具有 Lévy 扩散的非局部演化方程：适定性与极限行为
Non-local evolution equations with Lévy diffusion: Well-posedness and limiting behavior
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中文总结 AI 辅助
研究具有 Lévy 扩散的时间非局部演化方程，通过从属原理和松弛函数理论，在相关条件下探讨非局部线性演化问题经典解存在性、向广义 Rayleigh - Stokes 方程的极限情况及非线性问题温和解的性质。
中文摘要 AI 辅助
在本笔记中，我们关注一类具有 Lévy 扩散的时间非局部演化方程，它们作为单向粘弹性流体流动和具有记忆的物理现象的模型出现。首先，我们在允许完全正性的相关记忆核条件下，考虑一个非局部线性演化问题经典解的存在性。接着，我们研究当 Lévy 扩散指数集中在 2 附近时，该模型向广义 Rayleigh - Stokes 方程的极限情况，证明了具有 Lévy 扩散的时间非局部问题的解一致收敛到广义 Rayleigh - Stokes 方程的解并揭示了收敛性。此外，还建立了具有非线性的非局部演化问题温和解的存在性和极限行为。证明基于从属原理和松弛函数理论。
英文摘要
In this note we focus our attention on a class of nonlocal-in-time evolution equations with Lévy diffusion, they arise as models of unidirectional viscoelastic fluid flow and physical phenomena with memory effect.We first consider the existence of the classical solution to a nonlocal linear evolution problem under conditions on the involved memory kernels which allows complete positivity. Then we investigate the limit of this model to a generalized Rayleigh-Stokes equation, as the index of Lévy diffusion gets concentrated near two, we prove that the solution of nonlocal-in-time problem with Lévy diffusion uniformly converges to that of the generalized Rayleigh-Stokes equation and reveal the convergence rate.Finally, the existence and limiting behavior of the mild solution to a nonlocal evolution problem with nonlinearity are established. The proofs are based on subordination principle and relaxation function theory.
