## 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.

### MSC:

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