-
Previous Article
A multi-group SIR epidemic model with age structure
- DCDS-B Home
- This Issue
-
Next Article
Error estimates of the aggregation-diffusion splitting algorithms for the Keller-Segel equations
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 |
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
Tools
Metrics
Other articles
by authors
[Back to Top]