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Shangbin Cui

Publications and source records attributed to Shangbin Cui.

At least 19 recordsLinked to original sources
arXiv2026

Modification to Maxwell's Equations

A fixed electric charge is an electric current relative to a moving magnetic field, so that it is subjected to the force of the moving magnetic field. This means that not only time-varying magnetic field produces electric field, but moving magnetic field produces electric field as well. Maxwell neglected this fact in deriving his equations for the description of dynamical behavior of electromagnetic field so that the two equations (A) $\nabla\cdot\bfE=\epsln_0^{-1}\rho$ and (B) $\partial_t\bfE-\epsln_0^{-1}\mu_0^{-1}\nabla\times\bfB=-\epsln_0^{-1}\bfJ$ are incorrect. In this paper we modify the equation (A) into $\nabla\cdot(\bfE+\overline{\bfu}\times\bfB)= \epsln_0^{-1}\rho$, where $\overline{\bfu}$ denotes the mean velocity of the charges in the electric current, and the equation (B) is correspondingly modified. The modified equations are invariant under Galilean transformation. As a byproduct of this work, we see that Einstein's theory of special relativity is wrong.

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arXiv2019

On the Banach manifold of simple domains in the Euclidean space and applications to free boundary problems

In this paper we study the Banach manifold made up of simple $C^{m+\mu}$-domains in the Euclidean space $\mathbb{R}$. This manifold is merely a topological or a $C^0$ Banach manifold. It does not possess a differentiable structure. We introduce the concept of differentiable point in this manifold and prove that it is still possible to introduce the concept of tangent vector and tangent space at a differentiable point. Consequent, it is possible to consider differential equations in this Banach space. We show how to reduce some important free boundary problems into differential equations in such a manifold and then use the abstract result that we established earlier to study these free boundary problems.

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arXiv2019

Analysis of a free boundary problem modeling the growth of necrotic tumors

In this paper we make rigorous mathematical analysis to a free boundary problem modeling the growth of necrotic tumors. A remarkable feature of this free boundary problem is that it contains two different-type free surfaces: One is the tumor surface whose evolution is governed by an evolution equation and the other is the interface between the living shell of the tumor and the necrotic core which is an obstacle-type free surface, i.e., its evolution is not governed by an evolution equation but instead is determined by some stationary-type equation. In mathematics, the inner free surface is induced by discontinuity of the nonlinear reaction functions in this model, which causes the main difficulty of analysis of this free boundary problem. Previous work on this model studies spherically symmetric situation which is in essence an one-dimension free boundary problem. The purpose of this paper is to make rigorous analysis in general spherically asymmetric situation. By applying the Nash-Moser implicit function theorem, we prove that the inner free surface is smooth and depends on the outer free surface smoothly when it is a small perturbation of the surface of a sphere. By applying this result and some abstract results for parabolic differential equations in Banach manifolds we prove that the unique radial stationary solution of this free boundary problem is asymptotically stable under small non-radial perturbations.

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arXiv2019

Quasi-differentiable Banach manifold and phase-diagram of invariant parabolic differential equation in such manifold

The purpose of this paper is twofold. First we study a class of Banach manifolds which are not differentiable in traditional sense but they are quasi-differentiable in the sense that a such Banach manifold has an embedded submanifold such that all points in that submanifold are differentiable and tangent spaces at those points can be defined. It follows that differential calculus can be performed in that submanifold and, consequently, differential equations in a such Banach manifold can be considered. Next we study the structure of phase diagram near center manifold of a parabolic differential equation in Banach manifold which is invariant or quasi-invariant under a finite number of mutually quasi-commutative Lie group actions. We prove that under certain conditions, near the center manifold $\mathcal{M}_c$ the underline manifold is a homogeneous fibre bundle over $\mathcal{M}_c$, with fibres being stable manifolds of the differential equation. As an application, asymptotic behavior of the solution of a two-free-surface Hele-Shaw problem is also studied.

