论文标题
有界和无限域中的非交通性分数傅立叶定律
The noncommutative fractional Fourier law in bounded and unbounded domains
论文作者
论文摘要
使用有关$ s $ spectrum的光谱理论，可以定义大量矢量运算符的分数幂。这种可能性会导致新的分数扩散和进化问题，对于非均匀材料而言，傅立叶定律不仅是负梯度运算符，而且是表格$ t = \ sum _ {\ ell = 1}^3e__ \ ell a__ \ el a_ e _ el（x） x =（x_1，x_2，x_3）\ in \barΩ，$$，$ω$可以是有界或无界域中的$ \ mathbb {r}^3 $，其边界$ \partialΩ$合适地被认为是正常的，$ \barΩ$是$ the $ $ e y $ e y y y y Immition n y Immition the Imimite n y Imimy n y Immimy gy Ell = 1，y Immim y Immim y Imimy Imimy Imim = 1 ,， quaternions $ \ mathbb {h} $。运算符$ t_ \ ell：= a_ \ ell（x）\ partial_ {x_ \ ell} $，对于$ \ ell = 1,2,3 $，称为$ t $和$ t $和$ a_1 $，$ a_2 $，$ a_2 $，$ a_3：\ a_3：\barΩ $ t $。在本文中，我们研究了$ t $的分数力量的产生，当操作员$ t_ \ ell $时，$p_α（t）$表示$α\ in（0,1）$，以$ \ ell = 1,2,3 $的价格不上交。要定义$ t $的分数p_α（t）$p_α（t）$，我们必须考虑与$ t $的伪$ s $ s $ s $ s-resolvent运算符相关的合适边界价值问题。在本文中，我们考虑了两个不同的边界条件。如果$ω$是无限的，我们将考虑Dirichlet边界条件。如果$ω$有限，我们将考虑与$ t $的分数幂产生的天然罗宾型边界条件。
Using the spectral theory on the $S$-spectrum it is possible to define the fractional powers of a large class of vector operators. This possibility leads to new fractional diffusion and evolution problems that are of particular interest for nonhomogeneous materials where the Fourier law is not simply the negative gradient operator but it is a nonconstant coefficients differential operator of the form $$ T=\sum_{\ell=1}^3e_\ell a_\ell(x)\partial_{x_\ell}, \ \ \ x=(x_1,x_2,x_3)\in \barΩ, $$ where, $Ω$ can be either a bounded or an unbounded domain in $\mathbb{R}^3$ whose boundary $\partialΩ$ is considered suitably regular, $\barΩ$ is the closure of $Ω$ and $e_\ell$, for $\ell=1,2,3$ are the imaginary units of the quaternions $\mathbb{H}$. The operators $T_\ell:=a_\ell(x)\partial_{x_\ell}$, for $\ell=1,2,3$, are called the components of $T$ and $a_1$, $a_2$, $a_3: \barΩ \subset\mathbb{R}^3\to \mathbb{R}$ are the coefficients of $T$. In this paper we study the generation of the fractional powers of $T$, denoted by $P_α(T)$ for $α\in(0,1)$, when the operators $T_\ell$, for $\ell=1,2,3$ do not commute among themselves. To define the fractional powers $P_α(T)$ of $T$ we have to consider the weak formulation of a suitable boundary value problem associated with the pseudo $S$-resolvent operator of $T$. In this paper we consider two different boundary conditions. If $Ω$ is unbounded we consider Dirichlet boundary conditions. If $Ω$ is bounded we consider the natural Robin-type boundary conditions associated with the generation of the fractional powers of $T$.
