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April  2020, 40(4): 2061-2087. doi: 10.3934/dcds.2020106

Global well-posedness of the free-interface incompressible Euler equations with damping

Jiali Lian

School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, China

Received  December 2018 Revised  September 2019 Published  January 2020

Fund Project: This work was supported by the National Natural Science Foundation of China (No. 11771358).

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.

Keywords: Euler, free boundary problems, two-phase, damping, global well-posedness.
Mathematics Subject Classification: 35L60, 35Q35, 35R35, 76B03, 76B15.
Citation: Jiali Lian. Global well-posedness of the free-interface incompressible Euler equations with damping. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2061-2087. doi: 10.3934/dcds.2020106
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. 

[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. 

[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.

[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. 

[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. 

[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.

[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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