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.

[1]

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

[2]

A. Bernardini, J. Bragard, H. Mancini. Synchronization between two Hele-Shaw Cells. Mathematical Biosciences & Engineering, 2004, 1 (2) : 339-346. doi: 10.3934/mbe.2004.1.339

[3]

Xuming Xie. Analytic solution to an interfacial flow with kinetic undercooling in a time-dependent gap Hele-Shaw cell. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4663-4680. doi: 10.3934/dcdsb.2020307

[4]

Francesco Della Porta, Maurizio Grasselli. On the nonlocal Cahn-Hilliard-Brinkman and Cahn-Hilliard-Hele-Shaw systems. Communications on Pure and Applied Analysis, 2016, 15 (2) : 299-317. doi: 10.3934/cpaa.2016.15.299

[5]

Wenbin Chen, Wenqiang Feng, Yuan Liu, Cheng Wang, Steven M. Wise. A second order energy stable scheme for the Cahn-Hilliard-Hele-Shaw equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 149-182. doi: 10.3934/dcdsb.2018090

[6]

Chengchun Hao. Cauchy problem for viscous shallow water equations with surface tension. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 593-608. doi: 10.3934/dcdsb.2010.13.593

[7]

Min Chen, Nghiem V. Nguyen, Shu-Ming Sun. Solitary-wave solutions to Boussinesq systems with large surface tension. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1153-1184. doi: 10.3934/dcds.2010.26.1153

[8]

Samuel Walsh. Steady stratified periodic gravity waves with surface tension II: Global bifurcation. Discrete and Continuous Dynamical Systems, 2014, 34 (8) : 3287-3315. doi: 10.3934/dcds.2014.34.3287

[9]

Calin Iulian Martin. Dispersion relations for periodic water waves with surface tension and discontinuous vorticity. Discrete and Continuous Dynamical Systems, 2014, 34 (8) : 3109-3123. doi: 10.3934/dcds.2014.34.3109

[10]

Roman M. Taranets, Jeffrey T. Wong. Existence of weak solutions for particle-laden flow with surface tension. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 4979-4996. doi: 10.3934/dcds.2018217

[11]

Colette Calmelet, Diane Sepich. Surface tension and modeling of cellular intercalation during zebrafish gastrulation. Mathematical Biosciences & Engineering, 2010, 7 (2) : 259-275. doi: 10.3934/mbe.2010.7.259

[12]

Samuel Walsh. Steady stratified periodic gravity waves with surface tension I: Local bifurcation. Discrete and Continuous Dynamical Systems, 2014, 34 (8) : 3241-3285. doi: 10.3934/dcds.2014.34.3241

[13]

FRANCESCO DELLA PORTA, Maurizio Grasselli. Erratum: "On the nonlocal Cahn-Hilliard-Brinkman and Cahn-Hilliard-Hele-Shaw systems" [Comm. Pure Appl. Anal. 15 (2016), 299--317]. Communications on Pure and Applied Analysis, 2017, 16 (1) : 369-372. doi: 10.3934/cpaa.2017018

[14]

Jie Wang, Xiaoqiang Wang. New asymptotic analysis method for phase field models in moving boundary problem with surface tension. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3185-3213. doi: 10.3934/dcdsb.2015.20.3185

[15]

Shengfu Deng. Generalized pitchfork bifurcation on a two-dimensional gaseous star with self-gravity and surface tension. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3419-3435. doi: 10.3934/dcds.2014.34.3419

[16]

Marcelo M. Disconzi, Igor Kukavica. A priori estimates for the 3D compressible free-boundary Euler equations with surface tension in the case of a liquid. Evolution Equations and Control Theory, 2019, 8 (3) : 503-542. doi: 10.3934/eect.2019025

[17]

Grigor Nika, Bogdan Vernescu. Rate of convergence for a multi-scale model of dilute emulsions with non-uniform surface tension. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1553-1564. doi: 10.3934/dcdss.2016062

[18]

Khadijah Sharaf. A perturbation result for a critical elliptic equation with zero Dirichlet boundary condition. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1691-1706. doi: 10.3934/dcds.2017070

[19]

Ramzi Alsaedi. Perturbation effects for the minimal surface equation with multiple variable exponents. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 139-150. doi: 10.3934/dcdss.2019010

[20]

François Delarue, Franco Flandoli. The transition point in the zero noise limit for a 1D Peano example. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4071-4083. doi: 10.3934/dcds.2014.34.4071

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (106)
  • HTML views (0)
  • Cited by (0)

[Back to Top]