Journal article
Time-Dependent Moments From the Heat Equation and a Transport Equation
International mathematics research notices, Vol.2023(17), pp.14955-14990
08/2023
DOI: 10.1093/imrn/rnac244
Abstract
We present a new connection between the classical theory of full and truncated moment problems and the theory of partial differential equations, as follows. For the classical heat equation partial derivative(t)u = v Delta u, with initial data u(0) is an element of S(R-n), we first compute the moments s(alpha)(t) of the unique solution u is an element of S(R-n). These moments are polynomials in the time variable, of degree comparable to alpha, and with coefficients satisfying a recursive relation. This allows us to define the polynomials for any sequence, and prove that they preserve some of the features of the heat kernel. In the case of moment sequences, the polynomials trace a curve (which we call the heat curve), which remains in the moment cone for positive time, but may wander outside the moment cone for negative time. This provides a description of the boundary points of the moment cone, which are also moment sequences. We also study how the determinacy of a moment sequence behaves along the heat curve. Next, we consider the transport equation partial derivative(t)u = ax.del u and conduct a similar analysis. Along the way we incorporate several illustrating examples. We show that while partial derivative(t)u = nu Delta u + ax . del u has no explicit solution, the time-dependent moments can be explicitly calculated.
Details
- Title: Subtitle
- Time-Dependent Moments From the Heat Equation and a Transport Equation
- Creators
- Raul E. Curto - University of IowaPhilipp J. di Dio - Leipzig University
- Resource Type
- Journal article
- Publication Details
- International mathematics research notices, Vol.2023(17), pp.14955-14990
- Publisher
- Oxford Univ Press
- DOI
- 10.1093/imrn/rnac244
- ISSN
- 1073-7928
- eISSN
- 1687-0247
- Number of pages
- 36
- Grant note
- DI 2780/2-1 / Deutsche Forschungsgemeinschaft DFG; German Research Foundation (DFG) University of Konstanz
- Language
- English
- Electronic publication date
- 09/13/2022
- Date published
- 08/2023
- Academic Unit
- Mathematics
- Record Identifier
- 9984306760002771
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