Global Solutions To The Coupled Chemotaxis-Fluid Equations In High Dimensional Spaces

In this paper, we give a brief survey of the results studying chemotaxis model and cou-pled chemotaxis-fluid model. We consider a coupled chemotaxis-fluid model in high di-mensional space,and prove the global existence of weak solutions under certain conditions. Chemotaxis model arises from biology, and the fluid model is given in the form of Navier-Stokes equation or Stokes equation. This thesis reviews the history, recent research,and some relevant study results of chemotaxis-fluid model. We apply entropy estimate, Gagliardo-Nirenberg-Sobolev inequality and Aubin-Lions compactness theorem etc to prove the ex-istence of weak solutions for the following two models.The first model is: where Ω=RN(N>3).We give the following conditions: (1) χ(c),K(c) are continuous functions,κ(0)=0,κ’(c)>0,χ’(c)≥0,(2)▽φ∈L∞(Ω)ï¼›(3)0≤c0≤cM<∞,c0∈L1(Ω),â–½c0∈L2(Ω),▽φ(c0)∈L2(Ω), where ψ=(?)l;(4) n0≥0,n0(1+x+|ln n0|)∈L1(Ω)ï¼›(5)â–½.(?)(x)=0,(?)(x)∈L2(Ω).The main result is the following:Theorem:Suppose that m=2-2/N,(1)-(5)hold,then for all T>0,the problem(0.1) has a global weak solution.Moreover,we have the following estimatewhere||·||denotes the general L2norm.The second model is: We remark that it could be realistic to include the efect of gravity on cells, so we bringin the term (nφ) in the first equation of the first model to compose the new equation,â†'tn+u· n=δ nm·(χ(c)n c)+(n φ).We need more constrains toφ,(2)φ∈L∞(), φ∈L∞();When N≥2§and (1)(2)(3)(4)(5) hold, the conclusion of problem (0.2) is similar tothat of the previous model.