发表机构
Wroclaw University of Science and Technology（弗罗茨瓦夫科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究多维分数阶物质导数，推导其逐点表示及逆算子，分析相关线性偏微分方程，证明测度解的存在唯一性，并给出质量守恒条件及非负性和弱收敛条件。
AI 中文摘要
我们分析了一个在空间和时间上的非局部算子，称为多维分数阶物质导数。我们推导了它的逐点表示，这使我们能够研究其其他性质。我们定义了一个逆算子，称为分数阶物质积分，并推导了其逐点表示。此外，我们分析了一类线性偏微分方程，这些方程对应于多维莱维游走标度极限的确定性描述，其中输运由多维分数阶物质导数驱动，该导数与一个关于适当概率测度积分的速度向量以及一个分布源项相结合。利用傅里叶-拉普拉斯变换技术和直接的卷积核构造，我们证明了对于测度数据，存在唯一的指数有界测度解。此外，我们确定了源项上保持单位质量的必要且充分条件，并提供了非负性和弱收敛于$\delta_0$的分别充分条件。
英文摘要
We analyze a nonlocal operator in space and time, called the multidimensional fractional material derivative. We derive its pointwise representation, which allows us to study its other properties. We define an inverse operator, called fractional material integral, and derive its pointwise representation. Furthermore, we analyze a class of linear partial differential equations, which corresponds to deterministic descriptions of the scaling limits of multidimensional Lévy walks, in which transport is driven by a multidimensional fractional material derivative with a speed vector integrated with respect to a suitable probability measure and a distributional source term. Using Fourier-Laplace transform techniques and a direct convolution-kernel construction, we prove the existence and uniqueness of exponentially bounded measure solutions for measure data. Moreover, we identify a necessary and sufficient condition on the source term for conservation of unit mass and provide separate sufficient conditions for non-negativity and weak convergence to $δ_0$.
