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 & 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
[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

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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