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

[2211.04416] Time-dependent moments from partial differential equations and the time-dependent set of atoms Skip to main content Search Submit Donate Log in Mathematics > Functional Analysis arXiv:2211.04416 (math) [Submitted on 8 Nov 2022 ( v1 ), last revised 26 Aug 2026 (this version, v3)] Title: Time-dependent moments from partial differential equations and the time-dependent set of atoms Authors: Raúl E. Curto , Philipp J. di Dio , Milan Korda , Victor Magron View PDF 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 $\mathrm{div}\, g$. Comments: Extended Results Subjects: Functional Analysis (math.FA) MSC classes: Primary: 47A57, 44A60, Secondary: 30E05, 65D32 Cite as: arXiv:2211.04416 [math.FA] (or arXiv:2211.04416v3 [math.FA] for this version) https://doi.org/10.48550/arXiv.2211.04416 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Raul Curto [ view email ] [v1] Tue, 8 Nov 2022 18:00:15 UTC (24 KB) [v2] Mon, 13 Mar 2023 12:52:03 UTC (28 KB) [v3] Wed, 26 Aug 2026 19:18:24 UTC (29 KB) Full-text links: Access Paper: View PDF TeX Source view license Current browse context: math.FA < prev | next > new | recent | 2022-11 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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