209 citations · 1.4k across the 114 of their papers we have counts for
18 papers · 1 filter
On the best constant for Gagliardo-Nirenberg interpolation inequalities
Jian-Guo Liu, Jinhuan Wang
In this paper we derive the best constant for the following Gagliardo-Nirenberg interpolation inequality \begin{eqnarray*} \|u\|_{L^{m+1}}\leq C_{q,m,p} \|u\|^{1-θ}_{L^{q+1}}\|\nab…
On Runge-Kutta methods for the water wave equation and its simplified nonlocal hyperbolic model
Lei Li, Jian-Guo Liu, Zibu Liu +2
There is a growing interest in investigating numerical approximations of the water wave equation in recent years, whereas the lack of rigorous analysis of its time discretization i…
A note on one-dimensional time fractional ODEs
Yuanyuan Feng, Lei Li, Jian-Guo Liu +1
In this note, we prove or re-prove several important results regarding one dimensional time fractional ODEs following our previous work \cite{fllx17}. Here we use the definition of…
Cauchy problems for Keller-Segel type time-space fractional diffusion equation
Lei Li, Jian-Guo Liu, Li-zhen Wang
This paper investigates Cauchy problems for nonlinear fractional time-space generalized Keller-Segel equation , where Cap…
Online learning in optical tomography: a stochastic approach
Ke Chen, Qin Li, Jian-Guo Liu
We study the inverse problem of radiative transfer equation (RTE) using stochastic gradient descent method (SGD) in this paper. Mathematically, optical tomography amounts to recove…
Gradient flow approach to an exponential thin film equation: global existence and latent singularity
Yuan Gao, Jian-Guo Liu, Xin Yang Lu
In this work, we study a fourth order exponential equation, derived from thin film growth on crystal surface in multiple space dimensions. We use the gradient flow…
An accurate front capturing scheme for tumor growth models with a free boundary limit
Jian-Guo Liu, Min Tang, Li Wang +1
We consider a class of tumor growth models under the combined effects of density-dependent pressure and cell multiplication, with a free boundary model as its singular limit when t…
Some compactness criteria for weak solutions of time fractional PDEs
Lei Li, Jian-Guo Liu
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 crit…
Sensitivity analysis of Burgers' equation with shocks
Qin Li, Jian-Guo Liu, Ruiwen Shu
Generalized polynomial chaos (gPC) method has been extensively used in uncertainty quantification problems where equations contain random variables. For gPC to achieve high accurac…
A dispersive regularization for the modified Camassa-Holm equation
Yu Gao, Lei Li, Jian-Guo Liu
In this paper, we present a dispersive regularization for the modified Camassa-Holm equation (mCH) in one dimension, which is achieved through a double mollification for the system…
-Euler equations and -Navier-Stokes equations
Lei Li, Jian-Guo Liu
We propose in this work new systems of equations which we call -Euler equations and -Navier-Stokes equations. -Euler equations are derived as the Euler-Lagrange equations…
The modified Camassa-Holm equation in Lagrangian coordinates
Yu Gao, Jian-Guo Liu
In this paper, we study the modified Camassa-Holm (mCH) equation in Lagrangian coordinates. For some initial data , we show that classical solutions to this equation blow up i…
Maximal monotone operator theory and its applications to thin film equation in epitaxial growth on vicinal surface
Yuan Gao, Jian-Guo Liu, Xin Yang Lu +1
In this work we consider which is derived from a thin film equation for epitaxial growth on vicinal surface. We formulate the pr…
A vicinal surface model for epitaxial growth with logarithmic free energy
Yuan Gao, Hangjie Ji, Jian-Guo Liu +1
We study a continuum model for solid films that arises from the modeling of one-dimensional step flows on a vicinal surface in the attachment-detachment-limited regime. The resulti…
Asymmetry in crystal facet dynamics of homoepitaxy by a continuum model
Jian-Guo Liu, Jianfeng Lu, Dionisios Margetis +1
In the absence of external material deposition, crystal surfaces usually relax to become flat by decreasing their free energy. We study an asymmetry in the relaxation of macroscopi…
Patched peakon weak solutions of the modified Camassa-Holm equation
Yu Gao, Lei Li, Jian-Guo Liu
In this paper, we study traveling wave solutions and peakon weak solutions of the modified Camassa-Holm (mCH) equation with dispersive term for . We study t…
Continuous and discrete one dimensional autonomous fractional ODEs
Yuanyuan Feng, Lei Li, Jian-Guo Liu +1
In this paper, we study 1D autonomous fractional ODEs , where is the unknown function and is the generalized Caputo de…
Global existence of solutions to a tear film model with locally elevated evaporation rates
Yuan Gao, Hangjie Ji, Jian-Guo Liu +1
Motivated by a model proposed by Peng et al. [Advances in Coll. and Interf. Sci. 206 (2014)] for break-up of tear films on human eyes, we study the dynamics of a generalized thin f…