Well-posedness of the free-surface incompressible Euler equations with or without surface tension.(English)Zbl 1123.35038

The paper deals with local existence and uniqueness of solutions in Sobolev spaces to free boundary incompressible Euler equations in vacuum:
$\partial_t+\nabla_uu+\nabla p=0,\qquad \text{div}\;u=0\quad \text{in } Q,$
$p=\sigma H\text{ on}\;\partial Q,\quad \left(\partial_t+\nabla_u\right)|_{\partial Q}\in T(\partial Q),\quad u=u_0\text{ at }t=0, \quad Q(0)=\Omega,$
where $$Q(t)=\bigcup_{0\leq t\leq T} \{t\}\times\Omega(t)$$, $$\Omega(t)\in \mathbb R_n$$, $$n=2$$ or 3, $$\partial Q(t)=\bigcup_{0\leq t\leq T} \{t\}\times\partial\Omega(t)$$, $$\nabla_uu=u^j\partial u^i/\partial x_j$$, the vector field $$u$$ is the Eulerian or spatial velocity field defined on the time-dependent domain $$\Omega(t)$$, $$p$$ denotes the pressure function, $$H$$ is twice the mean curvature of the boundary of the fluid $$\partial\Omega(t)$$, and $$\sigma$$ is the surface tension.
The two following main theorems are proved.
Theorem 1.1. Let $$\sigma>0$$, $$\partial\Omega$$ be of class $$H^{5.5}$$, and $$u_0\in H^{4.5}(\Omega)$$. Then, there exists $$T>0$$ and a solution $$u(t)$$, $$p(t)$$, $$\Omega(t)$$ of the problem with $$u\in L^\infty(0,T;H^{4.5}(\Omega(t)))$$, $$p\in L^\infty(0,T;H^4(\Omega(t)))$$. The solution is unique if $$u_0\in H^{5.5}(\Omega)$$ and $$\partial\Omega\in H^{6.5}$$.
Theorem 1.2. Let $$\sigma=0$$, $$\partial\Omega$$ be of class $$H^3$$, and $$u_0\in H^3(\Omega)$$ and condition $$\nabla p\cdot n<0$$ on $$\partial Q$$ holds at $$t=0$$. Then, there exists $$T>0$$ and a unique solution $$u(t)$$, $$p(t)$$, $$\Omega(t)$$ of the problem with $$u\in L^\infty(0,T;H^3(\Omega(t)))$$, $$p\in L^\infty(0,T;H^{3.5}(\Omega(t)))$$, and $$\partial\Omega(t)\in H^3$$.
In the proof, the Eulerian problem, set on the moving domain $$\Omega(t)$$, is converted to a system on the fixed domain $$\Omega$$, by using Lagrangian variables.

MSC:

 35Q05 Euler-Poisson-Darboux equations 76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids 35Q35 PDEs in connection with fluid mechanics 35R35 Free boundary problems for PDEs
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References:

