论文标题
部分偏微分方程和时间依赖的原子集的时间相关矩
Time-dependent moments from partial differential equations and the time-dependent set of atoms
论文作者
论文摘要
我们研究由部分微分方程$ \ partial_t f =νδf + g \ cdot \ nabla f + h \ cdot f $产生的时间依赖的矩和相关的多项式。对于热方程式，我们发现，在有限的时间内，几个不是正方形总和的非阴性多项式成为正方形的总和。我们表明，$ \ mathbb {r} [x，y，z] _ {\ leq 4} $中的每个非负多项式在有限的时间内成为一个正方形的总和。我们解决了方程式下移动原子的问题，$ \ partial_t f = g \ cdot \ nabla f + h \ cdot f $，$ f_0 =μ_0$是有限的原子量度量。时间演变$μ_t= \ sum_ {i = 1}^k c_i（t）\cdotΔ_{x_i（t）$的$ x_i（t）$描述由运输$ g \ g \ cdot \ cdot \ nabla $和时间相关的系数$ c_i（t）$ c_i（t） $ h $和$ \ mathrm {div} \，g $。
We study the time-dependent moments and associated polynomials arising from the partial differential equation $\partial_t f = νΔf + g\cdot\nabla f + h\cdot f$, 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 $f_0 = μ_0$ being a finitely atomic measure. The time evolution $μ_t = \sum_{i=1}^k c_i(t)\cdot δ_{x_i(t)}$ 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 $\mathrm{div}\, g$.
