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December  2012, 1(2): 297-314. doi: 10.3934/eect.2012.1.297

On the 2D free boundary Euler equation

Igor Kukavica 1, and  Amjad Tuffaha 2,

1. 

Department of Mathematics, University of Southern California, Los Angeles, CA 90089
2. 

Department of Mathematics, The Petroleum Institute, Abu Dhabi

Received  May 2012 Revised  July 2012 Published  October 2012

We provide a new simple proof of local-in-time existence of regular solutions to the Euler equation on a domain with a free moving boundary and without surface tension in 2 space dimensions. We prove the existence under the condition that the initial velocity belongs to the Sobolev space $H^{2.5+δ}$ where $\delta>0$ is arbitrary.
Keywords: surface water waves, shallow water, Free moving boundary problem, free boundary euler equations with no surface tension., incompressible inviscid flow.
Mathematics Subject Classification: Primary: 35Q31, 74J15; Secondary: 35R3.
Citation: Igor Kukavica, Amjad Tuffaha. On the 2D free boundary Euler equation. Evolution Equations & Control Theory, 2012, 1 (2) : 297-314. doi: 10.3934/eect.2012.1.297
References:
[1]

T. Alazard, N. Burq and C. Zuily, On the water-wave equations with surface tension, Duke Math. J., 158 (2011), 413-499.  Google Scholar

[2]

T. Alazard, N. Burq and C. Zuily, Low regularity Cauchy theory for the water-waves problem: canals and swimming pools, Journeés Équations aux Dérivées Partielles, Biarritz 6 Juin-10 Juin (2011), Exposé no. III, p. 20. Google Scholar

[3]

D. M. Ambrose and N. Masmoudi, The zero surface tension limit of two-dimensional water waves, Comm. Pure Appl. Math., 58 (2005), 1287-1315.  Google Scholar

[4]

D. M. Ambrose and N. Masmoudi, The zero surface tension limit of three-dimensional water waves, Indiana Univ. Math. J., 58 (2009), 479-521.  Google Scholar

[5]

J. Thomas Beale, The initial value problem for the Navier-Stokes equations with a free surface, Comm. Pure Appl. Math., 34 (1981), 359-392.  Google Scholar

[6]

J. T. Beale, T. Y. Hou, and J. S. Lowengrub, Growth rates for the linearized motion of fluid interfaces away from equilibrium, Comm. Pure Appl. Math., 46 (1993), 1269-1301.  Google Scholar

[7]

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.  Google Scholar

[8]

D. Coutand and S. Shkoller, A simple proof of well-posedness for the free-surface incompressible Euler equations, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 429-449.  Google Scholar

[9]

D. Christodoulou and H. Lindblad, On the motion of the free surface of a liquid, Comm. Pure Appl. Math., 53 (2000), 1536-1602.  Google Scholar

[10]

D. G. Ebin, The equations of motion of a perfect fluid with free boundary are not well posed, Comm. Partial Differential Equations, 12 (1987), 1175-1201.  Google Scholar

[11]

T. Iguchi, Well-posedness of the initial value problem for capillary-gravity waves, Funkcial. Ekvac., 44 (2001), 219-241.  Google Scholar

[12]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  Google Scholar

[13]

C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347.  Google Scholar

[14]

I. Kukavica and A. Tuffaha, Solutions to a fluid-structure interaction free boundary problem, Discrete Contin. Dyn. Syst., 32 (2012), 1355-1389.  Google Scholar

[15]

D. Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc., 18 (2005), 605-654 (electronic).  Google Scholar

[16]

H. Lindblad, Well-posedness for the linearized motion of an incompressible liquid with free surface boundary, Comm. Pure Appl. Math., 56 (2003), 153-197.  Google Scholar

[17]

H. Lindblad, Well-posedness for the motion of an incompressible liquid with free surface boundary, Ann. of Math. (2), 162 (2005), 109-194.  Google Scholar

[18]

V. I. Nalimov, The Cauchy-Poisson problem, Dinamika Splošn. Sredy no. Vyp. 18 Dinamika Zidkost. so Svobod. Granicami,(1974), 104-210, 254 .  Google Scholar

[19]

M. Ogawa and A. Tani, Free boundary problem for an incompressible ideal fluid with surface tension, Math. Models Methods Appl. Sci., 12 (2002), 1725-1740.  Google Scholar

[20]

B. Schweizer, On the three-dimensional Euler equations with a free boundary subject to surface tension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 753-781.  Google Scholar

[21]

J. Shatah and C. Zeng, Geometry and a priori estimates for free boundary problems of the Euler equation, Comm. Pure Appl. Math., 61 (2008), 698-744.  Google Scholar

[22]

A. I. Shnirelman, The geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid, Mat. Sb. (N.S.), 128(170) (1985), 82-109, 144.  Google Scholar

[23]

A. Tani, Small-time existence for the three-dimensional Navier-Stokes equations for an incompressible fluid with a free surface, Arch. Rational Mech. Anal., 133 (1996), 299-331.  Google Scholar

[24]

T. Tao, Harmonic analysis,, , ().   Google Scholar

[25]

S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in $2$-D, Invent. Math., 130 (1997), 39-72.  Google Scholar

[26]

S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc., 12 (1999), 445-495.  Google Scholar

[27]

L. Xu and Z. Zhang, On the free boundary problem to the two viscous immiscible fluids, J. Differential Equations, 248 (2010), 1044-1111.  Google Scholar

