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.Comment: 35 page

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.Comment: This is a revised version of a submission with the same title to the journal "Acta Applicandae Mathematicae" on November 1, 2018. arXiv admin note: text overlap with arXiv:1606.0939

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.Comment: arXiv admin note: substantial text overlap with arXiv:1606.0939

Weak Solutions for the Navier-Stokes Equations for B∞∞−1(ln)+BX˙r−1+r,21−r+L2{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 Calder\'on 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), Lemari\'e-Rieusset extended this result of Calder\'on 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 Lemari\'e-Rieusset to the larger initial value space ${B}^{-1(ln)}_{\infty\infty}(\mathbb{R}^N)+{B}_{\widetilde{\dot{X}}_r}^{-1+r,\frac{2}{1-r}}(\mathbb{R}^N)+L^2(\mathbb{R}^N)$ ($0<r<1$).Comment: 24 pages. This new version of the manuscript repairs some mistakes contained in the previous version of this manuscrip

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 $D_A(\theta)$ or $D_A(\theta,\infty)$ 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 $\gamma$ is larger than a threshold value $\gamma^\ast$ 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 $\gamma< \gamma^\ast$ then this stationary solution is unstable

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.Comment: Some mistakes in the previous version are fixed . arXiv admin note: text overlap with arXiv:1606.0939

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

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).Comment: 18page

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 $\gamma_*>0$ for the surface tension coefficient $\gamma$, such that in the case $\gamma>\gamma_*$ this radially symmetric stationary solution is asymptotically stable under small non-radial perturbations, whereas in the opposite case it is unstable

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 $\omega\in\Omega$, where $\Omega$ is the sample space of a probability space $(\Omega,\mathcal{A},p)$, for the randomized initial data $\vec{f}^\omega\in\mathcal{H}_{\sigma}^{s}(\mathbb{T}^N)$ with $-1<s<0$, such a problem has a unique local solution.Comment: 12 pages, no figure