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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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

J. T. Beale, Large-time regularity of viscous surface waves, Arch. Ration. Mech. Anal., 84 (1983/84), 307-252.  doi: 10.1007/BF00250586.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

G. Gallavotti, Foundations of Fluid Dynamics, Texts and Monographs in Physics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04670-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

J.-C. Saut, Remarks on the damped stationary Euler equations, Differ. Integral Equ., 3 (1990), 801-812.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

H. Stommel, The westward intensification of wind-driven ocean currents, Trans. Amer. Geophys. Union, 29 (1948), 202-206.  doi: 10.1029/TR029i002p00202.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

J. T. Beale, Large-time regularity of viscous surface waves, Arch. Ration. Mech. Anal., 84 (1983/84), 307-252.  doi: 10.1007/BF00250586.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

G. Gallavotti, Foundations of Fluid Dynamics, Texts and Monographs in Physics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04670-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

J.-C. Saut, Remarks on the damped stationary Euler equations, Differ. Integral Equ., 3 (1990), 801-812.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

H. Stommel, The westward intensification of wind-driven ocean currents, Trans. Amer. Geophys. Union, 29 (1948), 202-206.  doi: 10.1029/TR029i002p00202.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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