[28]

H. Yosihara, Gravity waves on the free surface of an incompressible perfect fluid of finite depth, Publ. Res. Inst. Math. Sci., 18 (1982), 49-96.  Google Scholar

[29]

H. Yosihara, Capillary-gravity waves for an incompressible ideal fluid, J. Math. Kyoto Univ., 23 (1983), 649-694.  Google Scholar

[30]

P. Zhang and Z. Zhang, On the free boundary problem of three-dimensional incompressible Euler equations, Comm. Pure Appl. Math., 61 (2008), 877-940.  Google Scholar

show all references

References:
[1]

T. Alazard, N. Burq and C. Zuily, On the water-wave equations with surface tension, Duke Math. J., 158 (2011), 413-499.  Google Scholar

[2]

T. Alazard, N. Burq and C. Zuily, Low regularity Cauchy theory for the water-waves problem: canals and swimming pools, Journeés Équations aux Dérivées Partielles, Biarritz 6 Juin-10 Juin (2011), Exposé no. III, p. 20. Google Scholar

[3]

D. M. Ambrose and N. Masmoudi, The zero surface tension limit of two-dimensional water waves, Comm. Pure Appl. Math., 58 (2005), 1287-1315.  Google Scholar

[4]

D. M. Ambrose and N. Masmoudi, The zero surface tension limit of three-dimensional water waves, Indiana Univ. Math. J., 58 (2009), 479-521.  Google Scholar

[5]

J. Thomas Beale, The initial value problem for the Navier-Stokes equations with a free surface, Comm. Pure Appl. Math., 34 (1981), 359-392.  Google Scholar

[6]

J. T. Beale, T. Y. Hou, and J. S. Lowengrub, Growth rates for the linearized motion of fluid interfaces away from equilibrium, Comm. Pure Appl. Math., 46 (1993), 1269-1301.  Google Scholar

[7]

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.  Google Scholar

[8]

D. Coutand and S. Shkoller, A simple proof of well-posedness for the free-surface incompressible Euler equations, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 429-449.  Google Scholar

[9]

D. Christodoulou and H. Lindblad, On the motion of the free surface of a liquid, Comm. Pure Appl. Math., 53 (2000), 1536-1602.  Google Scholar

[10]

D. G. Ebin, The equations of motion of a perfect fluid with free boundary are not well posed, Comm. Partial Differential Equations, 12 (1987), 1175-1201.  Google Scholar

[11]

T. Iguchi, Well-posedness of the initial value problem for capillary-gravity waves, Funkcial. Ekvac., 44 (2001), 219-241.  Google Scholar

[12]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  Google Scholar

[13]

C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347.  Google Scholar

[14]

I. Kukavica and A. Tuffaha, Solutions to a fluid-structure interaction free boundary problem, Discrete Contin. Dyn. Syst., 32 (2012), 1355-1389.  Google Scholar

[15]

D. Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc., 18 (2005), 605-654 (electronic).  Google Scholar

[16]

H. Lindblad, Well-posedness for the linearized motion of an incompressible liquid with free surface boundary, Comm. Pure Appl. Math., 56 (2003), 153-197.  Google Scholar

[17]

H. Lindblad, Well-posedness for the motion of an incompressible liquid with free surface boundary, Ann. of Math. (2), 162 (2005), 109-194.  Google Scholar

[18]

V. I. Nalimov, The Cauchy-Poisson problem, Dinamika Splošn. Sredy no. Vyp. 18 Dinamika Zidkost. so Svobod. Granicami,(1974), 104-210, 254 .  Google Scholar

[19]

M. Ogawa and A. Tani, Free boundary problem for an incompressible ideal fluid with surface tension, Math. Models Methods Appl. Sci., 12 (2002), 1725-1740.  Google Scholar

[20]

B. Schweizer, On the three-dimensional Euler equations with a free boundary subject to surface tension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 753-781.  Google Scholar

[21]

J. Shatah and C. Zeng, Geometry and a priori estimates for free boundary problems of the Euler equation, Comm. Pure Appl. Math., 61 (2008), 698-744.  Google Scholar

[22]

A. I. Shnirelman, The geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid, Mat. Sb. (N.S.), 128(170) (1985), 82-109, 144.  Google Scholar

[23]

A. Tani, Small-time existence for the three-dimensional Navier-Stokes equations for an incompressible fluid with a free surface, Arch. Rational Mech. Anal., 133 (1996), 299-331.  Google Scholar

[24]

T. Tao, Harmonic analysis,, , ().   Google Scholar

[25]

S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in $2$-D, Invent. Math., 130 (1997), 39-72.  Google Scholar

[26]

S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc., 12 (1999), 445-495.  Google Scholar

[27]

L. Xu and Z. Zhang, On the free boundary problem to the two viscous immiscible fluids, J. Differential Equations, 248 (2010), 1044-1111.  Google Scholar

[28]

H. Yosihara, Gravity waves on the free surface of an incompressible perfect fluid of finite depth, Publ. Res. Inst. Math. Sci., 18 (1982), 49-96.  Google Scholar

[29]

H. Yosihara, Capillary-gravity waves for an incompressible ideal fluid, J. Math. Kyoto Univ., 23 (1983), 649-694.  Google Scholar

[30]

P. Zhang and Z. Zhang, On the free boundary problem of three-dimensional incompressible Euler equations, Comm. Pure Appl. Math., 61 (2008), 877-940.  Google Scholar

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