Well-posedness for the motion of an incompressible liquid with free surface boundary. (English) Zbl 1095.35021

The author studies a model for the motion of the ocean or a star. A perfect incompressible fluid occupies a time depended domain \(\Omega(t)\subset \mathbb R^n\). The initial-boundary value problem is considered \[ \frac{\partial v}{\partial t}+(v\cdot\nabla)v+\nabla p=0,\quad\text{div}\,v=0 \quad \text{in}\quad\Omega(t),\quad 0<t<T,\tag{1} \]
\[ p=0\quad \text{on}\quad\partial\Omega(t),\quad 0<t<T,\tag{2} \]
\[ \partial\Omega(t)\text{ moves with the velocity \(v\) of the fluid particle at the boundary},\tag{3} \]
\[ \Omega(0)=\Omega_0,\quad v(x,0)=v_0(x),\quad x\in\Omega_0.\tag{4} \] Here \(v(x,t)\) is the velocity of the fluid, \(p\) is the pressure, \(\Omega_0\) is a given domain with smooth boundary, \(v_0(x)\) is a given vector.
It is supposed that \(p\) satisfies to a natural physical condition \[ \frac{\partial p}{\partial n}\leq-c_0<0\quad \text{on}\quad\partial\Omega(t),\quad 0<t<T,\tag{5} \] where \(n\) is the exterior normal to the free surface \(\partial\Omega(t)\). It means that the pressure \(p\) has to be positive in the interior of the fluid.
The local existence in Sobolev spaces to the problem (1) - (4) is proved under the assumption that (5) holds initially when \(t=0\). Lagrangian coordinates and Nash-Moser technique are used.


35Q05 Euler-Poisson-Darboux equations
76B07 Free-surface potential flows for incompressible inviscid fluids
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