Time-Dependent Moments From the Heat Equation and a Transport Equation
نویسندگان
چکیده
Abstract We present a new connection between the classical theory of full and truncated moment problems partial differential equations, as follows. For heat equation $\partial _t u = {\nu } \Delta u$, with initial data $u_0 \in {\mathcal {S}}(\mathds {R}^n)$, we first compute moments $s_{\alpha }(t)$ unique solution $u {R}^n)$. These are polynomials in time variable, degree comparable to $\alpha $, coefficients satisfying recursive relation. This allows us define for any sequence, prove that they preserve some features kernel. In case sequences, trace curve (which call curve), which remains cone positive time, but may wander outside negative time. provides description boundary points cone, also sequences. study how determinacy sequence behaves along curve. Next, consider transport ax \cdot \nabla u$ conduct similar analysis. Along way incorporate several illustrating examples. show while }\Delta + ax\cdot has no explicit solution, time-dependent can be explicitly calculated.
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ژورنال
عنوان ژورنال: International Mathematics Research Notices
سال: 2022
ISSN: ['1687-0247', '1073-7928']
DOI: https://doi.org/10.1093/imrn/rnac244