Acta mathematica scientia, Series B >
A NOTE ON THE DIPERNA-LIONS FLOWS
Received date: 2010-07-28
Online published: 2011-09-20
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
LIU Xin , HUANG Wen-Liang . A NOTE ON THE DIPERNA-LIONS FLOWS[J]. Acta mathematica scientia, Series B, 2011 , 31(5) : 1719 -1724 . DOI: 10.1016/S0252-9602(11)60356-1
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