REGULARITY OF SOLUTIONS FOR NONLOCAL DIFFUSION EQUATIONS ON PERIODIC DISTRIBUTIONS
REGULARITY OF SOLUTIONS FOR NONLOCAL DIFFUSION EQUATIONS ON PERIODIC DISTRIBUTIONS
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DOI：
10.1216/jie.2023.35.81
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发表时间：
2022-09
影响因子：
0.8
通讯作者：
I. Mustapha;Bacim Alali;Nathan Albin
中科院分区：
文献类型：
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作者：
I. Mustapha;Bacim Alali;Nathan Albin
本文研究了周期分布空间上一类非局部扩散方程解的正则性。非局部扩散方程的空间算子由具有紧支撑积分核的非局部拉普拉斯算子给出。我们遵循一个统一的方法的基础上的傅立叶乘数的非局部拉普拉斯算子，它允许研究的经常以及分布的解决方案的非局部扩散方程，可积以及奇异内核，在任何空间维度。此外，所得结果还从奇异核算子推广到了非局部超扩散算子。我们提出的结果的空间和时间的规律性的解决方案的规则性的初始数据或扩散源项。此外，非局部扩散方程的解决方案收敛到经典扩散方程的解决方案的两种类型的限制：作为空间非局部性消失或作为积分核的奇异性接近一定的临界奇异性，取决于空间维度。此外，我们表明，对于可积核的情况下，在初始数据的不连续性传播，并坚持在非局部扩散方程的解决方案。跳跃不连续性的大小随时间衰减。
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.