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arXiv2016

Linearized stability theorem for invariant and quasi-invariant parabolic differential equations in Banach manifolds with applications to free boundary problems

If a differential equation in a Banach manifold is invariant or quasi-invariant under the action of one or more Lie groups, then its stationary points cannot be isolated, so that classical linearized stability theorem does not apply to it. The first main purpose of this paper is to establish a linearized stability theorem for parabolic differential equations in Banach manifolds which are either invariant or quasi-invariant under actions of a number of Lie groups. The second purpose of this paper is to apply this theorem to analyze stability of stationary solutions of some free boundary problems. In order to apply the abstract result to concrete free boundary problems, Banach manifold made up of certain kind of domains such as simple domains in ${\mathbf{R}}^n$ is a fundamental tool which seems to have not been well-studied in the literature yet. Hence in this paper we also make some basic investigation to a such manifold. In Section 5 we use Nash-Moser implicit function theorem to prove an interesting result for an obstacle problem which says that if the domain $\Omega$ of this obstacle problem is a small perturbation of a sphere then its interface $\Gamma$ is smooth and depends on $\Omega$ smoothly. By using these results, in the last section we prove asymptotic stability of radial stationary solution of a free boundary problem modeling the growth of necrotic tumors, which has been kept open for over ten years.

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arXiv2015

Sharp well-posedness and ill-posedness in Fourier-Besov spaces for the viscous primitive equations of geophysics

We study well-posedness and ill-posedness for Cauchy problem of the three-dimensional viscous primitive equations describing the large scale ocean and atmosphere dynamics. By using the Littlewood-Paley analysis technique, in particular Chemin-Lerner's localization method, we prove that the Cauchy problem with Prandtl number $P=1$ is locally well-posed in the Fourier-Besov spaces $[\dot{FB}^{2-\frac{3}{p}}_{p,r}(\mathbb{R}^3)]^4$ for $1<p\leq\infty,1\leq r<\infty$ and $[\dot{FB}^{-1}_{1,r}(\mathbb{R}^3)]^4$ for $1\leq r\leq 2$, and globally well-posed in these spaces when the initial data $(u_0,\theta_0)$ are small. We also prove that such problem is ill-posed in $[\dot{FB}^{-1}_{1,r}(\mathbb{R}^3)]^4$ for $2<r\leq\infty$, showing that the results stated above are sharp.

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arXiv2015

Sharp well-posedness and ill-posedness of the Navier-Stokes initial value problem in Besov-type spaces

We prove that the Navier-Stokes initial value problem is well-posed in the logrithmically refined Besov spaces when the second index is not less than certain critical value, and ill-posed in such spaces when the second index is less than this critical value. The well-posedness result is proved by using some sharp bilinear estimates obtained from some Hardy-Littlewood type inequalities. The ill-posedness assertion is proved by refining the arguments of Wang [18] and Yoneda [20].

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arXiv2014

Mathematical analysis of population migration and its effects to spread of epidemics

In this paper we study some mathematical models describing evolution of population density and spread of epidemics in population systems in which spatial movement of individuals depends only on the departure and arrival locations and does not have apparent connection with the population density. We call such models as population migration models and migration epidemics models, respectively. We first apply the theories of positive operators and positive semigroups to make systematic investigation to asymptotic behavior of solutions of the population migration models as time goes to infinity, and next use such results to study asymptotic behavior of solutions of the migration epidemics models as time goes to infinity. Some interesting properties of solutions of these models are obtained.

