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Emergent dynamics of an orientation flocking model for multi-agent system
Global well-posedness of the free-interface incompressible Euler equations with damping
School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, China |
We prove the global well-posedness of the free interface problem for the two-phase incompressible Euler Equations with damping for the small initial data, where the effect of surface tension is included on the free surfaces. Moreover, the solution decays exponentially to the equilibrium.
References:
[1] |
D. M. Ambrose,
Well-posedness of vortex sheets with surface tension, SIAM J. Math. Anal., 35 (2003), 211-244.
doi: 10.1137/S0036141002403869. |
[2] |
D. M. Ambrose and N. Masmoudi,
Well-posedness of 3D vortex sheets with surface tension, Commun. Math. Sci., 5 (2007), 391-430.
doi: 10.4310/CMS.2007.v5.n2.a9. |
[3] |
V. Barcilon, P. Constantin and E. S. Titi,
Existence of solutions to the Stommel-Charney model of the gulf stream, SIAM J. Math. Anal., 19 (1988), 1355-1364.
doi: 10.1137/0519099. |
[4] |
J. T. Beale,
Large-time regularity of viscous surface waves, Arch. Ration. Mech. Anal., 84 (1983/84), 307-252.
doi: 10.1007/BF00250586. |
[5] |
R. E. Caflisch and O. F. Orellana,
Singular solutions and ill-posedness for the evolution of vortex sheets, SIAM J. Math. Anal., 20 (1989), 293-307.
doi: 10.1137/0520020. |
[6] |
J. G. Charney, The Gulf Stream as an inertial boundary layer, Proc. Nat. Acad. Sci. USA, 41 (1955), 731-740. Google Scholar |
[7] |
C.-H. A. Cheng, D. Coutand and S. Shkoller,
On the motion of vortex sheets with surface tension in three-dimensional Euler equations with vorticity, Comm. Pure Appl. Math., 61 (2008), 1715-1752.
doi: 10.1002/cpa.20240. |
[8] |
C.-H. A. Cheng, D. Coutand and S. Shkoller,
On the limit as the density ratio tends to zero for two perfect incompressible fluids separated by a surface of discontinuity, Comm. Partial Differential Equations, 35 (2010), 817-845.
doi: 10.1080/03605300903503115. |
[9] |
V. Chepyzhov and S. Zelik,
Infinite energy solutions for dissipative Euler equations in $\mathbb{R}^2$, J. Math. Fluid Mech., 17 (2015), 513-532.
doi: 10.1007/s00021-015-0213-x. |
[10] |
P. Constantin and F. Ramos,
Inviscid limit for damped and driven incompressible Navier-Stokes equations in $\mathbb{R}^2$, Comm. Math. Phys., 275 (2007), 529-551.
doi: 10.1007/s00220-007-0310-7. |
[11] |
J.-F. Coulombel and P. Secchi,
The stability of compressible vortex sheets in two space dimensions, Indiana Univ. Math. J., 53 (2004), 941-1012.
doi: 10.1512/iumj.2004.53.2526. |
[12] |
J.-F. Coulombel and P. Secchi, Nonlinear compressible vortex sheets in two space dimensions, Ann. Sci. Éc. Norm. Supér. (4), 41 (2008), 85–139.
doi: 10.24033/asens.2064. |
[13] |
D. Coutand and S. Shkoller,
Well-posedness of the free-surface incompressible Euler equations with or without surface tension, J. Amer. Math. Soc., 20 (2007), 829-930.
doi: 10.1090/S0894-0347-07-00556-5. |
[14] |
V. P. Dymnikov and A. N. Filatov, Mathematics of climate modelling, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 1997. |
[15] |
D. G. Ebin,
Ill-posedness of the Rayleigh-Taylor and Helmholtz problems for incompressible fluids, Comm. Partial Differential Equations, 13 (1988), 1265-1295.
doi: 10.1080/03605308808820576. |
[16] |
G. Gallavotti, Foundations of Fluid Dynamics, Texts and Monographs in Physics, Springer-Verlag, Berlin, 2002.
doi: 10.1007/978-3-662-04670-8. |
[17] |
P. Germain, N. Masmoudi and J. Shatah,
Global solutions for capillary waves equation, Comm. Pure Appl. Math., 68 (2015), 625-487.
