Blowup for regular solutions and $ C^{1} $ solutions of the two-phase model in $ \mathbb{R}^{N} $ with a free boundary

Abstract

In this paper, under the assumption of an initial bounded region $ \Omega(0) $, we establish the blowup phenomenon of the regular solutions and $ C^{1} $ solutions to the two-phase model in $ \mathbb{R}^{N} $. If the total energy $ E $ and the total mass $ M > 0 $ satisfy $ \begin{equation} \nonumber \max\limits_{\vec{x_{0}}\in\partial\Omega(0)}\sum\limits_{i = 1}^{N}u_{i}^{2}(0,\vec{x_{0}})<\frac{\min\{2,N(\Gamma-1),N(\gamma-1)\}E}{M}, \end{equation} $ where $ E = \int_{\Omega(0)}\left(\frac{1}{2}n\left\vert \vec{u}\right\vert ^{2} +\frac{1}{2}\rho\left\vert \vec{u}\right\vert ^{2}+\frac{1}{\Gamma-1}n^{\Gamma}+\frac{1}{\gamma-1}\rho^{\gamma}\right) dV $ and $ M = {\int_{\Omega(0)}} (n+\rho) dV > 0 $, then the blowup of the solutions to the two-phase model will be formed in finite time in $ \mathbb{R}^{N} $. Furthermore, under the assumptions that the radially symmetric initial data and initial density contain vacuum states, the blowup of the smooth solutions to the two-phase model will be formed in finite time in $ \mathbb{R}^{N} (N \geq2) $.

Keywords

• two-phase model
• singularity formation
• free boundary problems
• regular solutions
• vacuum