On the blowup of classical solutions to the 3-D pressure-gradient systems

The 3-D pressure-gradient system arises from the splitting of the three-dimensional compressible Euler system, which can be reduced into such a nonlinear wave equation , where , ∂i=∂xi (1⩽i⩽3) and x=(x1,x2,x3). In this paper, we will establish a blowup result of classical solutions to the 3-D pressure-gradient systems when the initial data are of small perturbations with respect to constant states.