Optimal Regularity Estimates To Solutions Of A Class Of Parabolic Partial Differential Equations

Regularity theory has long history, as early as1900international mathematics confer-ence was held in Paris. Mathematician D.Hilbert proposed23famous open problems, and two of them are about regularity theory of equation solution, that highlights the difficult and important significance of regularity theory.Regularity theory are one of the hottest and challenging issues in the field of partial differential equations. Regularity theory include Lp estimate, Schauder estimate, Krylov-Safanov estimate, De Giorgi-Nash estimate, and so on. Regularity theory play important roles in the field of partial differential equation, they are basis for the existence, uniqueness, and regularity of solutions. Sobolev space is one of the most powerful analysis tools in the20th century, and they are widely used and research in the field of Mathematic. Regularity theory of many partial differential equations are study in Sobolev spaces. But along with the introduction of Orlicz space, many scholars extend the regularity theory in Sobolev spaces to more generalization Orlicz spaces, that gradually become the focus of scholars at home and abroad.This paper is mainly about the regularity estimate of parabolic Schrodinger equation and high order polyharmonic equation. For parabolic Schrodinger equation through char-acterization of domain of parabolic Schrodinger operator, thereby without any restrictions on the space dimension, we finally obtained the optimal regularity estimate for parabolic Schrodinger equations; For using simplified iteration-covering approach to discuss the reg-ularity of high order ployharmonic equation in Orlicz spaces.This paper is organized as follows. In Chapter1, we firstly introduce the theory of the solution regularity of differential equations, then discuss the basis concepts and important conclusions of Orlicz spaces. In Chapter2, we introduce preliminary knowledge and main theorem of parabolic Schrodinger equations, then we give some important lemma and proof, finally we obtained the optimal regularity estimate of parabolic Schrodinger equations. In Chapter3, we introduce preliminary knowledge and main theorem of high order polyhar-monic equations, then we give some important lemma and proof, finally we obtained the regularity estimate of high order polyharmonic equations.