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 $\partial _t u = {\nu } \Delta u$, with initial data $u_0 \in {\mathcal {S}}(\mathds {R}^n)$, we first compute the moments $s_{\alpha }(t)$ of the unique solution $u \in {\mathcal {S}}(\mathds {R}^n)$. 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 $\partial _t u = ax \cdot \nabla u$ and conduct a similar analysis. Along the way we incorporate several illustrating examples. We show that while $\partial _t u = {\nu }\Delta u + ax\cdot \nabla u$ has no explicit solution, the time-dependent moments can be explicitly calculated.

中文翻译：

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热方程和输运方程的瞬态矩

我们提出了完整和截断矩问题的经典理论与偏微分方程理论之间的新联系，如下所示。对于经典热方程 $\partial _t u = {\nu } \Delta u$，初始数据 $u_0 \in {\mathcal {S}}(\mathds {R}^n)$，我们首先计算矩$s_{\alpha }(t)$ 的唯一解 $u \in {\mathcal {S}}(\mathds {R}^n)$。这些矩是时间变量中的多项式，其程度与 $\alpha $ 相当，并且具有满足递归关系的系数。这允许我们为任何序列定义多项式，并证明它们保留了热核的一些特征。在矩序列的情况下，多项式描绘了一条曲线（我们称之为热曲线），该曲线在正时间段内保持在矩锥中，但可能会在力矩锥外徘徊负时间。这提供了力矩锥的边界点的描述，它们也是力矩序列。我们还研究了矩序列的确定性如何沿热曲线表现。接下来，我们考虑输运方程 $\partial _t u = ax \cdot \nabla u$ 并进行类似的分析。在此过程中，我们结合了几个说明性示例。我们证明，虽然 $\partial _t u = {\nu }\Delta u + ax\cdot \nabla u$ 没有明确的解决方案，但可以明确计算与时间相关的矩。我们考虑输运方程 $\partial _t u = ax \cdot \nabla u$ 并进行类似的分析。在此过程中，我们结合了几个说明性示例。我们证明，虽然 $\partial _t u = {\nu }\Delta u + ax\cdot \nabla u$ 没有明确的解决方案，但可以明确计算与时间相关的矩。我们考虑输运方程 $\partial _t u = ax \cdot \nabla u$ 并进行类似的分析。在此过程中，我们结合了几个说明性示例。我们证明，虽然 $\partial _t u = {\nu }\Delta u + ax\cdot \nabla u$ 没有明确的解决方案，但可以明确计算与时间相关的矩。