分数阶趋化模型在临界Besov空间中解的整体存在性北大核心CSCD

该文主要研究如下的分数阶趋化模型:{■_(t)+(-△)^(α/2)=▽·(u▽v)(x,t)∈R^(n)×(0,∞),ε■_(t)v+(-△)^(β/2)v=u,(x,t)∈R^(n)×(0,∞),u(x,0)=u_(0)(x),v(x,0)=v_(0)(x),x∈R^(n)其中α∈[1,2],β∈(0,2],ε≥0.基于分数阶耗散方程在Chemin-Lerner混合时空空间中的线性估计和Fourier局部化方法,作者得到了如下结果:(1)当ε=0时,建立了次临界情形1<α≤2下该模型在Besov空间中的局部适定性和小初值问题的整体适定性,优化了[陈化,吕文斌,吴少华.分数阶趋化模型在Besov空间中解的存在性.中国科学:数学,2019,49(12):1-17]所得适定性结果中正则性和可积性指标的范围.并且还建立了临界情形α=1下该模型在Besov空间中小初值问题的整体适定性;(2)当ε>0时,利用特殊的迭代技巧,作者分别建立了次临界情形1<α≤2和临界情形α=1下该模型在Besov空间中的局部适定性和小初值问题的整体适定性.进一步,利用模型所特有的代数结构,作者还证明了对初值v0无小性条件下解的整体存在性.     

Boundedness in a quasilinear chemotaxis‐haptotaxis system with logistic source

This paper studies the chemotaxis‐haptotaxis system with nonlinear diffusion subject to the homogeneous Neumann boundary conditions and suitable initial conditions, where χ , ξ and μ are positive constants, and (n ?2) is a bounded and smooth domain. Here, we assume that D (u )?c _D u ^{m  ? 1} for all u  > 0 with some c _D  > 0 and m ?1. For the case of non‐degenerate diffusion, if μ  > μ ^?, where it is proved that the system possesses global classical solutions which are uniformly‐in‐time bounded. In the case of degenerate diffusion, we show that the system admits a global bounded weak solution under the same assumptions. Copyright © 2016 John Wiley & Sons, Ltd.     

Uniform regularity and vanishing viscosity limit for the chemotaxis‐Navier‐Stokes system in a 3D bounded domain

We investigate the uniform regularity and vanishing viscosity limit for the incompressible chemotaxis‐Navier‐Stokes system with Navier boundary condition for velocity field and Neumann boundary condition for cell density and chemical concentration in a 3D bounded domain. It is shown that there exists a unique strong solution of the incompressible chemotaxis‐Navier‐Stokes system in a finite time interval, which is independent of the viscosity coefficient. Moreover, this solution is uniformly bounded in a conormal Sobolev space, which allows us to take the vanishing viscosity limit to obtain the incompressible chemotaxis‐Euler system.     

An advected-field method for deformable entities under flow

We study dynamics of a deformable entity (such as a vesicles under hydrodynamical constraints). We show how the problem can be solved by means of Green's functions associated with the Stokes equations. A gauge-field invariant formulation makes the study of dynamics efficient. However, this procedure has its short-coming. For example, if the fluids are not Newtonian, then no Green's function is available in general. We introduce a new approach, the advected field one, which opens a new avenue of applications. For example, non-Newtonian entities can be handled without additional deal. In addition problems like budding, droplet break-up in suspensions, can naturally be treated without additional complication. We exemplify the method on vesicles filled by a fluid having a viscosity contrast with the external fluid, and submitted to a shear flow. We show that beyond a viscosity contrast (the internal fluid being more viscous), the vesicle undergoes a tumbling bifurcation, which has a saddle-node nature. This bifurcation is known for blood cells. Indeed red cells either align in a shear flow or tumble according to whether haematocrit concentration is high or low. Received 19 December 2001 / Received in final form 31 May 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: chaouqi.misbah@ujf-grenoble.fr     

A Constructive Approach to Traveling Waves in Chemotaxis

In this paper we study the existence of one-dimensional and multidimensional traveling wave solutions for general chemotaxis or so-called Keller-Segel models without reproduction of the chemotactic species. We present a constructive approach to give modelers a choice of chemotactic sensitivity functionals, production, and degradation terms for the chemical signal at hand. The main aim is to understand the type of functionals and the interplay between them that are needed for the traveling wave and pulse patterns to occur.     

