In this paper,we explore bound preserving and high-order accurate local discontinuous Galerkin(LDG)schemes to solve a class of chemotaxis models,including the classical Keller-Segel(KS)model and two other density-depe...In this paper,we explore bound preserving and high-order accurate local discontinuous Galerkin(LDG)schemes to solve a class of chemotaxis models,including the classical Keller-Segel(KS)model and two other density-dependent problems.We use the convex splitting method,the variant energy quadratization method,and the scalar auxiliary variable method coupled with the LDG method to construct first-order temporal accurate schemes based on the gradient flow structure of the models.These semi-implicit schemes are decoupled,energy stable,and can be extended to high accuracy schemes using the semi-implicit spectral deferred correction method.Many bound preserving DG discretizations are only worked on explicit time integration methods and are difficult to get high-order accuracy.To overcome these difficulties,we use the Lagrange multipliers to enforce the implicit or semi-implicit LDG schemes to satisfy the bound constraints at each time step.This bound preserving limiter results in the Karush-Kuhn-Tucker condition,which can be solved by an efficient active set semi-smooth Newton method.Various numerical experiments illustrate the high-order accuracy and the effect of bound preserving.展开更多
This paper deals with an attraction-repulsion chemotaxis model(ARC) in multi-dimensions. By Duhamel's principle, the implicit expression of the solution to(ARC)is given. With the method of Green's function, th...This paper deals with an attraction-repulsion chemotaxis model(ARC) in multi-dimensions. By Duhamel's principle, the implicit expression of the solution to(ARC)is given. With the method of Green's function, the authors obtain the pointwise estimates of solutions to the Cauchy problem(ARC) for small initial data, which yield the W s,p(1 ≤p≤∞) decay properties of solutions.展开更多
In this paper, the local existence and uniqueness of a chemotaxis model with a moving boundary are considered by the contraction mapping principle, and the explicit expression for the moving boundary is formulated. In...In this paper, the local existence and uniqueness of a chemotaxis model with a moving boundary are considered by the contraction mapping principle, and the explicit expression for the moving boundary is formulated. In addition, the finite-time blowup and chemotactic collapse of the solution for such kind of problem are discussed.展开更多
We establish the global well-posedness for the multidimensional chemotaxis model with some classes of large initial data,especially the case when the rate of variation of ln v0(v0 is the chemical concentration)contain...We establish the global well-posedness for the multidimensional chemotaxis model with some classes of large initial data,especially the case when the rate of variation of ln v0(v0 is the chemical concentration)contains high oscillation and the initial density near the equilibrium is allowed to have large oscillation in 3D.Besides,we show the optimal time-decay rates of the strong solutions under an additional perturbation assumption,which include specially the situations of d=2,3 and improve the previous time-decay rates.Our method mainly relies on the introduce of the effective velocity and the application of the localization in Fourier spaces.展开更多
This work studies the stability and metastability of stationary patterns in a diffusionchemotaxis model without cell proliferation.We first establish the interval of unstable wave modes of the homogeneous steady state...This work studies the stability and metastability of stationary patterns in a diffusionchemotaxis model without cell proliferation.We first establish the interval of unstable wave modes of the homogeneous steady state,and show that the chemotactic flux is the key mechanism for pattern formation.Then,we treat the chemotaxis coefficient as a bifurcation parameter to obtain the asymptotic expressions of steady states.Based on this,we derive the sufficient conditions for the stability of one-step pattern,and prove the metastability of two or more step patterns.All the analytical results are demonstrated by numerical simulations.展开更多
In this paper, we use contraction mapping principle, operator-theoretic approach and some uniform estimates to establish local solvability of the parabolic-hyperbolic type chemotaxis system with fixed boundary in 1-di...In this paper, we use contraction mapping principle, operator-theoretic approach and some uniform estimates to establish local solvability of the parabolic-hyperbolic type chemotaxis system with fixed boundary in 1-dimensional domain. In addition, local solvability of the free boundary problem is considered by straightening the free boundary.展开更多
This paper is concerned with the asymptotic behavior of solution to the following model involving two species all with chemotaxis：{αp/αt=Dp△（p△lnp/w）,αp/αt=Dq△（q△lnq/w）,αw/αt=βp-δw,p△ln（p/w）.^-n=q...This paper is concerned with the asymptotic behavior of solution to the following model involving two species all with chemotaxis：{αp/αt=Dp△（p△lnp/w）,αp/αt=Dq△（q△lnq/w）,αw/αt=βp-δw,p△ln（p/w）.^-n=q△ln（q/w）.^-n=0.We prove that the solution exists globally asβ ≥ 0. Asβ 〈 0, whether the solution exists globally or not depends on the initial data. By function transformation and compari- son, the asymptotical behavior of the solution is studied.展开更多
we prove the local existence and uniqueness of a moving boundary prob- lem modeling chemotactic phenomena. We also get the explicit representative for the moving boundary in a special case.
