Preprint
Time-dependent moments from partial differential equations and the time-dependent set of atoms
ArXiv.org
11/08/2022
DOI: 10.48550/arXiv.2211.04416
Abstract
We study the time-dependent moments $s_\alpha(t) = \int x^\alpha\cdot
f(x,t)~\mathrm{d} x$ of the solution $f$ of the partial differential equation
$\partial_t f = \nu\Delta f + g\cdot\nabla f + h\cdot f$ with initial Schwartz
function data $f_0\in\mathcal{S}(\mathbb{R}^n)$. At first we describe the dual
action on the polynomials, i.e., the time-evolution of $f$ is completely moved
to the polynomial side $s_\alpha(t) = \int p(x,t)\cdot f_0(x)~\mathrm{d} x$. We
investigate the special case of 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. Finally, 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. We find that in the time evolution
$\mu_t = \sum_{i=1}^k c_i(t)\cdot \delta_{x_i(t)}$ the atom positions $x_i(t)$
are governed only by the transport term $g\cdot\nabla$ and that the
time-dependent coefficients $c_i(t)$ have an analytic solution depending on
$x_i(t)$.
Details
- Title: Subtitle
- Time-dependent moments from partial differential equations and the time-dependent set of atoms
- Creators
- Raúl E CurtoPhilipp J di DioMilan KordaVictor Magron
- Resource Type
- Preprint
- Publication Details
- ArXiv.org
- DOI
- 10.48550/arXiv.2211.04416
- ISSN
- 2331-8422
- Language
- English
- Date posted
- 11/08/2022
- Academic Unit
- Mathematics
- Record Identifier
- 9984311558802771
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