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

TL;DR

This paper investigates the evolution of moments and polynomials under certain PDEs, showing how non-negative polynomials become sums of squares over time and providing explicit solutions for atomic measures' dynamics.

Contribution

It introduces new results on sum of squares representations under heat equations and explicitly solves atom movement in PDEs with finitely atomic initial measures.

Findings

Non-negative polynomials become sums of squares in finite time under the heat equation.

Explicit formulas for the evolution of atomic measures in PDEs.

Finite-time sum of squares property for polynomials in three variables of degree up to four.

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 $\mathrm{div}\, g$.