doi: 10.1002/cpa.21535. |
[18] |
Y. Guo and I. Tice,
Linear Rayleigh-Taylor instability for viscous, compressible fluids, SIAM J. Math. Anal., 42 (2010), 1688-1720.
doi: 10.1137/090777438. |
[19] |
Y. Guo and I. Tice,
Local well-posedness of the viscous surface wave problem without surface tension, Anal. PDE, 6 (2013), 287-369.
doi: 10.2140/apde.2013.6.287. |
[20] |
Y. Guo and I. Tice,
Almost exponential decay of periodic viscous surface waves without surface tension, Arch. Ration. Mech. Anal., 207 (2013), 459-531.
doi: 10.1007/s00205-012-0570-z. |
[21] |
Y. Guo and I. Tice,
Decay of viscous surface waves without surface tension in horizontally infinite domains, Anal. PDE, 6 (2013), 1429-1533.
doi: 10.2140/apde.2013.6.1429. |
[22] |
A. A. Ilyin, The Euler equations with dissipation, Mat. Sb., 182 (1991), 1729-1739. Google Scholar |
[23] |
J. H. Jang, I. Tice and Y. J. Wang,
The compressible viscous surface-internal wave problem: Local well-posedness, SIAM J. Math. Anal., 48 (2016), 2602-2673.
doi: 10.1137/15M1036026. |
[24] |
J. H. Jang, I. Tice and Y. J. Wang,
The compressible viscous surface-internal wave problem: Stability and vanishing surface tension limit, Comm. Math. Phys., 343 (2016), 1039-1113.
doi: 10.1007/s00220-016-2603-1. |
[25] |
J. H. Jang, I. Tice and Y. J. Wang,
The compressible viscous surface-internal wave problem: Nonlinear Rayleigh-Taylor instability, Arch. Ration. Mech. Anal., 221 (2016), 215-272.
doi: 10.1007/s00205-015-0960-0. |
[26] |
R. H. Pan and K. Zhao,
The 3D compressible Euler equations with damping in a bounded domain, J. Differential Equations, 246 (2009), 581-596.
doi: 10.1016/j.jde.2008.06.007. |
[27] |
J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York, 1979. Google Scholar |
[28] |
F. Pusateri,
On the limit as the surface tension and density ratio tend to zero for the two-phase Euler equations, J. Hyperbolic Differ. Equ., 8 (2011), 347-373.
doi: 10.1142/S021989161100241X. |
[29] |
J.-C. Saut,
Remarks on the damped stationary Euler equations, Differ. Integral Equ., 3 (1990), 801-812.
|
[30] |
J. Shatah and C. C. Zeng,
A priori estimates for fluid interface problems, Comm. Pure Appl. Math., 61 (2008), 848-876.
doi: 10.1002/cpa.20241. |
[31] |
J. Shatah and C. C. Zeng,
Local well-posedness for fluid interface problems, Arch. Ration. Mech. Anal., 199 (2011), 653-705.
doi: 10.1007/s00205-010-0335-5. |
[32] |
T. C. Sideris, B. Thomases and D. H. Wang,
Long time behavior of solutions to the 3D compressible Euler equations with damping, Comm. Partial Differential Equations, 28 (2003), 795-816.
doi: 10.1081/PDE-120020497. |
[33] |
B. Stevens,
Short-time structural stability of compressible vortex sheets with surface tension, Arch. Ration. Mech. Anal., 222 (2016), 603-730.
doi: 10.1007/s00205-016-1009-8. |
[34] |
H. Stommel,
The westward intensification of wind-driven ocean currents, Trans. Amer. Geophys. Union, 29 (1948), 202-206.
doi: 10.1029/TR029i002p00202. |
[35] |
Y. J. Wang and I. Tice,
The viscous surface-internal wave problem: Nonlinear Rayleigh-Taylor instability, Comm. Partial Differential Equations, 37 (2012), 1967-2028.
doi: 10.1080/03605302.2012.699498. |
[36] |
Y. J. Wang, I. Tice and C. Kim,
The viscous surface-internal wave problem: Global well-posedness and decay, Arch. Rational Mech. Anal., 212 (2014), 1-92.
