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

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

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.
Citation: Nataliya Vasylyeva, Vitalii Overko. The Hele-Shaw problem with surface tension in the case of subdiffusion. Communications on Pure and 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, Frac. Calcul. and Appl. Analysis, 15 (2012), 207-221. doi: 10.2478/s13540-012-0015-2.

[2]

B. V. Bazaliy, Stefan problem for the Laplace equation with regard for the curvature of the free boundary, Ukr. Math. J., 40 (1997), 1465-1484.

[3]

B. V. Bazaliy and A. Friedman, The Hele-Shaw problem with surface tension in a half-plane: a model problem, J. Diff. Equs., 216 (2005), 387-438. doi: 10.1016/j.jde.2005.03.007.

[4]

B. V. Bazaliy and A. Friedman, The Hele-Shaw problem with surface tension in a half-plane, J. Diff. Equs., 216 (2005), 439-469. doi: 10.1016/j.jde.2005.03.017.

[5]

B. V. Bazaliy and N. Vasylyeva, The two-phase Hele-Shaw problem with a nonregular initial interface and without surface tension, J. Math. Phys., Anal., Geom., 10 (2014), 3-43.

[6]

J.-P. Bouchard and A. Georges, Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications, Phys. Rep., 195 (1990), 127-293. doi: 10.1016/0370-1573(90)90099-N.

[7]

A. Cartea and D. del Castillo-Negrete, Fractional diffusion models of option prices in markets with jumps, Physica A, 374 (2007), 749-763.

[8]

X. Chen, The Hele-Shaw problem and area preserving curve shortening motions, Arch. Rat. Mech. Anal., 123 (1993), 117-151. doi: 10.1007/BF00695274.

[9]

S. Chen, B. Merriman, S. Osher and P. Smereka, A simple level set method for solving Stefan problems, J. Comput. Phys., 135 (1997), 8-29. doi: 10.1006/jcph.1997.5721.

[10]

X. Chen and F. Reitich, Local existence and uniqueness of solutions of the Stefan problem with surface tension and kinetic indercooling, J. Math. Anal. Appl., 164 (1992), 350-362. doi: 10.1016/0022-247X(92)90119-X.

[11]

K. Diethelm, N. J. Ford and A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam., 29 (2002), 3-22. doi: 10.1023/A:1016592219341.

[12]

G. Drazer and D. H. Zanette, Experimental evidence of power-law trapping-time distributions in porous media, Phys. Rev. E, 60 (1999), 5858-5864.

[13]

A. Erdélyi, Higher Transcendental Functions v.3, Mc Graw-Hill, New York, 1955.

[14]

J. Escher and G. Simonett, Classical solutions multidimensional Hele-Shaw models, SIAM J. Math. Anal., 28 (1997), 1028-1047. doi: 10.1137/S0036141095291919.

[15]

A. Friedman and F. Reitich, Analysis of a mathematical model for the growth of tumors, J. Math. Bio., 38 (1999), 262-284. doi: 10.1007/s002850050149.

[16]

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

[17]

E. I. Hanzava, Classical solution of the Stefan problem, Tohoku Math. J., 33 (1981), 297-335.

[18]

H. S. Hele-Shaw, The flow of water, Nature, 58 (1898), 34-36.

[19]

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

[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).

[21]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006.

[22]

A. N. Kochubei, Fractional-parabolic systems, Potential Analysis, 37 (2012), 1-30. doi: 10.1007/s11118-011-9243-z.

[23]

M. Krasnoschok and N. Vasylyeva, Existence and uniqueness of the solutions for some initial-boundary value problems with the fractional dynamic boundary condition, Inter. J. PDE, 2013 (2013), Article ID 796430, 20p.

[24]

O. A. Ladyzhenskaia, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasilinear Parabolic Equations, Academic Press, New York, 1968.

[25]

O. A. Ladyzhenskaia and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.

[26]

T. A. M. Langlands and B. I. Henry, Fractional chemotaxis diffusion equation, Physical Review E, 81 (2010), 051102. doi: 10.1103/PhysRevE.81.051102.

[27]

B.-T. Liu and J.-P. Hsu, Theoretical analysis on diffusional release from ellipsoidal drug delivery devices, Chem. Eng. Sci., 61 (2006), 1748-1752.

[28]

J. Y. Liu and M. Xu, An exact solution to the moving boundary problem with fractional anomalous diffusion in drug release devices, Z. Angew. Math. Mech., 84 (2004), 22-28. doi: 10.1002/zamm.200410074.

[29]

J. Y. Liu and M. Xu, Some exact solutions to Stefan problems with fractional differential equations, J. Math. Anal. Appl., 351 (2009), 536-542. doi: 10.1016/j.jmaa.2008.10.042.

[30]

A. Logg, K. A. Mardal and G. Wells eds., Automated Solution of Differential Equations by the Finite Element Method: The Fenics Book, 84, Springer, 2012. doi: 10.1007/978-3-642-23099-8.

[31]

A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and their Applications 16, Basel: Birkhäuser Verlag, 1995. doi: 10.1007/978-3-0348-9234-6.

