On the use of the Riesz transforms to determine the pressure term in the incompressible Navier-Stokes equations on the whole space

Abstract

We give some conditions under which the pressure term in the incompressible Navier-Stokes equations on the entire d-dimensional Euclidean space is determined by the formula ∇p " ∇ ˜d ÿ i,j"1 R i R j pu i u j ´Fi,j q ¸, where d P t2, 3u, u :" pu 1 , . . . , u d q is the fluid velocity, F :" pF i,j q 1ďi,jďd is the forcing tensor, and for all k P t1, . . . , du, R k is the k-th Riesz transform.