doi: 10.1007/s00205-013-0700-2. |
show all references
References:
[1] |
D. M. Ambrose,
Well-posedness of vortex sheets with surface tension, SIAM J. Math. Anal., 35 (2003), 211-244.
doi: 10.1137/S0036141002403869. |
[2] |
D. M. Ambrose and N. Masmoudi,
Well-posedness of 3D vortex sheets with surface tension, Commun. Math. Sci., 5 (2007), 391-430.
doi: 10.4310/CMS.2007.v5.n2.a9. |
[3] |
V. Barcilon, P. Constantin and E. S. Titi,
Existence of solutions to the Stommel-Charney model of the gulf stream, SIAM J. Math. Anal., 19 (1988), 1355-1364.
doi: 10.1137/0519099. |
[4] |
J. T. Beale,
Large-time regularity of viscous surface waves, Arch. Ration. Mech. Anal., 84 (1983/84), 307-252.
doi: 10.1007/BF00250586. |
[5] |
R. E. Caflisch and O. F. Orellana,
Singular solutions and ill-posedness for the evolution of vortex sheets, SIAM J. Math. Anal., 20 (1989), 293-307.
doi: 10.1137/0520020. |
[6] |
J. G. Charney, The Gulf Stream as an inertial boundary layer, Proc. Nat. Acad. Sci. USA, 41 (1955), 731-740. Google Scholar |
[7] |
C.-H. A. Cheng, D. Coutand and S. Shkoller,
On the motion of vortex sheets with surface tension in three-dimensional Euler equations with vorticity, Comm. Pure Appl. Math., 61 (2008), 1715-1752.
doi: 10.1002/cpa.20240. |
[8] |
C.-H. A. Cheng, D. Coutand and S. Shkoller,
On the limit as the density ratio tends to zero for two perfect incompressible fluids separated by a surface of discontinuity, Comm. Partial Differential Equations, 35 (2010), 817-845.
doi: 10.1080/03605300903503115. |
[9] |
V. Chepyzhov and S. Zelik,
Infinite energy solutions for dissipative Euler equations in $\mathbb{R}^2$, J. Math. Fluid Mech., 17 (2015), 513-532.
doi: 10.1007/s00021-015-0213-x. |
[10] |
P. Constantin and F. Ramos,
Inviscid limit for damped and driven incompressible Navier-Stokes equations in $\mathbb{R}^2$, Comm. Math. Phys., 275 (2007), 529-551.
doi: 10.1007/s00220-007-0310-7. |
[11] |
J.-F. Coulombel and P. Secchi,
The stability of compressible vortex sheets in two space dimensions, Indiana Univ. Math. J., 53 (2004), 941-1012.
doi: 10.1512/iumj.2004.53.2526. |
[12] |
J.-F. Coulombel and P. Secchi, Nonlinear compressible vortex sheets in two space dimensions, Ann. Sci. Éc. Norm. Supér. (4), 41 (2008), 85–139.
doi: 10.24033/asens.2064. |
[13] |
D. Coutand and S. Shkoller,
Well-posedness of the free-surface incompressible Euler equations with or without surface tension, J. Amer. Math. Soc., 20 (2007), 829-930.
doi: 10.1090/S0894-0347-07-00556-5. |
[14] |
V. P. Dymnikov and A. N. Filatov, Mathematics of climate modelling, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 1997. |
[15] |
D. G. Ebin,
Ill-posedness of the Rayleigh-Taylor and Helmholtz problems for incompressible fluids, Comm. Partial Differential Equations, 13 (1988), 1265-1295.
doi: 10.1080/03605308808820576. |
[16] |
G. Gallavotti, Foundations of Fluid Dynamics, Texts and Monographs in Physics, Springer-Verlag, Berlin, 2002.
doi: 10.1007/978-3-662-04670-8. |
[17] |
P. Germain, N. Masmoudi and J. Shatah,
Global solutions for capillary waves equation, Comm. Pure Appl. Math., 68 (2015), 625-487.