[32]

F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, in Fractals and Fractional Calculus in Continuum Mechanics (A. Garpinteri and F. Mainardi Eds.), Springer-Verlag, New York, (1997), 291-348. doi: 10.1007/978-3-7091-2664-6_7.

[33]

G. Prokert, Existence results for Hele-Shaw problem driven by surface tension, European J. Appl. Math., 9 (1998), 195-221. doi: 10.1017/S0956792597003276.

[34]

A. V. Pskhu, Partial Differential Equations of the Fractional Order, Nauka, Moscow, 2005 (in Russian).

[35]

A. V. Pskhu, The fundamental solution of a diffusion-wave equation of fractional order, Izvestia RAN, 73 (2009), 141-182 (in Russian). doi: 10.1070/IM2009v073n02ABEH002450.

[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, J. Statistical Physics, 140 (2010), 797-811. doi: 10.1007/s10955-010-0007-8.

[37]

W. Tan, C. Fu, C. Fu, W. Xie and H. Cheng, An anomalous subdiffusion model for calcium spark in cardiac myocytes, Appl. Phys. Letter, 91 (2007), 183901.

[38]

V. R. Voller, An exact solution of a limit case Stefan problem governed by a fractional diffusion equation, Internat. J. Heat and Mass Transf., 53 (2010), 5622-5625.

[39]

V. R. Voller, F. Falcini and R. Garra, Fractional Stefan problems exibiting lumped and distributed latent-heat memory effects, Phys. Rew. E, 87 (2013), 042401.

[40]

V. R. Voller, An overview of numerical methods for solving phase change problems, in Advances in Numerical Heat Transfer (W. J. Minkowycz and E. M. Sparrow Eds.), Taylor & Francis Washington, DC, 1 (1996), 341-375.

[41]

N. Vasylyeva and L. Vynnytska, On a multidimensional moving boundary problem governed by anomalous diffusion: analytical and numerical study, Nonlinear Differ. Equ. Appl., 22 (2015), 543-577. doi: 10.1007/s00030-014-0295-9.

[42]

N. Vasylyeva, Local solvability of a linear system with a fractional derivative in time in a boundary condition, Frac. Calc. Appl. Anal., 18 (2015), 982-1005. doi: 10.1515/fca-2015-0058.

[43]

N. Vasylyeva, On a local solvability of the multidimensional Muscat problem with a fractional derivative in time on the boundary condition, J. Frac. Differ. Calc., 4 (2014), 80-124. doi: 10.7153/fdc-04-06.

show all references

References:
[1]

C. Atkinson, Moving boundary problems for time fractional and composition dependent diffusion, Frac. Calcul. and Appl. Analysis, 15 (2012), 207-221. doi: 10.2478/s13540-012-0015-2.

[2]

B. V. Bazaliy, Stefan problem for the Laplace equation with regard for the curvature of the free boundary, Ukr. Math. J., 40 (1997), 1465-1484.

[3]

B. V. Bazaliy and A. Friedman, The Hele-Shaw problem with surface tension in a half-plane: a model problem, J. Diff. Equs., 216 (2005), 387-438. doi: 10.1016/j.jde.2005.03.007.

[4]

B. V. Bazaliy and A. Friedman, The Hele-Shaw problem with surface tension in a half-plane, J. Diff. Equs., 216 (2005), 439-469. doi: 10.1016/j.jde.2005.03.017.

[5]

B. V. Bazaliy and N. Vasylyeva, The two-phase Hele-Shaw problem with a nonregular initial interface and without surface tension, J. Math. Phys., Anal., Geom., 10 (2014), 3-43.

[6]

J.-P. Bouchard and A. Georges, Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications, Phys. Rep., 195 (1990), 127-293. doi: 10.1016/0370-1573(90)90099-N.

[7]

A. Cartea and D. del Castillo-Negrete, Fractional diffusion models of option prices in markets with jumps, Physica A, 374 (2007), 749-763.

[8]

X. Chen, The Hele-Shaw problem and area preserving curve shortening motions, Arch. Rat. Mech. Anal., 123 (1993), 117-151. doi: 10.1007/BF00695274.

[9]

S. Chen, B. Merriman, S. Osher and P. Smereka, A simple level set method for solving Stefan problems, J. Comput. Phys., 135 (1997), 8-29. doi: 10.1006/jcph.1997.5721.

[10]

X. Chen and F. Reitich, Local existence and uniqueness of solutions of the Stefan problem with surface tension and kinetic indercooling, J. Math. Anal. Appl., 164 (1992), 350-362. doi: 10.1016/0022-247X(92)90119-X.

[11]

K. Diethelm, N. J. Ford and A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam., 29 (2002), 3-22. doi: 10.1023/A:1016592219341.

[12]

G. Drazer and D. H. Zanette, Experimental evidence of power-law trapping-time distributions in porous media, Phys. Rev. E, 60 (1999), 5858-5864.

