Cauchy problem of semi-linear parabolic equations with weak data in homogeneous space and application to the Navier-Stokes equations  

作　　者：苗长兴 

机构地区：[1]Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

出　　处：《Science China Mathematics》2003年第5期641-661,共21页中国科学：数学（英文版）

基　　金：This work was supported by the National Natural Science Foundation of China(Grant No.19971001);the Special Funds for Major State Basic Research Projects of China.

摘　　要：In this paper we study the Cauchy problem for a class of semi-linear parabolic type equations with weak data in the homogeneous spaces. We give a method which can be used to construct local mild solutions of the abstract Cauchy problem in? σ,s,p andL q([0, T);H s,p) by introducing the concept of both admissible quintuplet and compatible space and establishing time-space estimates for solutions to the linear parabolic type equations. For the small data, we prove that these results can be extended globally in time. We also study the regularity of the solution to the abstract Cauchy problem for nonlinear parabolic type equations in ?σ,s,p. As an application, we obtain the same result for Navier-Stokes equations with weak initial data in homogeneous Sobolev spaces.In this paper we study the Cauchy problem for a class of semi-linear parabolic type equations withweak data in the homogeneous spaces. We give a method which can be used to construct local mild solutionsof the abstract Cauchy problem in Cσ,s,p and Lq([O, T);Hs,p) by introducing the concept of both admissiblequintuplet and compatible space and establishing time-space estimates for solutions to the linear parabolic typeequations. For the small data, we prove that these results can be extended globally in time. We also study theregularity of the solution to the abstract Cauchy problem for nonlinear parabolic type equations in Cσ,s,p. Asan application, we obtain the same result for Navier-Stokes equations with weak initial data in homogeneousSobolev spaces.

关 键 词：CAUCHY problem  parabolic equation  Naiver-Stokes equation  ADMISSIBLE quintuplet compatible space  time-space estimates. 

分 类 号：O175.26[理学—数学]