## Abstract

We study the two-dimensional quasistationary Stefan probem (the Hele-Shaw problem) in which the motion of the free boundary is described by a “fractional” Darcy law. We prove the existence and uniqueness of a classical solution to the free boundary problem for a small time interval.

This is a preview of subscription content, access via your institution.

## References

- 1.
S. Antontsev, C. Gonçalves, and A. Meirmanov, “Exact estimates for the classical solutions to the freeboundary problem in the Hele-Shaw cell,” Adv. Differential Equations

**8**, 1259 (2003). - 2.
C. Atkinson, “Moving boundary problems for time fractional and composition dependent diffusion,” Fract. Calc. Appl. Anal.

**15**, 207 (2012). - 3.
B. V. Bazalii, “On one proof of the classical solvability of theHele–Shawproblemwith free boundary,” Ukrain. Mat. Zh.

**50**, 1452 (1998) [Ukrainian Math. J.**50**, 1659 (1998)]. - 4.
B. V. Bazalii and S. P. Degtyarev, “On the classical solvability of the multidimensional Stefan problem for convective motion of a viscous incompressible fluid,” Mat. Sb. (N. S.)

**132**(174), No 1, 3 (1987) [Math. USSR, Sb.**60**, 1 (1988)]. - 5.
B. V. Bazaliy and N. N. Vasylyeva, “The two-phase Hele-Shaw problem with a nonregular initial interface and without surface tension,” J. Math. Phys. Anal. Geom.

**10**, 3 (2014). - 6.
G. I Bizhanova and V. A. Solonnikov, “Free boundary problems for second order parabolic equations,” Algebra Anal.

**12**(6), 98 (2000) [St. Petersbg. Math. J.**12**, 949 (2001)]. - 7.
J. -P. Bouchaud and A. Georges, “Anomalous diffusion in disordered media: statistical mechanisms, models, and physical applications,” Phys. Rep.

**195**(4–5), 127. - 8.
P. Clément, S, -O. Londen, and G. Simonett, “Quasilinear evolutionary equations and continuous interpolation spaces,” J. Differential Equations

**196**, 418 (2004). - 9.
S. D. Eidelman and A. N. Kochubei, “Cauchy problem for fractional diffusion equations,” J. Differential Equations

**199**, 211 (2004)]. - 10.
J. Escher and G. Simonett, “Classical solutions of multidimensional Hele–Shaw models,” SIAM J. Math. Anal.

**28**, 1028 (1997). - 11.
E. V. Frolova, “Quasistationary approximation for the Stefan problem,” Probl. Mat. Anal.

**31**, 167 (2005) [J. Math. Sci., New York**132**, 562 (2006)]. - 12.
D. Gilbarg and N. Trudinger,

*Elliptic Partial Differential Equations of Second Order*(Springer-Verlag, Berlin etc., 1983; Nauka, Moscow, 1989). - 13.
S. D. Howison,

*Bibliography of Free and Moving Boundary Problems in Hele-Shaw and Stokes Flow*, http://www.maths.ox.ac.uk/howison/Hele-Shaw (2006). - 14.
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). - 15.
M. Krasnoschok and N. Vasylyeva, “On nonclassical fractional boundary-value problem for the Laplace operator,” J. Differential Equations

**257**, 1814 (2014). - 16.
O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva,

*Linear and Quasi-Linear Equations of Parabolic Type*(Nauka, Moscow, 1967; AMS, Providence, RI, 1967). - 17.
O. A. Ladyzhenskaya and N. N. Ural’tseva,

*Linear and Quasilinear Elliptic Equations*(Nauka, Moscow, 1967; Academic Press, New York–London, 1968). - 18.
J. Liu J. and M. Xu, “An exact solution to the moving boundary problem with fractional anomalous diffusion in drug release devices,” Z. Angew. Math. Mech.

**84**, 22 (2004). - 19.
A. Lunardi,

*Analytic Semigroups and Optimal Regularity in Parabolic Problems*(Birkhäuser, Basel, 1995). - 20.
R. Metzler. and J. Klafter, “Boundary value problems for fractional diffusion equations,” Phys. A.

**278**(1–2), 107 (2000). - 21.
P. B. Mucha, “On weak solutions to the Stefan problem with Gibbs–Thomson correction,” Differential Integral Equations

**20**, 769 (2007). - 22.
P. Ya. Polubarinova-Kochina,

*Theory of Ground Water Movement*(Nauka, Moscow, 1977; Princeton University Press, Princeton, 1962). - 23.
A. V. Pskhu, “The fundamental solution of a diffusion-wave equation of fractional order,” Izv. Ross. Akad. Nauk, Ser. Mat.

**73**(2), 141 (2009) [Izv. Math.**73**, 351 (2009)]. - 24.
K. Sakamoto and M. Yamamoto, “Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems,” J. Math. Anal. Appl.

**382**, 426 (2011). - 25.
V. A. Solonnikov, “Estimates for the solution of the second initial boundary-value problem for the Stokes system in the spaces of functions having Hölder continuous derivatives with respect to the spatial variables,” Zap. Nauchn. Semin. POMI

**259**, 254 (1999) [J. Math. Sci., New York**109**, 1997 (2002)]. - 26.
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**, 5622 (2010). - 27.
V. R. Voller, F. Falcini, and R. Garra, “Fractional Stefan problems exhibiting lumped and distributed latentheat memory effects,” Phys. Rev. E.

**87**, 042401 (2013). - 28.
A. A. Voroshilov and A. A. Kilbas, “Conditions for the existence of a classical solution of a Cauchy type problem for the diffusion equation with a Riemann–Liouville partial derivative,” Differ. Uravn.

**44**, 768 (2008) [Differ. Equ.**44**, 789 (2008)]. - 29.
F. Yi, “Global classical solution of quasi-stationary Stefan free boundary problem,” Appl. Math. Comput.

**160**, 797 (2005).

## Author information

### Affiliations

### Corresponding author

## Additional information

Original Russian Text © N.V. Vasil’eva, N.V. Krasnoshchek, 2014, published in Matematicheskie Trudy, 2014, Vol. 17, No. 2, pp. 102–131.

## About this article

### Cite this article

Vasil’eva, N.V., Krasnoshchek, N.V. On the local solvability of the two-dimensional Hele-Shaw problem with fractional derivative with respect to time.
*Sib. Adv. Math.* **25, **276–296 (2015). https://doi.org/10.3103/S1055134415040057

Received:

Published:

Issue Date:

### Keywords

- quasistationary Stefan problem
- anomalous diffusion
- Caputo derivative
- regularizer
- coercive estimate