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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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