文摘In this paper,we explore bound preserving and high-order accurate local discontinuous Galerkin(LDG)schemes to solve a class of chemotaxis models,including the classical Keller-Segel(KS)model and two other density-dependent problems.We use the convex splitting method,the variant energy quadratization method,and the scalar auxiliary variable method coupled with the LDG method to construct first-order temporal accurate schemes based on the gradient flow structure of the models.These semi-implicit schemes are decoupled,energy stable,and can be extended to high accuracy schemes using the semi-implicit spectral deferred correction method.Many bound preserving DG discretizations are only worked on explicit time integration methods and are difficult to get high-order accuracy.To overcome these difficulties,we use the Lagrange multipliers to enforce the implicit or semi-implicit LDG schemes to satisfy the bound constraints at each time step.This bound preserving limiter results in the Karush-Kuhn-Tucker condition,which can be solved by an efficient active set semi-smooth Newton method.Various numerical experiments illustrate the high-order accuracy and the effect of bound preserving.
基金supported by the National Natural Science Foundation of China(No.11171213)supported by the National Natural Science Foundation of China(No.11231006)the National Research Foundation for the Doctoral Program of Higher Education of China(No.20130073110073)
文摘This paper deals with an attraction-repulsion chemotaxis model(ARC) in multi-dimensions. By Duhamel's principle, the implicit expression of the solution to(ARC)is given. With the method of Green's function, the authors obtain the pointwise estimates of solutions to the Cauchy problem(ARC) for small initial data, which yield the W s,p(1 ≤p≤∞) decay properties of solutions.
文摘In this paper, the local existence and uniqueness of a chemotaxis model with a moving boundary are considered by the contraction mapping principle, and the explicit expression for the moving boundary is formulated. In addition, the finite-time blowup and chemotactic collapse of the solution for such kind of problem are discussed.
基金Supported by the National Natural Science Foundation of China(Grant No.12071043)the National Key Research and Development Program of China(Grant No.2020YFA0712900)。
文摘We establish the global well-posedness for the multidimensional chemotaxis model with some classes of large initial data,especially the case when the rate of variation of ln v0(v0 is the chemical concentration)contains high oscillation and the initial density near the equilibrium is allowed to have large oscillation in 3D.Besides,we show the optimal time-decay rates of the strong solutions under an additional perturbation assumption,which include specially the situations of d=2,3 and improve the previous time-decay rates.Our method mainly relies on the introduce of the effective velocity and the application of the localization in Fourier spaces.
基金Manjun Ma was supported by the National Natural Science Foundation of China(Nos.12071434 and 11671359).
文摘This work studies the stability and metastability of stationary patterns in a diffusionchemotaxis model without cell proliferation.We first establish the interval of unstable wave modes of the homogeneous steady state,and show that the chemotactic flux is the key mechanism for pattern formation.Then,we treat the chemotaxis coefficient as a bifurcation parameter to obtain the asymptotic expressions of steady states.Based on this,we derive the sufficient conditions for the stability of one-step pattern,and prove the metastability of two or more step patterns.All the analytical results are demonstrated by numerical simulations.
基金Supported by the National Natural Science Foundation of China(11131005)the Fundamental Research Funds for the Central Universities(2014201020202)
文摘In this paper, we use contraction mapping principle, operator-theoretic approach and some uniform estimates to establish local solvability of the parabolic-hyperbolic type chemotaxis system with fixed boundary in 1-dimensional domain. In addition, local solvability of the free boundary problem is considered by straightening the free boundary.
基金This work is supported by the National Natural Science Foundation of China （10471108）.
文摘This paper is concerned with the asymptotic behavior of solution to the following model involving two species all with chemotaxis：{αp/αt=Dp△（p△lnp/w）,αp/αt=Dq△（q△lnq/w）,αw/αt=βp-δw,p△ln（p/w）.^-n=q△ln（q/w）.^-n=0.We prove that the solution exists globally asβ ≥ 0. Asβ 〈 0, whether the solution exists globally or not depends on the initial data. By function transformation and compari- son, the asymptotical behavior of the solution is studied.
文摘we prove the local existence and uniqueness of a moving boundary prob- lem modeling chemotactic phenomena. We also get the explicit representative for the moving boundary in a special case.
