Some exact solutions that describe an ideal fluid flow with a free boundary

The study is aimed at constructing exact solutions that describe unsteady flows of an ideal fluid with a free boundary. Currently existing examples of solutions of this kind have a linear or quadratic initial field of velocities. The object of the present study is flows with a cubic initial velocity field. Several exact solutions are obtained by using a method based on the complex Hopf equation for the complex velocity. One of the main problems in solving the Euler equations in the domain with a free boundary is to study the positions of singular points because these points can reach the free boundary with time, which may lead to solution breakdown. In the present case, singularities can be found explicitly. However, as the initial velocity field is defined by a third-power polynomial, one has to study the Riemann surfaces for the function of complex velocity in order to determine the mutual positions of the singularities and flow domain. It is shown that any initial velocity field defined by a cubic polynomial yields one of seven flows described in the paper with time. It is of interest that there are examples of flows that exist for a finite time and break down when the singularities reach the free boundary. Using a certain modification of the Hopf equation, we obtain a two-parameter family of exact solutions that describe fluid flows with a free boundary. Two examples of this family are considered.