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 & Continuous Dynamical Systems - B, 2016, 21 (10) : 3479-3514. doi: 10.3934/dcdsb.2016108
References:
[1]

European J. Appl. Math., 15 (2004), 597-607. doi: 10.1017/S0956792504005662.  Google Scholar

[2]

Port. Math., 59 (2002), 435-452.  Google Scholar

[3]

Adv. Differential Equations, 8 (2003), 1259-1280. Google Scholar

[4]

Nonlinear Anal., 11 (1987), 17-47. doi: 10.1016/0362-546X(87)90024-1.  Google Scholar

[5]

Interfaces Free Bound, 16 (2014), 297-338. doi: 10.4171/IFB/321.  Google Scholar

[6]

American journal of mathematics, 129 (2007), 527-582. doi: 10.1353/ajm.2007.0008.  Google Scholar

[7]

Indiana University Mathematics Journal, 58 (2009), p2765. Google Scholar

[8]

Phys. D, 47 (1991), 450-460. doi: 10.1016/0167-2789(91)90042-8.  Google Scholar

[9]

Nonlinearity, 6 (1993), 393-415. doi: 10.1088/0951-7715/6/3/004.  Google Scholar

[10]

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 88 (1981), 93-107. doi: 10.1017/S0308210500017315.  Google Scholar

[11]

Math. Res. Lett., 3 (1996), 467-474. doi: 10.4310/MRL.1996.v3.n4.a5.  Google Scholar

[12]

Adv. Differential Equations, 2 (1997), 619-642. Google Scholar

[13]

SIAM J. Math. Anal., 28 (1997), 1028-1047. Google Scholar

[14]

Dokl. Akad. Nauk USSR, 47 (1945), 246-249.  Google Scholar

[15]

vol. 2, Springer, 2008.  Google Scholar

[16]

Ark. Mat., 22 (1984), 251-268. doi: 10.1007/BF02384382.  Google Scholar

[17]

Birkhäuser, 2006.  Google Scholar

[18]

arXiv preprint, arXiv:1112.5817, (2011). Google Scholar

[19]

The Journal of Geometric Analysis, 15 (2005), 641-667. doi: 10.1007/BF02922248.  Google Scholar

[20]

Nonlinear Analysis: Theory, Methods & Applications, 64 (2006), 2817-2831. doi: 10.1016/j.na.2005.09.021.  Google Scholar

[21]

vol. 1, Cambridge Univ. Press, 2009.  Google Scholar

[22]

vol. 2, Cambridge Univ. Press, 2009.  Google Scholar

[23]

vol. 96, Amer. Math. Soc., 2008. doi: 10.1090/gsm/096.  Google Scholar

[24]

Doklady Akad. Nauk SSSR (N. S.), 60 (1948), 1333-1334.  Google Scholar

[25]

European J. Appl. Math., 13 (2002), 431-447. doi: 10.1017/S0956792502004874.  Google Scholar

[26]

Dokl. Akad. Nauk USSR, 47 (1945), 254-257. Google Scholar

[27]

European J. Appl. Math., 9 (1998), 195-221. doi: 10.1017/S0956792597003276.  Google Scholar

[28]

Ark. Mat., 31 (1993), 101-116. doi: 10.1007/BF02559501.  Google Scholar

[29]

SIAM journal on mathematical analysis, 24 (1993), 341-364. doi: 10.1137/0524023.  Google Scholar

[30]

Philos. Trans. Roy. Soc. London Ser. A, 343 (1993), 155-204. doi: 10.1098/rsta.1993.0049.  Google Scholar

[31]

J. Nonlinear Sci., 10 (2000), 275-290. doi: 10.1007/s003329910011.  Google Scholar

[32]

Nature, 58 (1898), 34-36. Google Scholar

[33]

Complex Anal. Oper. Theory, 3 (2009), 551-585. doi: 10.1007/s11785-008-0104-8.  Google Scholar

[34]

Akad. Nauk SSSR Prikl. Mat. Meh., 12 (1948), 181-198.  Google Scholar

[35]

IMA J. Appl. Math., 76 (2011), 219-241. doi: 10.1093/imamat/hxp037.  Google Scholar

[36]

