# American Institute of Mathematical Sciences

December  2016, 21(10): 3479-3514. doi: 10.3934/dcdsb.2016108

## The vanishing surface tension limit for the Hele-Shaw problem

 1 Department of Mathematics, Pohang University of Science and Technology, Pohang 790-784, South Korea, South Korea 2 Instituto de Ciencias Matemáticas (ICMAT), C/Nicolás Cabrera, 28049 Madrid, Spain

Received  July 2015 Revised  August 2016 Published  November 2016

In this paper, we show the existence of solutions of the Hele-Shaw problem in two dimensions in the presence of surface tension and for a general class of initial data. The limit problem from nonzero to zero surface tension will also be investigated. In the case of injection and when volume conservation holds, for a sufficiently small surface tension, we prove the existence and uniqueness of perturbed solutions with nonzero surface tension near solutions with zero surface tension. We also show that solutions with nonzero surface tension exist up to a finite time before a possible singularity occurs in which solutions with zero surface tension are well defined. In addition, in the finite time interval, we prove that the solutions with nonzero surface tension approach the solutions with zero surface tension as the surface tension coefficient goes to zero. In the case of suction, for sufficiently small surface tension, we prove the existence of perturbed solutions near solutions with zero surface tension in any initially smooth domains. In this case, the local existence time depends on the surface tension coefficient.
Citation: Hyung Ju Hwang, Youngmin Oh, Marco Antonio Fontelos. The vanishing surface tension limit for the Hele-Shaw problem. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3479-3514. doi: 10.3934/dcdsb.2016108
##### References:
 [1] D. M. Ambrose, Well-posedness of two-phase Hele-Shaw flow without surface tension, European J. Appl. Math., 15 (2004), 597-607. doi: 10.1017/S0956792504005662. [2] S. N. Antontsev, C. Gonçalves and A. M. Meirmanov, Local existence of classical solutions to the well-posed Hele-Shaw problem, Port. Math., 59 (2002), 435-452. [3] ________, Exact estimates for the classical solutions to the free-boundary problem in the Hele-Shaw cell, Adv. Differential Equations, 8 (2003), 1259-1280. [4] H. G. W. Begehr and R. P. Gilbert, Non-Newtonian Hele-Shaw flows in $n\geq 2$ dimensions, Nonlinear Anal., 11 (1987), 17-47. doi: 10.1016/0362-546X(87)90024-1. [5] C.-H. A. Cheng, D. Coutand and S. Shkoller, Global existence and decay for solutions of the Hele-Shaw flow with injection, Interfaces Free Bound, 16 (2014), 297-338. doi: 10.4171/IFB/321. [6] S. Choi, D. Jerison and I. Kim, Regularity for the one-phase hele-shaw problem from a lipschitz initial surface, American journal of mathematics, 129 (2007), 527-582. doi: 10.1353/ajm.2007.0008. [7] ________, Local regularization of the one-phase hele-shaw flow, Indiana University Mathematics Journal, 58 (2009), p2765. [8] P. Constantin and L. P. Kadanoff, Dynamics of a complex interface, Phys. D, 47 (1991), 450-460. doi: 10.1016/0167-2789(91)90042-8. [9] P. Constantin and M. Pugh, Global solutions for small data to the Hele-Shaw problem, Nonlinearity, 6 (1993), 393-415. doi: 10.1088/0951-7715/6/3/004. [10] C. M. Elliott and V. Janovskỳ, A variational inequality approach to hele-shaw flow with a moving boundary, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 88 (1981), 93-107. doi: 10.1017/S0308210500017315. [11] J. Escher and G. Simonett, On Hele-Shaw models