带有logistic源的三维趋化系统弱解的最终光滑性

The Final Smoothness of the Weak Solution of the Three-dimensional Chemotaxis System with Logistic Source

基于三维趋化系统弱解的整体存在性的结果,进一步考虑了一类带有logistic源抛物-抛物型趋化方程组在三维情形时对应的齐次诺依曼初边值条件下弱解的最终光滑性;通过构造能量泛函并利用Sobolev最大正则性理论、Sobolev嵌入定理、Gagliardo-Nirenberg不等式、Young不等式、Hölder不等式、Poincaré不等式、紧嵌入定理以及Gronwall不等式得到解的高阶正则性估计;结果表明:对于任意非负且适当的初始值,可以证明到系统的弱解在一定的等待时间后变成经典解。

Based on the results of the global existence of weak solutions of three-dimensional chemotaxis system,this paper further considers the final smoothness of the weak solutions of a class of parabolic chemotaxis equations with logistic source under the corresponding homogeneous Neumann initial boundary value conditions in three-dimensional case.In this paper,the higher-order regularity estimation of the solution is obtained by constructing the energy functional and using Sobolev maximum regularity theory,Sobolev embedding theorem,Gagliardo Nirenberg inequality,Young inequality,Hölder inequality,Poincaréinequality,compact embedding theorem and Gronwall inequality.The results show that for any nonnegative and appropriate initial value,it can be proved that the weak solution of the system becomes a classical solution after a certain waiting time.