. Fu, W. Xie and H. Cheng, An anomalous subdiffusion model for calcium spark in cardiac myocytes,, \emph{Appl. Phys. Letter}, 91 (2007).   Google Scholar

[38]

V. R. Voller, An exact solution of a limit case Stefan problem governed by a fractional diffusion equation,, \emph{Internat. J. Heat and Mass Transf.}, 53 (2010), 5622.   Google Scholar

[39]

V. R. Voller, F. Falcini and R. Garra, Fractional Stefan problems exibiting lumped and distributed latent-heat memory effects,, \emph{Phys. Rew. E}, 87 (2013).   Google Scholar

[40]

V. R. Voller, An overview of numerical methods for solving phase change problems,, in \emph{Advances in Numerical Heat Transfer} (W. J. Minkowycz and E. M. Sparrow Eds.), 1 (1996), 341.   Google Scholar

[41]

N. Vasylyeva and L. Vynnytska, On a multidimensional moving boundary problem governed by anomalous diffusion: analytical and numerical study,, \emph{Nonlinear Differ. Equ. Appl.}, 22 (2015), 543.  doi: 10.1007/s00030-014-0295-9.  Google Scholar

[42]

N. Vasylyeva, Local solvability of a linear system with a fractional derivative in time in a boundary condition,, \emph{Frac. Calc. Appl. Anal.}, 18 (2015), 982.  doi: 10.1515/fca-2015-0058.  Google Scholar

[43]

N. Vasylyeva, On a local solvability of the multidimensional Muscat problem with a fractional derivative in time on the boundary condition,, \emph{J. Frac. Differ. Calc.}, 4 (2014), 80.  doi: 10.7153/fdc-04-06.  Google Scholar

show all references

References:
[1]

C. Atkinson, Moving boundary problems for time fractional and composition dependent diffusion,, \emph{Frac. Calcul. and Appl. Analysis}, 15 (2012), 207.  doi: 10.2478/s13540-012-0015-2.  Google Scholar

[2]

B. V. Bazaliy, Stefan problem for the Laplace equation with regard for the curvature of the free boundary,, \emph{Ukr. Math. J.}, 40 (1997), 1465.   Google Scholar

[3]

B. V. Bazaliy and A. Friedman, The Hele-Shaw problem with surface tension in a half-plane: a model problem,, \emph{J. Diff. Equs.}, 216 (2005), 387.  doi: 10.1016/j.jde.2005.03.007.  Google Scholar

[4]

B. V. Bazaliy and A. Friedman, The Hele-Shaw problem with surface tension in a half-plane,, \emph{J. Diff. Equs.}, 216 (2005), 439.  doi: 10.1016/j.jde.2005.03.017.  Google Scholar

[5]

B. V. Bazaliy and N. Vasylyeva, The two-phase Hele-Shaw problem with a nonregular initial interface and without surface tension,, \emph{J. Math. Phys., 10 (2014), 3.   Google Scholar

[6]

J.-P. Bouchard and A. Georges, Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications,, \emph{Phys. Rep.}, 195 (1990), 127.  doi: 10.1016/0370-1573(90)90099-N.  Google Scholar

[7]

A. Cartea and D. del Castillo-Negrete, Fractional diffusion models of option prices in markets with jumps,, \emph{Physica A}, 374 (2007), 749.   Google Scholar

[8]

X. Chen, The Hele-Shaw problem and area preserving curve shortening motions,, \emph{Arch. Rat. Mech. Anal.}, 123 (1993), 117.  doi: 10.1007/BF00695274.  Google Scholar

[9]

S. Chen, B. Merriman, S. Osher and P. Smereka, A simple level set method for solving Stefan problems,, \emph{J. Comput. Phys.}, 135 (1997), 8.  doi: 10.1006/jcph.1997.5721.  Google Scholar

[10]

X. Chen and F. Reitich, Local existence and uniqueness of solutions of the Stefan problem with surface tension and kinetic indercooling,, \emph{J. Math. Anal. Appl.}, 164 (1992), 350.  doi: 10.1016/0022-247X(92)90119-X.  Google Scholar

[11]

K. Diethelm, N. J. Ford and A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations,, \emph{Nonlinear Dynam.}, 29 (2002), 3.  doi: 10.1023/A:1016592219341.  Google Scholar

[12]

G. Drazer and D. H. Zanette, Experimental evidence of power-law trapping-time distributions in porous media,, \emph{Phys. Rev. E}, 60 (1999), 5858.   Google Scholar

[13]

A. Erdélyi, Higher Transcendental Functions v.3,, Mc Graw-Hill, (1955).   Google Scholar

[14]

J. Escher and G. Simonett, Classical solutions multidimensional Hele-Shaw models,, \emph{SIAM J. Math. Anal.}, 28 (1997), 1028.  doi: 10.1137/S0036141095291919.  Google Scholar

[15]

A. Friedman and F. Reitich, Analysis of a mathematical model for the growth of tumors,, \emph{J. Math. Bio.}, 38 (1999), 262.  doi: 10.1007/s002850050149.  Google Scholar

[16]

D. Gilbarg and D. Ttrudinger, Elliptic Partial Differential Equations of Second Order,, Springer-Verlag Berlin Heidelberg, (1989).   Google Scholar

[17]

E. I. Hanzava, Classical solution of the Stefan problem,, \emph{Tohoku Math. J.}, 33 (1981), 297.   Google Scholar

[18]

H. S. Hele-Shaw, The flow of water,, \emph{Nature}, 58 (1898), 34.   Google Scholar

[19]

