Semigroup Estimates Of The Three-Dimensional Patlak-Keller-Segel Model

We consider the Cauchy problem of the three-dimensional Patlak-Keller-Segel chemotactic model.In particular,the initial data we focus on is almost a Dirac mass supported on a straight line.Our conjecture is that if the data is sufficiently close to a straight line,then global well-posedness holds for the Patlak-Keller-Segel model.There’re two points lying in the significance of this conjecture.On the one hand,the data is of low regularity,so we can understand to a great extent the scheme of wellposedness in Patlak-Keller-Segel model.On the other hand,the conjecture is relative to how the motion of cells,supported nearly on a strip,is affected by chemo-attractant,hence there is some pratical meaning in our research.To prove the conjecture,the idea is to decompose the solution into a core part,which is induced by the Dirac mass,and a background part,which is related to the perturbation term.Then we’ll establish the semigroup estimates for these two parts.Finally we’ll use the contraction mapping principle to complete our proof.In this thesis,we state the three-dimensional semigroup estimates of these two parts and give a explicit proof.Meanwhile,we recall the two-dimensional semigroup estimates in the core part.When dealing with the core part,we deploy the Fourier transform to reduce the model to two-dimentional case,then the long time estimates can be derived by the two-dimensional semigroup estimates,which in turn can be combined with several short time estimates to prove the three-dimensional estimates.As for the background part,there are results claiming that the semigroup is well-defined,then we can repeat the mechanism in the proof of the core part to finish the corresponding estimate.Unfortunately,despite there are certain semigroup estimates in these two parts,the exponential decay estimate in the core part is not known,hence for now the estimates are not enough to construct the contraction mapping,and the validity of the conjecture remains an open problem.