with surface tension, Math. Res. Lett., 3 (1996), 467-474. doi: 10.4310/MRL.1996.v3.n4.a5. [12] ________, Classical solutions for Hele-Shaw models with surface tension, Adv. Differential Equations, 2 (1997), 619-642. [13] ________, Classical solutions of multidimensional Hele-Shaw models, SIAM J. Math. Anal., 28 (1997), 1028-1047. [14] L. A. Galin, Unsteady filtration with a free surface, Dokl. Akad. Nauk USSR, 47 (1945), 246-249. [15] L. Grafakos, Classical Fourier Analysis, vol. 2, Springer, 2008. [16] B. Gustafsson, On a differential equation arising in a Hele-Shaw flow moving boundary problem, Ark. Mat., 22 (1984), 251-268. doi: 10.1007/BF02384382. [17] B. Gustafsson and A. Vasil'ev, Conformal and Potential Analysis in Hele-Shaw Cells, Birkhäuser, 2006. [18] M. Hadzic and S. Shkoller, Well-posedness for the classical stefan problem and the zero surface tension limit, arXiv preprint, arXiv:1112.5817, (2011). [19] D. Jerison and I. Kim, The one-phase hele-shaw problem with singularities, The Journal of Geometric Analysis, 15 (2005), 641-667. doi: 10.1007/BF02922248. [20] I. Kim, Long time regularity of solutions of the hele-shaw problem, Nonlinear Analysis: Theory, Methods & Applications, 64 (2006), 2817-2831. doi: 10.1016/j.na.2005.09.021. [21] F. W. King, Hilbert Transforms, vol. 1, Cambridge Univ. Press, 2009. [22] ________, Hilbert Transforms, vol. 2, Cambridge Univ. Press, 2009. [23] N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, vol. 96, Amer. Math. Soc., 2008. doi: 10.1090/gsm/096. [24] P. P. Kufarev, A solution of the boundary problem for an oil well in a circle, Doklady Akad. Nauk SSSR (N. S.), 60 (1948), 1333-1334. [25] A. M. Meirmanov and B. Zaltzman, Global in time solution to the Hele-Shaw problem with a change of topology, European J. Appl. Math., 13 (2002), 431-447. doi: 10.1017/S0956792502004874. [26] P. Y. Polubarinova-Kochina, On a problem of the motion of the contour of a petroleum shell, Dokl. Akad. Nauk USSR, 47 (1945), 254-257. [27] G. Prokert, Existence results for Hele-Shaw flow driven by surface tension, European J. Appl. Math., 9 (1998), 195-221. doi: 10.1017/S0956792597003276. [28] M. Reissig and L. von Wolfersdorf, A simplified proof for a moving boundary problem for Hele-Shaw flows in the plane, Ark. Mat., 31 (1993), 101-116. doi: 10.1007/BF02559501. [29] M. Sakai, Regularity of boundaries of quadrature domains in two dimensions, SIAM journal on mathematical analysis, 24 (1993), 341-364. doi: 10.1137/0524023. [30] S. Tanveer, Evolution of Hele-Shaw interface for small surface tension, Philos. Trans. Roy. Soc. London Ser. A, 343 (1993), 155-204. doi: 10.1098/rsta.1993.0049. [31] F.-R. Tian, Hele-Shaw problems in multidimensional spaces, J. Nonlinear Sci., 10 (2000), 275-290. doi: 10.1007/s003329910011. [32] H. S. Hele Shaw, The flow of water, Nature, 58 (1898), 34-36. [33] A. Vasil'ev, From the Hele-Shaw experiment to integrable systems: A historical overview, Complex Anal. Oper. Theory, 3 (2009), 551-585. doi: 10.1007/s11785-008-0104-8. [34] Y. P. Vinogradov and P. P. Kufarev, On a problem of filtration, Akad. Nauk SSSR Prikl. Mat. Meh., 12 (1948), 181-198. [35] E. Vondenhoff, Large time behaviour of Hele-Shaw flow with injection or suction for perturbations of balls in $\mathbb{R}^N2$, IMA J. Appl. Math., 76 (2011), 219-241. doi: 10.1093/imamat/hxp037. [36] J. Ye and S. Tanveer, Global existence for a translating near-circular Hele-Shaw bubble with surface tension, SIAM J. Math. Anal., 43 (2011), 457-506. doi: 10.1137/100786332.

