Acta mathematica scientia,Series B ›› 2011, Vol. 31 ›› Issue (5): 1719-1724.doi: 10.1016/S0252-9602(11)60356-1

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A NOTE ON THE DIPERNA-LIONS FLOWS

 LIU Xin, HUANG Wen-Liang*   

  1. School of Computer Science and Technology, Huazhong University of Science and Technology, Wuhan 430074, China; Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China
  • Received:2010-07-28 Online:2011-09-20 Published:2011-09-20
  • Contact: HUANG Wen-Liang,WenliangHuang1@gmail.com E-mail:hustlx@yahoo.cn;WenliangHuang1@gmail.com

Abstract:

In this note, we give a short proof for the DiPerna-Lions flows associated to ODEs following the method of Crippa and De Lellis [3]. More precisely, assume that [divb]− ∈L loc(Rd), |b|/(1 + |x| log |x|) ∈L∞(Rd) and |∇b| ·(|∇b|) ∈L loc(Rd), where (r) = log ··· log(r + c), c > 0. Then, there exists a unique regular Lagrangian flow associated with the ODE ˙X (t, x) = b(X(t, x)), X(0, x) = x.

Key words: DiPerna-Lions flow, Hardy-Littlewood maximal function

CLC Number: 

  • 60H10
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