Regularity For Sub-elliptic Parabolic Systems With Different Structural Coefficients

The regularity theory of weak solutions is an important and difficult problem in the field of modern par-tial differential equations,which has a long history of research.Since L.H(?)rmander published the foundation work about subellipticity,the regularity of weak solutions for subelliptic systems consisted of noncommuta-tive vector fields has become one of the hot topics in the study of partial differential equations.In this paper,we consider the regularity of weak solutions for nonlinear subelliptic parabolic systems with different structural coefficients on the Heisenberg group under controlled growth and natural growth conditions,includes Dini continuity conditions and VMO discontinuity coefficients.The content of this paper is as follows:In the first chapter,we briefly introduce the research background,research status,research contents,research methods and innovation points.In the second chapter,we elaborate on the basic knowledge of Heisenberg group,basic lemma and inequality,which is convenient for later use.In the third chapter,we are concerned about the weak solutions of subelliptic parabolic systems with Dini continuous coefficients.We take advantage of the A-caloric approximation method in Heisenberg group to get the optimal partial C~1regularity for weak solutions under the superquadratic controllable growth and natural growth conditions.In the fourth chapter,we study the subelliptic parabolic equations with VMO discontinuous coeffi-cients.We use the A-caloric approximation technique to obtain the optimal partial H(?)lder regularity for weak solutions under the superquadratic controllable growth and natural growth conditions.