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

## On the 2D free boundary Euler equation

 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.
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.   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, (2011).   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.   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.   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.   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.   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.   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.   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.   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.   Google Scholar [11] T. Iguchi, Well-posedness of the initial value problem for capillary-gravity waves,, Funkcial. Ekvac., 44 (2001), 219.   Google Scholar [12] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations,, Comm. Pure Appl. Math., 41 (1988), 891.   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.   Google Scholar [14] I. Kukavica and A. Tuffaha, Solutions to a fluid-structure interaction free boundary problem,, Discrete Contin. Dyn. Syst., 32 (2012), 1355.   Google Scholar [15] D. Lannes, Well-posedness of the water-waves equations,, J. Amer. Math. Soc., 18 (2005), 605.   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.   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.   Google Scholar [18] V. I. Nalimov, The Cauchy-Poisson problem,, Dinamika Splošn. Sredy no. Vyp. 18 Dinamika Zidkost. so Svobod. Granicami, (1974), 104.   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.   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.   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.   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.   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.   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.   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.   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.   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.   Google Scholar [29] H. Yosihara, Capillary-gravity waves for an incompressible ideal fluid,, J. Math. Kyoto Univ., 23 (1983), 649.   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.   Google Scholar

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##### References:
 [1] T. Alazard, N. Burq and C. Zuily, On the water-wave equations with surface tension,, Duke Math. J., 158 (2011), 413.   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, (2011).   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.   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.   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.   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.   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.   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.   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.   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.   Google Scholar [11] T. Iguchi, Well-posedness of the initial value problem for capillary-gravity waves,, Funkcial. Ekvac., 44 (2001), 219.   Google Scholar [12] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations,, Comm. Pure Appl. Math., 41 (1988), 891.   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.   Google Scholar [14] I. Kukavica and A. Tuffaha, Solutions to a fluid-structure interaction free boundary problem,, Discrete Contin. Dyn. Syst., 32 (2012), 1355.   Google Scholar [15] D. Lannes, Well-posedness of the water-waves equations,, J. Amer. Math. Soc., 18 (2005), 605.   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.   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.   Google Scholar [18] V. I. Nalimov, The Cauchy-Poisson problem,, Dinamika Splošn. Sredy no. Vyp. 18 Dinamika Zidkost. so Svobod. Granicami, (1974), 104.   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.   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.   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.   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.   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.   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.   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.   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.   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.   Google Scholar [29] H. Yosihara, Capillary-gravity waves for an incompressible ideal fluid,, J. Math. Kyoto Univ., 23 (1983), 649.   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.   Google Scholar
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