show all references

##### References:
 [1] D. M. Ambrose, Well-posedness of two-phase Hele-Shaw flow without surface tension, European J. Appl. Math., 15 (2004), 597-607. doi: 10.1017/S0956792504005662. [2] S. N. Antontsev, C. Gonçalves and A. M. Meirmanov, Local existence of classical solutions to the well-posed Hele-Shaw problem, Port. Math., 59 (2002), 435-452. [3] ________, Exact estimates for the classical solutions to the free-boundary problem in the Hele-Shaw cell, Adv. Differential Equations, 8 (2003), 1259-1280. [4] H. G. W. Begehr and R. P. Gilbert, Non-Newtonian Hele-Shaw flows in $n\geq 2$ dimensions, Nonlinear Anal., 11 (1987), 17-47. doi: 10.1016/0362-546X(87)90024-1. [5] C.-H. A. Cheng, D. Coutand and S. Shkoller, Global existence and decay for solutions of the Hele-Shaw flow with injection, Interfaces Free Bound, 16 (2014), 297-338. doi: 10.4171/IFB/321. [6] S. Choi, D. Jerison and I. Kim, Regularity for the one-phase hele-shaw problem from a lipschitz initial surface, American journal of mathematics, 129 (2007), 527-582. doi: 10.1353/ajm.2007.0008. [7] ________, Local regularization of the one-phase hele-shaw flow, Indiana University Mathematics Journal, 58 (2009), p2765. [8] P. Constantin and L. P. Kadanoff, Dynamics of a complex interface, Phys. D, 47 (1991), 450-460. doi: 10.1016/0167-2789(91)90042-8. [9] P. Constantin and M. Pugh, Global solutions for small data to the Hele-Shaw problem, Nonlinearity, 6 (1993), 393-415. doi: 10.1088/0951-7715/6/3/004. [10] C. M. Elliott and V. Janovskỳ, A variational inequality approach to hele-shaw flow with a moving boundary, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 88 (1981), 93-107. doi: 10.1017/S0308210500017315. [11] J. Escher and G. Simonett, On Hele-Shaw models with surface tension, Math. Res. Lett., 3 (1996), 467-474. doi: 10.4310/MRL.1996.v3.n4.a5. [12] ________, Classical solutions for Hele-Shaw models with surface tension, Adv. Differential Equations, 2 (1997), 619-642. [13] ________, Classical solutions of multidimensional Hele-Shaw models, SIAM J. Math. Anal., 28 (1997), 1028-1047. [14] L. A. Galin, Unsteady filtration with a free surface, Dokl. Akad. Nauk USSR, 47 (1945), 246-249. [15] L. Grafakos, Classical Fourier Analysis, vol. 2, Springer, 2008. [16] B. Gustafsson, On a differential equation arising in a Hele-Shaw flow moving boundary problem, Ark. Mat., 22 (1984), 251-268. doi: 10.1007/BF02384382. [17] B. Gustafsson and A. Vasil'ev, Conformal and Potential Analysis in Hele-Shaw Cells, Birkhäuser, 2006. [18] M. Hadzic and S. Shkoller, Well-posedness for the classical stefan problem and the zero surface tension limit, arXiv preprint, arXiv:1112.5817, (2011). [19] D. Jerison and I. Kim, The one-phase hele-shaw problem with singularities, The Journal of Geometric Analysis, 15 (2005), 641-667. doi: 10.1007/BF02922248. [20] I. Kim, Long time regularity of solutions of the hele-shaw problem, Nonlinear Analysis: Theory, Methods & Applications, 64 (2006), 2817-2831. doi: 10.1016/j.na.2005.09.021. [21] F. W. King, Hilbert Transforms, vol. 1, Cambridge Univ. Press, 2009. [22] ________, Hilbert Transforms, vol. 2, Cambridge Univ. Press, 2009. [23] N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, vol. 96, Amer. Math. Soc., 2008. doi: 10.1090/gsm/096. [24] P. P. Kufarev, A solution of the boundary problem for an oil well in a circle, Doklady Akad. Nauk SSSR (N. S.), 60 (1948), 1333-1334. [25] A. M. Meirmanov and B. Zaltzman, Global in time solution to the Hele-Shaw problem with a change of topology, European J. Appl. Math., 13 (2002), 431-447. doi: 10.1017/S0956792502004874. [26] P. Y. Polubarinova-Kochina, On a problem of the motion of the contour of a petroleum shell, Dokl. Akad. Nauk USSR, 47 (1945), 254-257. [27] G. Prokert, Existence results for Hele-Shaw flow driven by surface tension, European J. Appl. Math., 9 (1998), 195-221. doi: 10.1017/S0956792597003276. [28] M. Reissig and L. von Wolfersdorf, A simplified proof for a moving boundary problem for Hele-Shaw flows in the plane, Ark. Mat., 31 (1993), 101-116. doi: 10.1007/BF02559501. [29] M. Sakai, Regularity of boundaries of quadrature domains in two dimensions, SIAM journal on mathematical analysis, 24 (1993), 341-364. doi: 10.1137/0524023. [30] S. Tanveer, Evolution of Hele-Shaw interface for small surface tension, Philos. Trans. Roy. Soc. London Ser. A, 343 (1993), 155-204. doi: 10.1098/rsta.1993.0049. [31] F.-R. Tian, Hele-Shaw problems in multidimensional spaces, J. Nonlinear Sci., 10 (2000), 275-290. doi: 10.1007/s003329910011. [32] H. S. Hele Shaw, The flow of water, Nature, 58 (1898), 34-36. [33] A. Vasil'ev, From the Hele-Shaw experiment to integrable systems: A historical overview, Complex Anal. Oper. Theory, 3 (2009), 551-585. doi: 10.1007/s11785-008-0104-8. [34] Y. P. Vinogradov and P. P. Kufarev, On a problem of filtration, Akad. Nauk SSSR Prikl. Mat. Meh., 12 (1948), 181-198. [35] E. Vondenhoff, Large time behaviour of Hele-Shaw flow with injection or suction for perturbations of balls in $\mathbb{R}^N2$, IMA J. Appl. Math., 76 (2011), 219-241. doi: 10.1093/imamat/hxp037. [36] J. Ye and S. Tanveer, Global existence for a translating near-circular Hele-Shaw bubble with surface tension, SIAM J. Math. Anal., 43 (2011), 457-506. doi: 10.1137/100786332.
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