In fluid mechanics and astrophysics, relativistic Euler equations can be used to describe the effects of special relativity which are an extension of the classical Euler equations. In this paper, we will consider the initial value problem of relativistic Euler equations in an initial bounded region of $\mathbb{R}^{N}$. If the initial velocity satisfies

$$\mathop{\max}_{\vec{x}_{0}\in \partial\Omega(0)}\sum_{i=1}^{N}v_{i}^{2}(0,\vec{x}_{0})<\frac{c^{2}A_{1}}{2},$$

where $A_{1}$ is a positive constant depend on some sufficiently large $T^{*}$, then we can get the non-global existence of the regular solution for the $N$-dimensional relativistic Euler equations.