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

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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

Abstract
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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 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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