关键词: DiPerna-Lions flow, Hardy-Littlewood maximal function

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(R^d), |b|/(1 + |x| log |x|) ∈L^∞(R^d) and |∇_b| ·(|∇_b|) ∈L^∞ loc(R^d), 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