 [1] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. · Zbl 0314.46030 [2] David M. Ambrose, Well-posedness of vortex sheets with surface tension, SIAM J. Math. Anal. 35 (2003), no. 1, 211 – 244. · Zbl 1107.76010 · doi:10.1137/S0036141002403869 [3] David M. Ambrose and Nader Masmoudi, The zero surface tension limit of two-dimensional water waves, Comm. Pure Appl. Math. 58 (2005), no. 10, 1287 – 1315. · Zbl 1086.76004 · doi:10.1002/cpa.20085 [4] J. Thomas Beale, Thomas Y. Hou, and John S. Lowengrub, Growth rates for the linearized motion of fluid interfaces away from equilibrium, Comm. Pure Appl. Math. 46 (1993), no. 9, 1269 – 1301. · Zbl 0796.76041 · doi:10.1002/cpa.3160460903 [5] Walter Craig, An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits, Comm. Partial Differential Equations 10 (1985), no. 8, 787 – 1003. · Zbl 0577.76030 · doi:10.1080/03605308508820396 [6] Guido Schneider and C. Eugene Wayne, The long-wave limit for the water wave problem. I. The case of zero surface tension, Comm. Pure Appl. Math. 53 (2000), no. 12, 1475 – 1535. , https://doi.org/10.1002/1097-0312(200012)53:123.0.CO;2-V Demetrios Christodoulou and Hans Lindblad, On the motion of the free surface of a liquid, Comm. Pure Appl. Math. 53 (2000), no. 12, 1536 – 1602. , https://doi.org/10.1002/1097-0312(200012)53:123.3.CO;2-H [7] Daniel Coutand and Steve Shkoller, Motion of an elastic solid inside an incompressible viscous fluid, Arch. Ration. Mech. Anal. 176 (2005), no. 1, 25 – 102. · Zbl 1064.74057 · doi:10.1007/s00205-004-0340-7 [8] Daniel Coutand and Steve Shkoller, The interaction between quasilinear elastodynamics and the Navier-Stokes equations, Arch. Ration. Mech. Anal. 179 (2006), no. 3, 303 – 352. · Zbl 1138.74325 · doi:10.1007/s00205-005-0385-2 [9] Stefan Ebenfeld, \?²-regularity theory of linear strongly elliptic Dirichlet systems of order 2\? with minimal regularity in the coefficients, Quart. Appl. Math. 60 (2002), no. 3, 547 – 576. · Zbl 1030.35045 · doi:10.1090/qam/1914441 [10] David G. Ebin, The equations of motion of a perfect fluid with free boundary are not well posed, Comm. Partial Differential Equations 12 (1987), no. 10, 1175 – 1201. · Zbl 0631.76018 · doi:10.1080/03605308708820523 [11] David Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc. 18 (2005), no. 3, 605 – 654. · Zbl 1069.35056 [12] Hans Lindblad, Well-posedness for the linearized motion of an incompressible liquid with free surface boundary, Comm. Pure Appl. Math. 56 (2003), no. 2, 153 – 197. · Zbl 1025.35017 · doi:10.1002/cpa.10055 [13] Hans Lindblad, Well-posedness for the motion of an incompressible liquid with free surface boundary, Ann. of Math. (2) 162 (2005), no. 1, 109 – 194. · Zbl 1095.35021 · doi:10.4007/annals.2005.162.109 [14] V. I. Nalimov, The Cauchy-Poisson problem, Dinamika Splošn. Sredy Vyp. 18 Dinamika Židkost. so Svobod. Granicami (1974), 104 – 210, 254 (Russian). [15] J. Shatah, C. Zeng, GEOMETRY AND A PRIORI ESTIMATES FOR FREE BOUNDARY PROBLEMS OF THE EULER’S EQUATION, preprint, (2006). [16] Ben Schweizer, On the three-dimensional Euler equations with a free boundary subject to surface tension, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), no. 6, 753 – 781. · Zbl 1148.35071 · doi:10.1016/j.anihpc.2004.11.001 [17] V. A. Solonnikov, Solvability of a problem on the evolution of a viscous incompressible fluid, bounded by a free surface, on a finite time interval, Algebra i Analiz 3 (1991), no. 1, 222 – 257 (Russian); English transl., St. Petersburg Math. J. 3 (1992), no. 1, 189 – 220. [18] V. A. Solonnikov and V. E. Ščadilov, A certain boundary value problem for the stationary system of Navier-Stokes equations, Trudy Mat. Inst. Steklov. 125 (1973), 196 – 210, 235 (Russian). Boundary value problems of mathematical physics, 8. [19] Michael E. Taylor, Partial differential equations, Texts in Applied Mathematics, vol. 23, Springer-Verlag, New York, 1996. Basic theory. Michael E. Taylor, Partial differential equations. I, Applied Mathematical Sciences, vol. 115, Springer-Verlag, New York, 1996. Basic theory. Michael E. Taylor, Partial differential equations. II, Applied Mathematical Sciences, vol. 116, Springer-Verlag, New York, 1996. Qualitative studies of linear equations. Michael E. Taylor, Partial differential equations. III, Applied Mathematical Sciences, vol. 117, Springer-Verlag, New York, 1997. Nonlinear equations; Corrected reprint of the 1996 original. · Zbl 0869.35001 [20] Sijue Wu, Well-posedness in Sobolev spaces of the full water wave problem in 2-D, Invent. Math. 130 (1997), no. 1, 39 – 72. · Zbl 0892.76009 · doi:10.1007/s002220050177 [21] Sijue Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc. 12 (1999), no. 2, 445 – 495. · Zbl 0921.76017 [22] Hideaki Yosihara, Gravity waves on the free surface of an incompressible perfect fluid of finite depth, Publ. Res. Inst. Math. Sci. 18 (1982), no. 1, 49 – 96. · Zbl 0493.76018 · doi:10.2977/prims/1195184016
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