Time-dependent moments from partial differential equations and the time-dependent set of atoms

Abstract

We study the time-dependent moments and associated polynomials arising from the partial differential equation $\partial_t f = \nu\Delta f + g\cdot\nabla f + h\cdot f$, and consider in detail the dual equation. For the heat equation we find that several non-negative polynomials which are not sums of squares become sums of squares under the heat equation in finite time. We show that every non-negative polynomial in $\mathbb{R}[x,y,z]_{\leq 4}$ becomes a sum of squares in finite time under the heat equation. We solve the problem of moving atoms under the equation $\partial_t f = g\cdot\nabla f + h\cdot f$ with $f_0 = \mu_0$ being a finitely atomic measure. The time evolution $\mu_t = \sum_{i=1}^k c_i(t)\cdot \delta_{x_i(t)}$ of the atom positions $x_i(t)$ are described by the transport term $g\cdot\nabla$ and the time-dependent coefficients $c_i(t)$ have an explicit solution depending on $x_i(t)$, $h$, and $\mathrm{div}\, g$.