Global Well-posedness Of Chemotaxis System With Rotational Sensitivity

Because it can well explain some important phenomena and laws in physics,chem-istry,biology and other fields,the theory and application of partial differential equations have become an important mathematical research direction.These theories include the ex-istence,uniqueness,boundedness,finite time blow-up and large time asymptotic behavior of solution for the equations.In particular,chemotaxis is the directed movement of cells or organisms in response to chemical stimuli,and plays an important role in a variety of biological processes such as embryo development,wound healing,and tumor invasion.The following three chemotactic systems are studied in this dissertation:(?)attraction-repulsion chemotaxis system with general rotational sensitivity(?)(?)chemotaxis system with indirect signal production and rotational sensitivity(?)(?)chemotaxis-(Navier-)Stokes system with indirect signal production in a fluid environ-ment(?)where ?(?)Rd is a bounded domain with smooth boundary,the matrix-valued functions S1 ? Rd×D,S2 ? Rd×d and S ? Rd×d are rotational sensitivity,the smooth function f ?Wloc 1,?([0,?)),the gravitational potential ? ? W,1,?(?).The specific research contents and the main research results are as follows:1.For the model(?),under boundary conditions(?n-nS1(x,n,c,v)·?c+nS2(x,n,c,v)·?)·v=?c·v=?v·v=0,where v denotes the unit outer normal of(?)?,the global existence of generalized solutions are established for general large initial data and arbitrary dimension(d>1)by means of a new energy method proposed by Winkler(SIAM J.Math.Anal.,2015).In particular,we remove the smallness assumption on the initial data showed by Dong-Li(Math.Methods Appl.Sci.,2017).2.For the model(?),let the boundary conditions be(?n-nS(x,n,c,v)·?c)·v=?c·v=?v·v=0,and suppose that the smooth function f satisfiesf(0)?0,f(s)?r1-r2sa,s?0,where r1?0,r2>0 and ?>1.When a>d/4+1/2(d?2),we prove that the global existence and boundedness of classical solution,and analyse that the interaction of indirect signal production,logistic source and rotational sensitivity on the solution regularity of chemotaxis system.3.For the model(?),let the boundary conditions be?n·v=Vc·v=?v·v=0,u=0.For d=2,?=1 and d=3,?=0,when r? 0 and ?>0,the global existence and boundedness of classical solutions are proved by using the properties of Neumann semigroup and Stokes operator,and the large time asymptotic behavior of the solutions is further proved.These results imply that the indirect signal production mechanism can make arbitrarily small quadratic degradation of cells is sufficient to rule out blow-up of solutions.This dissertation improves the result of Tao-Winkler(Z.Angew.Math.Phys.,2015)requiring,u>23.