Math @ Duke

Publications [#338623] of JianGuo Liu
Papers Published
 Li, L; Liu, JG, Some Compactness Criteria for Weak Solutions of Time Fractional PDEs,
Siam Journal on Mathematical Analysis, vol. 50 no. 4
(January, 2018),
pp. 39633995, Society for Industrial & Applied Mathematics (SIAM) [doi]
(last updated on 2019/02/19)
Abstract: © 2018 Society for Industrial and Applied Mathematics. The AubinLions 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 AubinLions lemma for the existence of weak solutions to time fractional PDEs. We first define the weak Caputo derivatives of order γ ϵ (0; 1) for functions valued in general Banach spaces, consistent with the traditional definition if the space is Rd and functions are absolutely continuous. Based on a Volterratype 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 NavierStokes equations with constant density and time fractional KellerSegel equations in R2 are then proved as model problems. This work provides a framework for studying weak solutions of nonlinear time fractional PDEs.


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