Nonlocal Fourier Laws for Heat Propagation via Fractional powers of Vector Operators

Abstract

The present work is devoted to the study of fractional powers of vector operators, with particular emphasis on the gradient operator with non-constant coefficients. Within the setting of Clifford algebra $\mathbb{R}_n$, this operator turns out to have bisectorial properties. By applying the spectral theory on the $S$-spectrum, we address a fundamental mathematical challenge: unlike sectorial operators, bisectorial operators involve fractional powers that are not analytic on the negative real line. To circumvent this, we introduce a novel definition of the fractional power function in this setting. Building upon previous works on bisectorial vector operators and weak solutions, we extend the definition of fractional powers to abstract vector operators. The core contribution of this work is the application of the functional calculus for vector operators to the gradient operator, showing that these fractional powers provide a rigorous mathematical foundation for nonlocal Fourier laws in heat propagation.