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- W4308672432 abstract "We study the time-dependent moments and associated polynomials arising from the partial differential equation $partial_t f = nuDelta f + gcdotnabla f + hcdot f$, 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 $partial_t f = gcdotnabla f + hcdot f$ 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 $gcdotnabla$ and the time-dependent coefficients $c_i(t)$ have an explicit solution depending on $x_i(t)$, $h$, and $mathrm{div}, g$." @default.
- W4308672432 created "2022-11-14" @default.
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- W4308672432 date "2022-11-08" @default.
- W4308672432 modified "2023-10-16" @default.
- W4308672432 title "Time-dependent moments from partial differential equations and the time-dependent set of atoms" @default.
- W4308672432 doi "https://doi.org/10.48550/arxiv.2211.04416" @default.
- W4308672432 hasPublicationYear "2022" @default.
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