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arXiv2013

Linearized eigenvalues for a free boundary problem modeling two-phase tumor growth

In this paper we study a linearized eigenvalue problem derived from a a free boundary problem modeling the growth of a tumor containing two species of cells: proliferating cells and quiescent cells. The reduced form of this eigenvalue problem is a $2$-system of a first-order nonlocal singular differential-integral equation in a ball coupled by a third-order elliptic pseudo-differential equation in the unit sphere. The singularity joined with non-localness of the first-order equation causes the main difficulty of this problem. By using Fourier expansion via a basis of spherical harmonic functions and some techniques for solving singular differential integral equations developed in some previous literature, we prove that there exists a null sequence $\{\gamma_k\}_{k=2}^{\infty}$ for the surface tension coefficient $\gamma$, with each of them being an eigenvalue of the linearized problem, i.e., if $\gamma=\gamma_k$ for some $k\geq 2$ then the linearized problem has extra nontrivial solutions besides the standard nontrivial solutions, and if $\gamma\neq\gamma_k$ for all $k\geq 2$ then the linearized problem does not have other nontrivial solutions than the standard nontrivial solutions. Invertibility of some linear operators related to the linearized problem in suitable function spaces is also studied.

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arXiv2013

Linearized stability for a multi-dimensional free boundary problem modeling two-phase tumor growth

This paper is concerned with a multi-dimensional free boundary problem modeling the growth of a tumor with two species of cells: proliferating cells and quiescent cells. This free boundary problem has a unique radial stationary solution. By using the Fourier expansion of functions on unit sphere via spherical harmonics, we establish some decay estimates for the solution of the linearized system of this tumor model at the radial stationary solution, so that proving that the radial stationary solution is linearly asymptotically stable when neglecting translations.

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arXiv2013

Asymptotic stability of the stationary solution for a parabolic-hyperbolic free boundary problem modeling tumor growth

This paper studies asymptotic behavior of solutions of a free boundary problem modeling the growth of tumors with two species of cells: proliferating cells and quiecent cells. In previous literatures it has been proved that this problem has a unique stationary solution which is asymptotically stable in the limit case $\varepsilon=0$. In this paper we consider the more realistic case $0<\varepsilon<<1$. In this case, after suitable reduction the model takes the form of a coupled system of a parabolic equation and a hyperbolic system, so that it is more difficult than the limit case $\varepsilon=0$. By using some unknown variable transform as well as the similarity transform technique developed in our previous work, we prove that the stationary solution is also asymptotically stable in the case $0<\varepsilon<<1$.

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arXiv2012

Weak Solutions for the Navier-Stokes Equations for ${B}^{-1(ln)}_{\infty\infty}+{B}_{\dot{X}_r}^{-1+r,\frac{2}{1-r}}+L^2$ Initial Data

In 1934 Leray proved that the Navier-Stokes equations have global weak solutions for initial data in $L^2(\mathbb{R}^N)$. In 1990 Calderón extended this result to the initial value spaces $L^p(\mathbb{R}^N)$ ($2\leq p<\infty$). In the book "{\em Recent developments in the Navier-Stokes problems}" (2002), Lemarié-Rieusset extended this result of Calderón to the space $B_{\widetilde{X}_r}^{-1+r,\frac{2}{1-r}}(\mathbb{R}^N)+L^2(\mathbb{R}^N)$ ($0<r<1$), where ${X}_r$ is the space of functions whose pointwise products with $H^r$ functions belong to $L^2$, $\widetilde{X}_r$ denotes the closure of $C_0^\infty(\mathbb{R}^N)$ in ${X}_r$, and $B_{\widetilde{X}_r}^{-1+r,\frac{2}{1-r}}(\mathbb{R}^N)$ is the Besov space over $\widetilde{X}_r$. In this paper we further extend this result of Lemarié-Rieusset to the larger initial value space ${B}^{-1(ln)}_{\infty\infty}(\mathbb{R}^N)+{B}_{\widetilde{\dot{X}}_r}^{-1+r,\frac{2}{1-r}}(\mathbb{R}^N)+L^2(\mathbb{R}^N)$ ($0<r<1$).

