115 citations · 240 across the 21 of their papers we have counts for
18 papers · 1 filter
Lower Bounds for Advection-Diffusion Equations: An Exploration with AI-Generated Proofs
Chenyang An, Xiaoqian Xu
We establish explicit lower bounds for advection-diffusion equations in three settings: a polynomial bound for inviscid shears with , a uni…
Data-Driven Parameter Identification for Tumor Growth Models
Liu Liu, Yifei Wang, Qinyu Xu +1
Modeling tumor growth accurately is essential for understanding cancer progression and informing treatment strategies. To estimate the parameters in the tumor growth model describe…
Sequential Exponential Lower Bounds for the Advection-Diffusion Equation with Shear Flows
Yupei Huang, Xiaoqian Xu
In this paper, we prove that the norm of spatial mean-free solutions to the advection--diffusion equation on with shear drifts satisfies an \emph{exponential l…
Phase separation for the 2D Cahn-Hilliard equation with a background shear flow
Yu Feng, Yuanyuan Feng, Anna L. Mazzucato +1
We consider the Cahn-Hilliard equation, which models phase separation in binary fluids, on the two-dimen\-sional torus in the presence of advection by a given background shear flow…
A Three-dimensional tumor growth model and its boundary instability
Jian-Guo Liu, Thomas Witelski, Xiaoqian Xu +1
In this paper, we investigate the tumor instability by employing both analytical and numerical techniques to validate previous results and extend the analytical findings presented…
Suppression of blow-up in Patlak-Keller-Segel-Navier-Stokes system via the Poiseuille flow
Hao Li, Zhaoyin Xiang, Xiaoqian Xu
In this paper, we investigate the two-dimensional Patlak-Keller-Segel-Navier-Stokes system perturbed around the Poiseuille flow and show that the solutions to thi…
Existence of weak solutions to -Navier-Stokes equations
Yuanyuan Feng, Lei Li, Jian-Guo Liu +1
We study the existence of weak solutions to the -Navier-Stokes equations with a symmetric -Laplacian on bounded domains. We construct a particular Schauder basis in $W_0^{1,p…
Dissipation enhancement of cellular flows in general advection diffusion equations
Yu Feng, Xiaoqian Xu
The main contribution of this paper is twofold: (1) Recently, Iyer, Xu, and Zlatoš studied the dissipation enhancement by cellular flows based on standard advection-diffusion equat…
Tumor boundary instability induced by nutrient consumption and supply
Yu Feng, Min Tang, Xiaoqian Xu +1
We investigate the tumor boundary instability induced by nutrient consumption and supply based on a Hele-Shaw model derived from taking the incompressible limit of a cell density m…
Global existence of a non-local semilinear parabolic equation with advection and applications to shear flow
Yu Feng, Bingyang Hu, Xiaoqian Xu +1
In this paper, we consider the following non-local semi-linear parabolic equation with advection: for , \begin{equation*} \begin{cases} u_t+v \cdot \nabla u-…
Dissipation enhancement for a degenerated parabolic equation
Yu Feng, Bingyang Hu, Xiaoqian Xu
In this paper, we quantitatively consider the enhanced-dissipation effect of the advection term to the parabolic -Laplacian equations. More precisely, we show the mixing propert…
Suppression of epitaxial thin film growth by mixing
Yu Feng, Bingyang Hu, Xiaoqian Xu
We consider following fourth-order parabolic equation with gradient nonlinearity on the two-dimensional torus with and without advection of an incompressible vector field in the ca…
Convection-Induced Singularity Suppression in the Keller-Segel and Other Non-linear PDEs
Gautam Iyer, Xiaoqian Xu, Andrej Zlatoš
In this paper we study the effect of the addition of a convective term, and of the resulting increased dissipation rate, on the growth of solutions to a general class of non-linear…
Stability of Blow Up for a 1D model of Axisymmetric 3D Euler Equation
Tam Do, Alexander Kiselev, Xiaoqian Xu
The question of the global regularity vs finite time blow up in solutions of the 3D incompressible Euler equation is a major open problem of modern applied analysis. In this paper,…
Suppression of chemotactic explosion by mixing
Alexander Kiselev, Xiaoqian Xu
Chemotaxis plays a crucial role in a variety of processes in biology and ecology. In many instances, processes involving chemical attraction take place in fluids. One of the most s…
One-dimensional model equations for hyperbolic fluid flow
Tam Do, Vu Hoang, Maria Radosz +1
In this paper we study the singularity formation for two nonlocal 1D active scalar equations, focusing on the hyperbolic flow scenario. Those 1D equations can be regarded as simpli…
Fast growth of the vorticity gradient in symmetric smooth domains for 2D incompressible ideal flow
Xiaoqian Xu
We construct an initial data for the two-dimensional Euler equation in a bounded smooth symmetric domain such that the gradient of vorticity in grows as a double expon…
Lower bounds on the mix norm of passive scalars advected by incompressible enstrophy-constrained flows
Gautam Iyer, Alexander Kiselev, Xiaoqian Xu
Consider a diffusion-free passive scalar being mixed by an incompressible flow on the torus . Our aim is to study how well this scalar can be mixed under an e…