On page 16 of the book "Turbulence" written by Uriel Frisch, the author mentions that "...It is now sufficient to impose the divergence condition $\partial_j v_j=0$ at $t=0$, since (2.13) will propagate this condition to all times." However, I could not understand why we could make such a claim. I guess it should be proved by letting $\partial_t \partial_j v_j=0$, but I do not know how it can be done, especially how to handle the second term of the (2.13). The context and equation (2.13) are provided in the following figure.
The Navier-Stokes equation $(2.13)$ is $$\partial_tv_i+\left(\delta_{i\ell}-\partial_{i\ell}\nabla^{-2}\right)\partial_j(v_jv_\ell)=\nu\nabla^2v_i.$$ We apply $\partial_i$ to both sides of $(2.13)$ and use the Einstein's summation convention, then $$\partial_t(\partial_iv_i)+\left(\delta_{i\ell}-\partial_{i\ell}\nabla^{-2}\right)\partial_{ij}(v_jv_\ell)=\nu\nabla^2(\partial_iv_i).$$ Note that for fixed $j,\ell$ we have $$\partial_{i\ell}\nabla^{-2}\partial_{ij}=\nabla^{-2}\partial_{ii}\partial_{j\ell}=\partial_{j\ell},$$ hence $$\left(\delta_{i\ell}-\partial_{i\ell}\nabla^{-2}\right)\partial_{ij}(v_jv_\ell)=\partial_{j\ell}(v_jv_\ell)-\partial_{j\ell}(v_jv_\ell)=0.$$ Therefore, we obtain the evolution equation for $\partial_jv_j$: $$\partial_t(\partial_jv_j)=\nu\nabla^2(\partial_jv_j),$$ which is a heat equation. So, if $\partial_jv_j=0$ at time $t=0$ then it will hold for all positive time $t$.