209 citations · 1.4k across the 114 of their papers we have counts for
12 papers · 1 filter
A generalized definition of Caputo derivatives and its application to fractional ODEs
Lei Li, Jian-Guo Liu
We extend in this paper the definition of Caputo derivatives of order in to a certain class of locally integrable functions using a convolution group. Our strategy is to de…
Partial regularity of weak solutions to a PDE system with cubic nonlinearity
Jian-Guo Liu, Xiangsheng Xu
In this paper we investigate regularity properties of weak solutions to a PDE system that arises in the study of biological transport networks. The system consists of a possibly si…
Fractional stochastic differential equations satisfying fluctuation-dissipation theorem
Lei Li, Jian-Guo Liu, Jianfeng Lu
We propose in this work a fractional stochastic differential equation (FSDE) model consistent with the over-damped limit of the generalized Langevin equation model. As a result of…
Analytical validation of a continuum model for the evolution of a crystal surface in multiple space dimensions
Jian-Guo Liu, Xiangsheng Xu
In this paper we are concerned with the existence of a weak solution to the initial boundary value problem for the equation .…
A Class of Functional Inequalities and their Applications to Fourth-Order Nonlinear Parabolic Equations
Jian-Guo Liu, Xiangsheng Xu
We study a class of fourth order nonlinear parabolic equations which include the thin-film equation and the quantum drift-diffusion model as special cases. We investigate these equ…
Positivity-preserving and asymptotic preserving method for 2D Keller-Segal equations
Jian-Guo Liu, Li Wang, Zhennan Zhou
We propose a semi-discrete scheme for 2D Keller-Segel equations based on a symmetrization reformation, which is equivalent to the convex splitting method and is free of any nonline…
Explicit and implicit TVD schemes for conservation laws with Caputo derivatives
Jian-Guo Liu, Zheng Ma, Zhennan Zhou
In this paper, we investigate numerical approximations of the scalar conservation law with the Caputo derivative, which introduces the memory effect. We construct the first order a…
Weak solution of a continuum model for vicinal surface in the attachment-detachment-limited regime
Yuan Gao, Jian-Guo Liu, Jianfeng Lu
We study in this work a continuum model derived from 1D attachment-detachment-limited (ADL) type step flow on vicinal surface, , where , considered as a f…
Continuum dynamics of the intention field under weakly cohesive social interactions
Pierre Degond, Jian-Guo Liu, Sara Merino-Aceituno +1
We investigate the long-time dynamics of an opinion formation model inspired by a work by Borghesi, Bouchaud and Jensen. Firstly, we derive a Fokker-Planck type equation under the…
Continuum limit of a mesoscopic model with elasticity of step motion on vicinal surfaces
Yuan Gao, Jian-Guo Liu, Jianfeng Lu
This work considers the rigorous derivation of continuum models of step motion starting from a mesoscopic Burton-Cabrera-Frank (BCF) type model following the work [Xiang, SIAM J. A…
Numerical approximation of a coagulation-Fragmentation Model for Animal Group Size Statistics
Pierre Degond, Maximilian Engel
We study numerically a coagulation-fragmentation model derived by Niwa and further elaborated by Degond et al., where a unique equilibrium distribution of group sizes is shown to e…
Least action principles for incompressible flows and geodesics between shapes
Jian-Guo Liu, Robert L. Pego, Dejan Slepčev
As V. I. Arnold observed in the 1960s, the Euler equations of incompressible fluid flow correspond formally to geodesic equations in a group of volume-preserving diffeomorphisms. W…