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arXiv2011

A Beale--Kato--Majda criterion for the 3-D Compressible Nematic Liquid Crystal Flows with Vacuum

In this paper, we prove a Beale--Kato--Majda blow-up criterion in terms of the gradient of the velocity only for the strong solution to the 3-D compressible nematic liquid crystal flows with nonnegative initial densities. More precisely, the strong solution exists globally if the $L^{1}(0,T;L^{\infty})$-norm of the gradient of the velocity $u$ is bounded. Our criterion improves the recent result of X. Liu and L. Liu (\cite{LL}, A blow-up criterion for the compressible liquid crystals system, arXiv:1011.4399v2 [math-ph] 23 Nov. 2010).

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arXiv2011

Existence of Solutions for the Debye-Hückel System with Low Regularity Initial Data

In this paper we study existence of solutions for the Cauchy problem of the Debye-Hückel system with low regularity initial data. By using the Chemin-Lerner time-space estimate for the heat equation, we prove that there exists a unique local solution if the initial data belongs to the Besov space $\dot{B}^{s}_{p,q}(\mathbb{R}^{n})$ for $-3/2<s\leq-2+\frac{n}{2}$, $p=\frac{n}{s+2}$ and $1\leq q\leq \infty$, and furthermore, if the initial data is sufficiently small then the solution is global. This result improves the regularity index of the initial data space in previous results on this model.

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arXiv2011

Well-posedness of the Viscous Boussinesq System in Besov Spaces of Negative Order Near Index $s=-1$

This paper is concerned with well-posedness of the Boussinesq system. We prove that the $n$ ($n\ge2$) dimensional Boussinesq system is well-psoed for small initial data $(\vec{u}_0,\theta_0)$ ($\nabla\cdot\vec{u}_0=0$) either in $({B}^{-1}_{\infty,1}\cap{B^{-1,1}_{\infty,\infty}})\times{B}^{-1}_{p,r}$ or in ${B^{-1,1}_{\infty,\infty}}\times{B}^{-1,\epsilon}_{p,\infty}$ if $r\in[1,\infty]$, $\epsilon>0$ and $p\in(\frac{n}{2},\infty)$, where $B^{s,\epsilon}_{p,q}$ ($s\in\mathbb{R}$, $1\leq p,q\leq\infty$, $\epsilon>0$) is the logarithmically modified Besov space to the standard Besov space $B^{s}_{p,q}$. We also prove that this system is well-posed for small initial data in $({B}^{-1}_{\infty,1}\cap{B^{-1,1}_{\infty,\infty}})\times({B}^{-1}_{\frac{n}{2},1}\cap{B^{-1,1}_{\frac{n}{2},\infty}})$.

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arXiv2011

Global Existence and Stability for a Hydrodynamic System in the Nematic Liquid Crystal Flows

In this paper we consider a coupled hydrodynamical system which involves the Navier-Stokes equations for the velocity field and kinematic transport equations for the molecular orientation field. By applying the Chemin-Lerner's time-space estimates for the heat equation and the Fourier localization technique, we prove that when initial data belongs to the critical Besov spaces with negative-order, there exists a unique local solution, and this solution is global when initial data is small enough. As a corollary, we obtain existence of global self-similar solutions. In order to figure out the relation between the solution obtained here and weak solution of standard sense, we establish a stability result, which yields in a direct way that all global weak solutions associated with the same initial data must coincide with the solution obtained here, namely, weak-strong uniqueness holds.

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arXiv2011

Global Well-posedness of the Stochastic Kuramoto-Sivashinsky Equation with Multiplicative Noise

Global well-posedness of the initial-boundary value problem for the stochastic Kuramoto-Sivashinsky equation in a bounded domain $D$ with a multiplicative noise is studied. It is shown that under suitable sufficient conditions, for any initial data $u_0\in L^2(D\times Ω)$ this problem has a unique global solution $u$ in the space $L^2(Ω,C([0,T],L^2({D})))$ for any $T>0$, and the solution map $u_0\mapsto u$ is Lipschitz continuous.

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