arXiv subjects

Shangbin Cui

Publications and source records attributed to Shangbin Cui.

25 records · Page 2Linked to original sources

arXiv2011

Random-data Cauchy Problem for the Periodic Navier-Stokes Equations with Initial Data in Negative-order Sobolev Spaces

In this paper we study existence of solutions of the initial-boundary value problems of the Navier-Stokes equations with a periodic boundary value condition for initial data in the Sobolev spaces $\mathcal{H}^{s}(\mathbb{T}^N)$ with a negative order $-1<s<0$, where $N=2, 3$. By using the randomization approach of N. Burq and N. Tzvetkov, we prove that for almost all $ω\inΩ$, where $Ω$ is the sample space of a probability space $(Ω,\mathcal{A},p)$, for the randomized initial data $\vec{f}^ω\in\mathcal{H}_σ^{s}(\mathbb{T}^N)$ with $-1<s<0$, such a problem has a unique local solution.

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arXiv2009

Weak Continuity of Dynamical Systems for the KdV and mKdV Equations

In this paper we study weak continuity of the dynamical systems for the KdV equation in H^{-3/4}(R) and the modified KdV equation in H^{1/4}(R). This topic should have significant applications in the study of other properties of these equations such as finite time blow-up and asymptotic stability and instability of solitary waves. The spaces considered here are borderline Sobolev spaces for the corresponding equations from the viewpoint of the local well-posedness theory. We first use a variant of the method of [5] to prove weak continuity for the mKdV, and next use a similar result for a mKdV system and the generalized Miura transform to get weak continuity for the KdV equation.

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arXiv2009

Weak Continuity of the Flow Map for the Benjamin-Ono Equation on the Line

In this paper we show that the floow map of the Benjamin-Ono equation on the line is weakly continuous in L2(R), using "local smoothing" estimates. L2(R) is believed to be a borderline space for the local well-posedness theory of this equation. In the periodic case, Molinet [27] has recently proved that the flow map of the Benjamin-Ono equation is not weakly continuous in L2(T). Our results are in line with previous work on the cubic nonlinear Schrodinger equation, where Goubet and Molinet [11] showed weak continuity in L2(R) and Molinet [28] showed lack of weak continuity in L2(T).

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arXiv2008

Asymptotic Stability of Stationary Solutions of a Free Boundary Problem Modeling the Growth of Tumors with Fluid Tissues

This paper aims at proving asymptotic stability of the radial stationary solution of a free boundary problem modeling the growth of nonnecrotic tumors with fluid-like tissues. In a previous paper we considered the case where the nutrient concentration $σ$ satisfies the stationary diffusion equation $Δσ=f(σ)$, and proved that there exists a threshold value $γ_*>0$ for the surface tension coefficient $γ$, such that the radial stationary solution is asymptotically stable in case $γ>γ_*$, while unstable in case $γ<γ_*$. In this paper we extend this result to the case where $σ$ satisfies the non-stationary diffusion equation $\epsln\partial_tσ=Δσ-f(σ)$. We prove that for the same threshold value $γ_*$ as above, for every $γ>γ_*$ there is a corresponding constant $\epsln_0(γ)>0$ such that for any $0<\epsln<\epsln_0(γ)$ the radial stationary solution is asymptotically stable with respect to small enough non-radial perturbations, while for $0<γ<γ_*$ and $\epsln$ sufficiently small it is unstable under non-radial perturbations.

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arXiv2008

Asymptotic Behavior of Solutions of a Free Boundary Problem Modelling the Growth of Tumors with Stokes Equations

We study a free boundary problem modelling the growth of non-necrotic tumors with fluid-like tissues. The fluid velocity satisfies Stokes equations with a source determined by the proliferation rate of tumor cells which depends on the concentration of nutrients, subject to a boundary condition with stress tensor effected by surface tension. It is easy to prove that this problem has a unique radially symmetric stationary solution. By using a functional approach, we prove that there exists a threshold value $γ_*>0$ for the surface tension coefficient $γ$, such that in the case $γ>γ_*$ this radially symmetric stationary solution is asymptotically stable under small non-radial perturbations, whereas in the opposite case it is unstable.

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arXiv2007

Lie Group Action and Stability Analysis of Stationary Solutions for a Free Boundary Problem Modelling Tumor Growth

In this paper we study asymptotic behavior of solutions for a multidimensional free boundary problem modelling the growth of nonnecrotic tumors. We first establish a general result for differential equations in Banach spaces possessing a local Lie group action which maps a solution into new solutions. We prove that a center manifold exists under certain assumptions on the spectrum of the linearized operator without assuming that the space in which the equation is defined is of either $D_A(θ)$ or $D_A(θ,\infty)$ type. By using this general result and making delicate analysis of the spectrum of the linearization of the stationary free boundary problem, we prove that if the surface tension coefficient $γ$ is larger than a threshold value $γ^\ast$ then the unique stationary solution is asymptotically stable modulo translations, provided the constant $c$ representing the ratio between the nutrient diffusion time and the tumor-cell doubling time is sufficiently small, whereas if $γ< γ^\ast$ then this stationary solution is unstable.

math.AP↗

arXiv2007

Asymptotic Stability of the Stationary Solution for a Hyperbolic Free Boundary Problem Modeling Tumor Growth

In this paper we study asymptotic behavior of solutions for a free boundary problem modeling the growth of tumors containing two species of cells: proliferating cells and quiescent cells. This tumor model was proposed by Pettet et al in {\em Bull. Math. Biol.} (2001). By using a functional approach and the $C_0$ semigroup theory, we prove that the unique stationary solution of this model ensured by the work of Cui and Friedman ({\em Trans. Amer. Math. Soc.}, 2003) is locally asymptotically stable in certain function spaces. Key techniques used in the proof include an improvement of the linear estimate obtained by the work of Chen et al ({\em Trans. Amer. Math. Soc.}, 2005), and a similarity transformation.

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