On the absence of splash singularities in the case of two-fluid interfaces (Q267525)

In this article the authors study the formation of singularities of two ideal fluids interfaces for 2D flow. It is assumed that the first fluid occupies an open time depended region \(\Omega^1(t)\subset\mathbb{R}^2\) and the second fluid occupies another open region \(\Omega^2(t)\subset\mathbb{R}^2\). The interface is \(\Gamma(t)=\overline{\Omega^1(t)}\cap\overline{\Omega^2(t)}\), \(\mathbb{R}^2=\Omega^1(t)\cup\Omega^2(t)\cup\Gamma(t)\). \(\Omega^1(t)\) and \(\Omega^2(t)\) are open, unbounded and simply-connected sets. \(\Gamma(t)\) is a smooth curve with parametrization NEWLINE\[NEWLINE \Gamma(t)=\{z(\alpha,t)=(z_1(\alpha,t),z_2(\alpha,t)):\alpha\in\mathbb{R} \}. NEWLINE\]NEWLINE Fluids have constant densities \(\rho^1,\rho^2> 0\) and flow with smooth velocities \(u^1,u^2\in C^3\).NEWLINENEWLINEThe interface function \(z\), the velocities \(u^1,u^2\) and the pressures \(p^1,p^2\) satisfy the equations NEWLINE\[NEWLINE\begin{cases} \rho^j\left(\frac{\partial u^j}{\partial t}+u^j\cdot\nabla u^j\right)+\nabla p^j= -g\rho^je_2,\quad x\in \Omega^j(t),\quad t\in[0,T],\quad j=1,2, \\ \text{div}\, u^j=0,\quad \text{curl}\, u^j=0,\quad x\in \Omega^j(t),\quad t\in[0,T],\quad j=1,2, \\ \left(\frac{\partial z}{\partial t}-u^j\left.\right|_{\Gamma(t)}\right)\cdot n=0,\quad t\in[0,T],\quad j=1,2, \\ (p^1-p^2)\left.\right|_{\Gamma(t)}=-\sigma K(z),\quad t\in[0,T]. \end{cases}NEWLINE\]NEWLINE Here \(g\) is the gravity constant, \(e_2\) is the unit vector of the \(x_2\)-axis, \(n\) is the outward normal vector to \(\partial\Omega^1\), \(\sigma\geq 0\) is the surface tension coefficient, \(K\) is the curvature of the curve \(\Gamma(t)\). It is assumed that \(z(\alpha,t)-(\alpha,0)\) is a \(2\pi\)-periodic function of \(\alpha\).NEWLINENEWLINEThe authors show that splash singularities cannot develop smoothly in the case of regular solutions of the system.