论文标题
定期分布的非局部扩散方程的解决方案的规律性
Regularity of Solutions for the Nonlocal Diffusion Equation on Periodic Distributions
论文作者
论文摘要
这项工作介绍了在周期性分布空间上非本地扩散方程的解决方案的规律性。非局部扩散方程的空间操作员由具有紧凑型积分内核的非局部拉普拉斯操作员给出。我们遵循一种基于非局部拉普拉斯操作员的傅立叶乘数的统一方法，该方法允许在任何空间维度中研究非局部扩散方程的常规和分布解决方案，以及可集成和奇异内核。此外，结果超出了具有奇异内核的运算符，到非本地超扩散操作员。我们根据初始数据的规律性或扩散源项的定期介绍了解决方案的空间和时间规律性的结果。此外，非局部扩散方程的溶液显示出对两种限制类型的经典扩散方程的溶液收敛：随着空间非局部性的消失或积分内核的奇异性接近某种临界奇异性，取决于空间尺寸。此外，我们表明，对于可集成的内核的情况，初始数据中的不连续性在非局部扩散方程的解决方案中持续并持续存在。跳跃不连续性的大小表明加时性衰减。
This work addresses the regularity of solutions for a nonlocal diffusion equation over the space of periodic distributions. The spatial operator for the nonlocal diffusion equation is given by a nonlocal Laplace operator with a compactly supported integral kernel. We follow a unified approach based on the Fourier multipliers of the nonlocal Laplace operator, which allows the study of regular as well as distributional solutions of the nonlocal diffusion equation, integrable as well as singular kernels, in any spatial dimension. In addition, the results extend beyond operators with singular kernels to nonlocal super-diffusion operators. We present results on the spatial and temporal regularity of solutions in terms of regularity of the initial data or the diffusion source term. Moreover, solutions of the nonlocal diffusion equation are shown to converge to the solution of the classical diffusion equation for two types of limits: as the spatial nonlocality vanishes or as the singularity of the integral kernel approaches a certain critical singularity that depends on the spatial dimension. Furthermore, we show that, for the case of integrable kernels, discontinuities in the initial data propagate and persist in the solution of the nonlocal diffusion equation. The magnitude of a jump discontinuity is shown to decay overtime.