SIAM J. Math. Anal., 43 (2011), 457-506. doi: 10.1137/100786332.  Google Scholar

show all references

References:
[1]

European J. Appl. Math., 15 (2004), 597-607. doi: 10.1017/S0956792504005662.  Google Scholar

[2]

Port. Math., 59 (2002), 435-452.  Google Scholar

[3]

Adv. Differential Equations, 8 (2003), 1259-1280. Google Scholar

[4]

Nonlinear Anal., 11 (1987), 17-47. doi: 10.1016/0362-546X(87)90024-1.  Google Scholar

[5]

Interfaces Free Bound, 16 (2014), 297-338. doi: 10.4171/IFB/321.  Google Scholar

[6]

American journal of mathematics, 129 (2007), 527-582. doi: 10.1353/ajm.2007.0008.  Google Scholar

[7]

Indiana University Mathematics Journal, 58 (2009), p2765. Google Scholar

[8]

Phys. D, 47 (1991), 450-460. doi: 10.1016/0167-2789(91)90042-8.  Google Scholar

[9]

Nonlinearity, 6 (1993), 393-415. doi: 10.1088/0951-7715/6/3/004.  Google Scholar

[10]

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 88 (1981), 93-107. doi: 10.1017/S0308210500017315.  Google Scholar

[11]

Math. Res. Lett., 3 (1996), 467-474. doi: 10.4310/MRL.1996.v3.n4.a5.  Google Scholar

[12]

Adv. Differential Equations, 2 (1997), 619-642. Google Scholar

[13]

SIAM J. Math. Anal., 28 (1997), 1028-1047. Google Scholar

[14]

Dokl. Akad. Nauk USSR, 47 (1945), 246-249.  Google Scholar

[15]

vol. 2, Springer, 2008.  Google Scholar

[16]

Ark. Mat., 22 (1984), 251-268. doi: 10.1007/BF02384382.  Google Scholar

[17]

Birkhäuser, 2006.  Google Scholar

[18]

arXiv preprint, arXiv:1112.5817, (2011). Google Scholar

[19]

The Journal of Geometric Analysis, 15 (2005), 641-667. doi: 10.1007/BF02922248.  Google Scholar

[20]

Nonlinear Analysis: Theory, Methods & Applications, 64 (2006), 2817-2831. doi: 10.1016/j.na.2005.09.021.  Google Scholar

[21]

vol. 1, Cambridge Univ. Press, 2009.  Google Scholar

[22]

vol. 2, Cambridge Univ. Press, 2009.  Google Scholar

[23]

vol. 96, Amer. Math. Soc., 2008. doi: 10.1090/gsm/096.  Google Scholar

[24]

Doklady Akad. Nauk SSSR (N. S.), 60 (1948), 1333-1334.  Google Scholar

[25]

European J. Appl. Math., 13 (2002), 431-447. doi: 10.1017/S0956792502004874.  Google Scholar

[26]

Dokl. Akad. Nauk USSR, 47 (1945), 254-257. Google Scholar

[27]

European J. Appl. Math., 9 (1998), 195-221. doi: 10.1017/S0956792597003276.  Google Scholar

[28]

Ark. Mat., 31 (1993), 101-116. doi: 10.1007/BF02559501.  Google Scholar

[29]

SIAM journal on mathematical analysis, 24 (1993), 341-364. doi: 10.1137/0524023.  Google Scholar

[30]

Philos. Trans. Roy. Soc. London Ser. A, 343 (1993), 155-204. doi: 10.1098/rsta.1993.0049.  Google Scholar

[31]

J. Nonlinear Sci., 10 (2000), 275-290. doi: 10.1007/s003329910011.  Google Scholar

[32]

Nature, 58 (1898), 34-36. Google Scholar

[33]

Complex Anal. Oper. Theory, 3 (2009), 551-585. doi: 10.1007/s11785-008-0104-8.  Google Scholar

[34]

Akad. Nauk SSSR Prikl. Mat. Meh., 12 (1948), 181-198.  Google Scholar

[35]

IMA J. Appl. Math., 76 (2011), 219-241. doi: 10.1093/imamat/hxp037.  Google Scholar

[36]

SIAM J. Math. Anal., 43 (2011), 457-506. doi: 10.1137/100786332.  Google Scholar

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