SOME COMPACTNESS CRITERIA FOR WEAK SOLUTIONS OF TIME FRACTIONAL PDEs
SOME COMPACTNESS CRITERIA FOR WEAK SOLUTIONS OF TIME FRACTIONAL PDEs
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DOI：
10.1137/17m1145549
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发表时间：
2018-01-01
影响因子：
2
通讯作者：
Liu, Jian-Guo
中科院分区：
文献类型：
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作者：
Li, Lei;Liu, Jian-Guo
Aubin-Lions引理及其变种对于非线性发展偏微分方程弱解的存在起着至关重要的作用。在这篇文章中，我们旨在建立一些紧性判据，这些判据类似于Aubin-Lions引理，证明了时间分数阶偏微分方程解的存在性。我们首先定义了一般Banach空间中取值函数的Gamma阶弱Caputo导数是(0，1)元，如果空间是R-d且函数是绝对连续的，则定义与传统定义一致。基于Volterra型积分形式，当弱Caputo导数在一定的空间内时，我们建立了函数的时间正则性估计。然后使用时间正则性估计建立紧凑性标准。在此基础上，证明了R-2中具有常密度的时间分数阶可压缩Navier-Stokes方程和时间分数阶Keller-Segel方程的弱解的存在性。这一工作为研究非线性时间分数阶偏微分方程解的弱解提供了一个框架。
The Aubin-Lions lemma and its variants play crucial roles for the existence of weak solutions of nonlinear evolutionary PDEs. In this paper, we aim to develop some compactness criteria that are analogies of the Aubin-Lions lemma for the existence of weak solutions to time fractional PDEs. We first de fine the weak Caputo derivatives of order gamma is an element of (0, 1) for functions valued in general Banach spaces, consistent with the traditional definition if the space is R-d and functions are absolutely continuous. Based on a Volterra-type integral form, we establish some time regularity estimates of the functions provided that the weak Caputo derivatives are in certain spaces. The compactness criteria are then established using the time regularity estimates. The existence of weak solutions for a special case of time fractional compressible Navier-Stokes equations with constant density and time fractional Keller-Segel equations in R-2 are then proved as model problems. This work provides a framework for studying weak solutions of nonlinear time fractional PDEs.