R. Hilfer, Applications of Fractional Calculus in Physics,, World Scientific, (2000).  doi: 10.1142/9789812817747.  Google Scholar

[20]

S. D. Howison, Bibliography of free an moving boundary problems in Hele-Shaw and Stokes flow,, http://www.maths.ox.ac.uk/howison/Hele-Shaw, (2006).   Google Scholar

[21]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations,, North-Holland Mathematics Studies, (2006).   Google Scholar

[22]

A. N. Kochubei, Fractional-parabolic systems,, \emph{Potential Analysis}, 37 (2012), 1.  doi: 10.1007/s11118-011-9243-z.  Google Scholar

[23]

M. Krasnoschok and N. Vasylyeva, Existence and uniqueness of the solutions for some initial-boundary value problems with the fractional dynamic boundary condition,, \emph{Inter. J. PDE}, 2013 (2013).   Google Scholar

[24]

O. A. Ladyzhenskaia, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasilinear Parabolic Equations,, Academic Press, (1968).   Google Scholar

[25]

O. A. Ladyzhenskaia and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations,, Academic Press, (1968).   Google Scholar

[26]

T. A. M. Langlands and B. I. Henry, Fractional chemotaxis diffusion equation,, \emph{Physical Review E}, 81 (2010).  doi: 10.1103/PhysRevE.81.051102.  Google Scholar

[27]

B.-T. Liu and J.-P. Hsu, Theoretical analysis on diffusional release from ellipsoidal drug delivery devices,, \emph{Chem. Eng. Sci.}, 61 (2006), 1748.   Google Scholar

[28]

J. Y. Liu and M. Xu, An exact solution to the moving boundary problem with fractional anomalous diffusion in drug release devices,, \emph{Z. Angew. Math. Mech.}, 84 (2004), 22.  doi: 10.1002/zamm.200410074.  Google Scholar

[29]

J. Y. Liu and M. Xu, Some exact solutions to Stefan problems with fractional differential equations,, \emph{J. Math. Anal. Appl.}, 351 (2009), 536.  doi: 10.1016/j.jmaa.2008.10.042.  Google Scholar

[30]

A. Logg, K. A. Mardal and G. Wells eds., Automated Solution of Differential Equations by the Finite Element Method: The Fenics Book,, \textbf{84}, 84 (2012).  doi: 10.1007/978-3-642-23099-8.  Google Scholar

[31]

A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems,, \emph{Progress in Nonlinear Differential Equations and their Applications 16}, (1995).  doi: 10.1007/978-3-0348-9234-6.  Google Scholar

[32]

F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics,, in \emph{Fractals and Fractional Calculus in Continuum Mechanics} (A. Garpinteri and F. Mainardi Eds.), (1997), 291.  doi: 10.1007/978-3-7091-2664-6_7.  Google Scholar

[33]

G. Prokert, Existence results for Hele-Shaw problem driven by surface tension,, \emph{European J. Appl. Math.}, 9 (1998), 195.  doi: 10.1017/S0956792597003276.  Google Scholar

[34]

A. V. Pskhu, Partial Differential Equations of the Fractional Order,, Nauka, (2005).   Google Scholar

[35]

A. V. Pskhu, The fundamental solution of a diffusion-wave equation of fractional order,, \emph{Izvestia RAN}, 73 (2009), 141.  doi: 10.1070/IM2009v073n02ABEH002450.  Google Scholar

[36]

S. Z. Rida, A. M. A. El-Sayed and A. A. M. Arafa, Effect of bacterial memory dependent growth by using fractional derivatives reaction-diffusion chemotactic model,, \emph{J. Statistical Physics}, 140 (2010), 797.  doi: 10.1007/s10955-010-0007-8.  Google Scholar

[37]

W. Tan, C. Fu, C. Fu, W. Xie and H. Cheng, An anomalous subdiffusion model for calcium spark in cardiac myocytes,, \emph{Appl. Phys. Letter}, 91 (2007).   Google Scholar

[38]

V. R. Voller, An exact solution of a limit case Stefan problem governed by a fractional diffusion equation,, \emph{Internat. J. Heat and Mass Transf.}, 53 (2010), 5622.   Google Scholar

[39]

V. R. Voller, F. Falcini and R. Garra, Fractional Stefan problems exibiting lumped and distributed latent-heat memory effects,, \emph{Phys. Rew. E}, 87 (2013).   Google Scholar

[40]

V. R. Voller, An overview of numerical methods for solving phase change problems,, in \emph{Advances in Numerical Heat Transfer} (W. J. Minkowycz and E. M. Sparrow Eds.), 1 (1996), 341.   Google Scholar

[41]

N. Vasylyeva and L. Vynnytska, On a multidimensional moving boundary problem governed by anomalous diffusion: analytical and numerical study,, \emph{Nonlinear Differ. Equ. Appl.}, 22 (2015), 543.  doi: 10.1007/s00030-014-0295-9.  Google Scholar

[42]

N. Vasylyeva, Local solvability of a linear system with a fractional derivative in time in a boundary condition,, \emph{Frac. Calc. Appl. Anal.}, 18 (2015), 982.  doi: 10.1515/fca-2015-0058.  Google Scholar

[43]

N. Vasylyeva, On a local solvability of the multidimensional Muscat problem with a fractional derivative in time on the boundary condition,, \emph{J. Frac. Differ. Calc.}, 4 (2014), 80.  doi: 10.7153/fdc-04-06.  Google Scholar

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