[13]

A. Erdélyi, Higher Transcendental Functions v.3, Mc Graw-Hill, New York, 1955.

[14]

J. Escher and G. Simonett, Classical solutions multidimensional Hele-Shaw models, SIAM J. Math. Anal., 28 (1997), 1028-1047. doi: 10.1137/S0036141095291919.

[15]

A. Friedman and F. Reitich, Analysis of a mathematical model for the growth of tumors, J. Math. Bio., 38 (1999), 262-284. doi: 10.1007/s002850050149.

[16]

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

[17]

E. I. Hanzava, Classical solution of the Stefan problem, Tohoku Math. J., 33 (1981), 297-335.

[18]

H. S. Hele-Shaw, The flow of water, Nature, 58 (1898), 34-36.

[19]

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

[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).

[21]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006.

[22]

A. N. Kochubei, Fractional-parabolic systems, Potential Analysis, 37 (2012), 1-30. doi: 10.1007/s11118-011-9243-z.

[23]

M. Krasnoschok and N. Vasylyeva, Existence and uniqueness of the solutions for some initial-boundary value problems with the fractional dynamic boundary condition, Inter. J. PDE, 2013 (2013), Article ID 796430, 20p.

[24]

O. A. Ladyzhenskaia, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasilinear Parabolic Equations, Academic Press, New York, 1968.

[25]

O. A. Ladyzhenskaia and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.

[26]

T. A. M. Langlands and B. I. Henry, Fractional chemotaxis diffusion equation, Physical Review E, 81 (2010), 051102. doi: 10.1103/PhysRevE.81.051102.

[27]

B.-T. Liu and J.-P. Hsu, Theoretical analysis on diffusional release from ellipsoidal drug delivery devices, Chem. Eng. Sci., 61 (2006), 1748-1752.

[28]

J. Y. Liu and M. Xu, An exact solution to the moving boundary problem with fractional anomalous diffusion in drug release devices, Z. Angew. Math. Mech., 84 (2004), 22-28. doi: 10.1002/zamm.200410074.

[29]

J. Y. Liu and M. Xu, Some exact solutions to Stefan problems with fractional differential equations, J. Math. Anal. Appl., 351 (2009), 536-542. doi: 10.1016/j.jmaa.2008.10.042.

[30]

A. Logg, K. A. Mardal and G. Wells eds., Automated Solution of Differential Equations by the Finite Element Method: The Fenics Book, 84, Springer, 2012. doi: 10.1007/978-3-642-23099-8.

[31]

A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and their Applications 16, Basel: Birkhäuser Verlag, 1995. doi: 10.1007/978-3-0348-9234-6.

[32]

F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, in Fractals and Fractional Calculus in Continuum Mechanics (A. Garpinteri and F. Mainardi Eds.), Springer-Verlag, New York, (1997), 291-348. doi: 10.1007/978-3-7091-2664-6_7.

[33]

G. Prokert, Existence results for Hele-Shaw problem driven by surface tension, European J. Appl. Math., 9 (1998), 195-221. doi: 10.1017/S0956792597003276.

[34]

A. V. Pskhu, Partial Differential Equations of the Fractional Order, Nauka, Moscow, 2005 (in Russian).

[35]

A. V. Pskhu, The fundamental solution of a diffusion-wave equation of fractional order, Izvestia RAN, 73 (2009), 141-182 (in Russian). doi: 10.1070/IM2009v073n02ABEH002450.

[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, J. Statistical Physics, 140 (2010), 797-811. doi: 10.1007/s10955-010-0007-8.

[37]

W. Tan, C. Fu, C. Fu, W. Xie and H. Cheng, An anomalous subdiffusion model for calcium spark in cardiac myocytes, Appl. Phys. Letter, 91 (2007), 183901.

[38]

V. R. Voller, An exact solution of a limit case Stefan problem governed by a fractional diffusion equation, Internat. J. Heat and Mass Transf., 53 (2010), 5622-5625.

[39]

V. R. Voller, F. Falcini and R. Garra, Fractional Stefan problems exibiting lumped and distributed latent-heat memory effects, Phys. Rew. E, 87 (2013), 042401.

[40]

V. R. Voller, An overview of numerical methods for solving phase change problems, in Advances in Numerical Heat Transfer (W. J. Minkowycz and E. M. Sparrow Eds.), Taylor & Francis Washington, DC, 1 (1996), 341-375.

[41]

N. Vasylyeva and L. Vynnytska, On a multidimensional moving boundary problem governed by anomalous diffusion: analytical and numerical study, Nonlinear Differ. Equ. Appl., 22 (2015), 543-577. doi: 10.1007/s00030-014-0295-9.

[42]

N. Vasylyeva, Local solvability of a linear system with a fractional derivative in time in a boundary condition, Frac. Calc. Appl. Anal., 18 (2015), 982-1005. doi: 10.1515/fca-2015-0058.

[43]

N. Vasylyeva, On a local solvability of the multidimensional Muscat problem with a fractional derivative in time on the boundary condition, J. Frac. Differ. Calc., 4 (2014), 80-124. doi: 10.7153/fdc-04-06.

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