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

Jingjie Wang; Xiaoyong Wen; Manwai Yuen; Jingjie Wang; Xiaoyong Wen; Manwai Yuen

doi:10.3934/math.2022839

AIMS Mathematics

2022, Volume 7, Issue 8: 15313-15330. doi: 10.3934/math.2022839

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Theory article

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

1.
School of Mathematical Sciences, Tongji University, Shanghai 200092, China
2.
School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China
3.
Department of Mathematics and Information Technology, The Education University of Hong Kong, Hong Kong

Received: 16 February 2022 Revised: 29 April 2022 Accepted: 12 May 2022 Published: 17 June 2022
MSC : 35B44, 35R35, 76T10

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) $.
- two-phase model,
- singularity formation,
- free boundary problems,
- regular solutions,
- vacuum
Citation: Jingjie Wang, Xiaoyong Wen, Manwai Yuen. Blowup for regular solutions and $ C^{1} $ solutions of the two-phase model in $ \mathbb{R}^{N} $ with a free boundary[J]. AIMS Mathematics, 2022, 7(8): 15313-15330. doi: 10.3934/math.2022839

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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) $.

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