doi: 10.1002/cpa.21535. |
[18] |
Y. Guo and I. Tice,
Linear Rayleigh-Taylor instability for viscous, compressible fluids, SIAM J. Math. Anal., 42 (2010), 1688-1720.
doi: 10.1137/090777438. |
[19] |
Y. Guo and I. Tice,
Local well-posedness of the viscous surface wave problem without surface tension, Anal. PDE, 6 (2013), 287-369.
doi: 10.2140/apde.2013.6.287. |
[20] |
Y. Guo and I. Tice,
Almost exponential decay of periodic viscous surface waves without surface tension, Arch. Ration. Mech. Anal., 207 (2013), 459-531.
doi: 10.1007/s00205-012-0570-z. |
[21] |
Y. Guo and I. Tice,
Decay of viscous surface waves without surface tension in horizontally infinite domains, Anal. PDE, 6 (2013), 1429-1533.
doi: 10.2140/apde.2013.6.1429. |
[22] |
A. A. Ilyin, The Euler equations with dissipation, Mat. Sb., 182 (1991), 1729-1739. Google Scholar |
[23] |
J. H. Jang, I. Tice and Y. J. Wang,
The compressible viscous surface-internal wave problem: Local well-posedness, SIAM J. Math. Anal., 48 (2016), 2602-2673.
doi: 10.1137/15M1036026. |
[24] |
J. H. Jang, I. Tice and Y. J. Wang,
The compressible viscous surface-internal wave problem: Stability and vanishing surface tension limit, Comm. Math. Phys., 343 (2016), 1039-1113.
doi: 10.1007/s00220-016-2603-1. |
[25] |
J. H. Jang, I. Tice and Y. J. Wang,
The compressible viscous surface-internal wave problem: Nonlinear Rayleigh-Taylor instability, Arch. Ration. Mech. Anal., 221 (2016), 215-272.
doi: 10.1007/s00205-015-0960-0. |
[26] |
R. H. Pan and K. Zhao,
The 3D compressible Euler equations with damping in a bounded domain, J. Differential Equations, 246 (2009), 581-596.
doi: 10.1016/j.jde.2008.06.007. |
[27] |
J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York, 1979. Google Scholar |
[28] |
F. Pusateri,
On the limit as the surface tension and density ratio tend to zero for the two-phase Euler equations, J. Hyperbolic Differ. Equ., 8 (2011), 347-373.
doi: 10.1142/S021989161100241X. |
[29] |
J.-C. Saut,
Remarks on the damped stationary Euler equations, Differ. Integral Equ., 3 (1990), 801-812.
|
[30] |
J. Shatah and C. C. Zeng,
A priori estimates for fluid interface problems, Comm. Pure Appl. Math., 61 (2008), 848-876.
doi: 10.1002/cpa.20241. |
[31] |
J. Shatah and C. C. Zeng,
Local well-posedness for fluid interface problems, Arch. Ration. Mech. Anal., 199 (2011), 653-705.
doi: 10.1007/s00205-010-0335-5. |
[32] |
T. C. Sideris, B. Thomases and D. H. Wang,
Long time behavior of solutions to the 3D compressible Euler equations with damping, Comm. Partial Differential Equations, 28 (2003), 795-816.
doi: 10.1081/PDE-120020497. |
[33] |
B. Stevens,
Short-time structural stability of compressible vortex sheets with surface tension, Arch. Ration. Mech. Anal., 222 (2016), 603-730.
doi: 10.1007/s00205-016-1009-8. |
[34] |
H. Stommel,
The westward intensification of wind-driven ocean currents, Trans. Amer. Geophys. Union, 29 (1948), 202-206.
doi: 10.1029/TR029i002p00202. |
[35] |
Y. J. Wang and I. Tice,
The viscous surface-internal wave problem: Nonlinear Rayleigh-Taylor instability, Comm. Partial Differential Equations, 37 (2012), 1967-2028.
doi: 10.1080/03605302.2012.699498. |
[36] |
Y. J. Wang, I. Tice and C. Kim,
The viscous surface-internal wave problem: Global well-posedness and decay, Arch. Rational Mech. Anal., 212 (2014), 1-92.
doi: 10.1007/s00205-013-0700-2. |
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