Journal ArticleDiscrete and Continuous Dynamical Systems - Series B · April 1, 2024
We study the existence of weak solutions to the p-Navier-Stokes equations with a symmetric p-Laplacian on bounded domains. We construct a particular Schauder basis in W01, p(Ω) with divergence free constraint and prove existence of weak solutions using the ...
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Journal ArticleZeitschrift fur Angewandte Mathematik und Physik · June 1, 2023
We investigate the tumor boundary instability induced by nutrient consumption and supply based on a Hele-Shaw model. The model describes the geometric evolution of the tumor region and is derived from taking the incompressible limit of a cell density model ...
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Journal ArticleCommunications in Mathematical Sciences · January 1, 2023
In this paper, we quantitatively consider the enhanced-dissipation effect of the advection term to the parabolic p-Laplacian equations. More precisely, we show the mixing property of flow for the passive scalar enhances the dissipation process of the p-Lap ...
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Journal ArticleJournal of Differential Equations · April 25, 2022
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 case 2
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Journal ArticleTransactions of the American Mathematical Society · January 1, 2021
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 parabolic PDEs. In particular, we show that blow-up in these models can alw ...
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Journal ArticleJournal of Nonlinear Science · December 1, 2018
The question of the global regularity versus finite- time blowup in solutions of the 3D incompressible Euler equation is a major open problem of modern applied analysis. In this paper, we study a class of one-dimensional models of the axisymmetric hyperbol ...
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Journal ArticleDiscrete and Continuous Dynamical Systems - Series B · October 1, 2018
In this paper, we study 1D autonomous fractional ODEs D c γu = f(u); 0 < γ < 1, where u : [0;∞) → R is the unknown function and D c is the generalized Caputo derivative introduced by Li and Liu ( arXiv:1612.05103). Based on the existence and uniqueness the ...
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Journal ArticleApplied Mathematics Letters · September 1, 2018
In this note, we prove or re-prove several important results regarding one dimensional time fractional ODEs following our previous work Feng et al. [15]. Here we use the definition of Caputo derivative proposed in Li and Liu (2017) [5,7] based on a convolu ...
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Journal ArticleJournal of Scientific Computing · April 1, 2017
The level set method commonly requires a reinitialization of the level set function due to interface motion and deformation. We extend the traditional technique for reinitializing the level set function to a method that preserves the interface gradient. Th ...
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Journal ArticleArchive for Rational Mechanics and Analysis · November 1, 2016
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 studied PDE models of chemotaxis is given by the Keller–Segel equation, which ...
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Journal ArticleDiscrete and Continuous Dynamical Systems - Series B · April 1, 2024
We study the existence of weak solutions to the p-Navier-Stokes equations with a symmetric p-Laplacian on bounded domains. We construct a particular Schauder basis in W01, p(Ω) with divergence free constraint and prove existence of weak solutions using the ...
Full textCite

Journal ArticleZeitschrift fur Angewandte Mathematik und Physik · June 1, 2023
We investigate the tumor boundary instability induced by nutrient consumption and supply based on a Hele-Shaw model. The model describes the geometric evolution of the tumor region and is derived from taking the incompressible limit of a cell density model ...
Full textCite

Journal ArticleCommunications in Mathematical Sciences · January 1, 2023
In this paper, we quantitatively consider the enhanced-dissipation effect of the advection term to the parabolic p-Laplacian equations. More precisely, we show the mixing property of flow for the passive scalar enhances the dissipation process of the p-Lap ...
Full textCite

Journal ArticleJournal of Differential Equations · April 25, 2022
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 case 2
Full textCite

Journal ArticleTransactions of the American Mathematical Society · January 1, 2021
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 parabolic PDEs. In particular, we show that blow-up in these models can alw ...
Full textCite

Journal ArticleJournal of Nonlinear Science · December 1, 2018
The question of the global regularity versus finite- time blowup in solutions of the 3D incompressible Euler equation is a major open problem of modern applied analysis. In this paper, we study a class of one-dimensional models of the axisymmetric hyperbol ...
Full textCite

Journal ArticleDiscrete and Continuous Dynamical Systems - Series B · October 1, 2018
In this paper, we study 1D autonomous fractional ODEs D c γu = f(u); 0 < γ < 1, where u : [0;∞) → R is the unknown function and D c is the generalized Caputo derivative introduced by Li and Liu ( arXiv:1612.05103). Based on the existence and uniqueness the ...
Full textCite

Journal ArticleApplied Mathematics Letters · September 1, 2018
In this note, we prove or re-prove several important results regarding one dimensional time fractional ODEs following our previous work Feng et al. [15]. Here we use the definition of Caputo derivative proposed in Li and Liu (2017) [5,7] based on a convolu ...
Full textCite

Journal ArticleJournal of Scientific Computing · April 1, 2017
The level set method commonly requires a reinitialization of the level set function due to interface motion and deformation. We extend the traditional technique for reinitializing the level set function to a method that preserves the interface gradient. Th ...
Full textCite

Journal ArticleArchive for Rational Mechanics and Analysis · November 1, 2016
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 studied PDE models of chemotaxis is given by the Keller–Segel equation, which ...
Full textCite

Journal ArticleJournal of Mathematical Analysis and Applications · July 15, 2016
We construct initial data for the two-dimensional Euler equation in a bounded smooth symmetric domain such that the gradient of vorticity in L∞ grows as a double exponential in time, for all time. Our construction is based on the recent result by Kiselev a ...
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Journal ArticleNonlinear Analysis, Theory, Methods and Applications · July 1, 2016
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 simplified models of some 2D fluid equations. ...
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Journal ArticleNonlinearity · January 1, 2014
Consider a diffusion-free passive scalar θ being mixed by an incompressible flow u on the torus d. Our aim is to study how well this scalar can be mixed under an enstrophy constraint on the advecting velocity field. Our main result shows that the mix-norm ...
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