An infinite linear hierarchy for the incompressible Navier-Stokes equation and application

This paper introduces an infinite linear hierarchy for the homogeneous, incompressible three-dimensional Navier-Stokes equation. The Cauchy problem of the hierarchy with a factorized divergence-free initial datum is shown to be equivalent to that of the incompressible Navier-Stokes equation in $\mathcal{H}^1.$ This allows us to present an explicit formula for solutions to the incompressible Navier-Stokes equation under consideration. The obtained formula is an expansion in terms of binary trees encoding the collision histories of the "particles" in a concise form. Precisely, each term in the summation of $n$ "particles" collision is expressed by a $n$-parameter singular integral operator with an explicit kernel in Fourier space, describing a kind of processes of two-body interaction of $n$ "particles". Therefore, this formula is a physical expression for the solutions of the incompressible Navier-Stokes equation.