Vesicles in haptotaxis with hydrodynamical dissipation

We analyze the problem of vesicle migration in haptotaxis (a motion directed by an adhesion gradient), though most of the reasoning applies to chemotaxis as well as to a variety of driving forces. A brief account has been published on this topic [#!Cantat99a!#]. We present an extensive analysis of this problem and provide a basic discussion of most of the relevant processes of migration. The problem allows for an arbitrary shape evolution which is compatible with the full hydrodynamical flow in the Stokes limit. The problem is solved within the boundary integral formulation based on the Oseen tensor. For the sake of simplicity we confine ourselves to 2D flows in the numerical analysis. There are basically two regimes (i) the tense regime where the vesicle behaves as a “droplet” with an effective contact angle. In that case the migration velocity is given by the Stokes law. (ii) The flask regime where the vesicle has a significant (on the scale of the vesicle size) contact curvature. In that case we obtain a new migration law which substantially differs from the Stokes law. We develop general arguments in order to extract analytical laws of migration. These are in good agreement with the full numerical analysis. Finally we mention several important future issues and open questions. Received 24 June 2002 and Received in final form 4 February 2003 Published online: 16 April 2003 RID="a" ID="a"e-mail: isabelle.cantat@univ-rennes1.fr     

Stochastic aspects of biological locomotion

Various aspects of random walks undertaken by motile bacteria and migrating leukocytes are discussed, including the motions of these cells when responding to gradients of chemoattractants. Brief reference also is made to studies of particle movements within the cytoplasm of eucaryotic cells.Presented at the Symposium on Random Walks, Gaithersburg, MD, June 1982.     

Hydrodynamics and cell motion during the rounding of two dimensional hydra cell aggregates

Using a two dimensional experimental geometry, we study hydrodynamics and cell motion during the rounding-up of three kinds of hydra aggregates (ectodermal, endodermal and mixed). The relaxation of initially elongated aggregates toward a circular shape is exponential and the relaxation time is proportional to the aggregate size as expected from hydrodynamics laws. As for viscous liquids, aggregate rounding is driven by tissue surface tensions σ and resisted by tissue viscosity η. The ratio η/σ is the same for the three kinds of aggregates. With a reasonable value σ = 1 mN/m, we obtain an estimate larger than 10⁴ Pa s for the tissue viscosity. Cell motion during rounding is strongly cooperative and cell displacements exhibit some specific patterns in each aggregate. These results point out the role of adhesive bonds in the observed kinetics. Received 21 December 2001     

A new approach toward boundedness in a two‐dimensional parabolic chemotaxis system with singular sensitivity

We consider the parabolic chemotaxis model in a smooth, bounded, convex two‐dimensional domain and show global existence and boundedness of solutions for χ∈(0,χ₀) for some χ₀>1, thereby proving that the value χ = 1 is not critical in this regard. Our main tool is consideration of the energy functional for a > 0, b≥0, where using nonzero values of b appears to be new in this context. Copyright © 2015 John Wiley & Sons, Ltd.     

Local and global existence of solutions to a quasilinear degenerate chemotaxis system with unbounded initial data

This paper is concerned with local and global existence of solutions to the parabolic‐elliptic chemotaxis system . Marinoschi (J. Math. Anal. Appl. 2013; 402:415–439) established an abstract approach using nonlinear m‐accretive operators to giving existence of local solutions to this system when 0 < D₀≤D′(r)≤D_∞<∞ and (r₁,r₂)?K(r₁,r₂)r₁ is Lipschitz continuous on , provided that the initial data is assumed to be small. The smallness assumption on the initial data was recently removed (J. Math. Anal. Appl. 2014; 419:756–774). However the case of non‐Lipschitz and degenerate diffusion, such as D(r) = r^m(m > 1), is left incomplete. This paper presents the local and global solvability of the system with non‐Lipschitz and degenerate diffusion by applying (J. Math. Anal. Appl. 2013; 402:415–439) and (J. Math. Anal. Appl. 2014; 419:756–774) to an approximate system. In particular, the result in the present paper does not require any properties of boundedness, smoothness and radial symmetry of initial data. This makes it difficult to deal with nonlinearity. Copyright © 2015 John Wiley & Sons, Ltd.