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Singularity formation for relativistic Euler and Euler-Poisson equations with repulsive force

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  • In this paper, we investigate the singularities formation for the relativistic Euler and Euler-Poisson equations with repulsive force, in spherical symmetry. We will show the non-trivial regular solution $(p^{\frac{\gamma-1}{2\gamma}}\in C^1$) with compact support in $[0,R]$, under a certain condition on initial data \begin{eqnarray} H_0:=\int_0^R rv_0dr>0, \end{eqnarray} will blow up, where $0 < R < \min\{\frac{8}{45},\frac{8(\gamma-1)}{5\gamma+23}\}$ is a constant. Since every term in the relativistic Euler-Poisson equations corresponding to $\rho, v$ in non-relativistic case [34] has a relativistic factor $\frac{1}{\sqrt{1-v^2/c^2}}$, we will separate variables and estimate some integration items instead of direct using integration method as [34].
    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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