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

Abstract

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 ∂ t 2 v − div ( e v ∇ v ) = 0 , where div ( e v ∇ v ) = ∑ i = 1 3 ∂ i ( e v ∂ i v ) , ∂ i = ∂ x i ( 1 ⩽ i ⩽ 3 ) and x = ( x 1 , x 2 , x 3 ) . 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.