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Estimates and regularity results for the DiPerna-Lions flow. (English) Zbl 1160.34004
The flow of a smooth bounded vector \(b:[0,1]\times \mathbb{R}^n\to \mathbb{R}^n\) is the smooth map \(X:[0,1]\times \mathbb{R}^n\to \mathbb{R}^n\) such that \[ \tfrac{dX}{dt}=b(t,X),\;X(0,x)=x. \] The existence, uniqueness and stability of regular Lagrangian flows for vector fields with bouned divergence have been proved by R. J. DiPerna and P. L. Lions [Invent. Math. 98, No. 3, 511–547 (1989; Zbl 0696.34049)].
In this paper, the authors show that some results of DiPerna and Lions can be recovered from simple a priori estimates, directly in the Lagrangian formulation. Assuming the existence of a regular Lagrangian flow \(X\), under various relaxed hypotheses, they establish estimates of some integral quantities depending on \(X(t,x)-X(t,y\)). Similar estimates are also established for \(X(t,x)-X'(t,X)\) or regular Lagrangian flows of different vector fields \(b\) and \(b'\). As direct corollaries of such estimates they derive existence, uniqueness, stability as well some mild regularity properties.

MSC:
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
37C10 Dynamics induced by flows and semiflows
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[1] DOI: 10.1007/s00222-004-0367-2 · Zbl 1075.35087
[2] Ambrosio L., Rend. Sem. Mat. Univ. Padova 114 pp 29– (2005)
[3] Bressan A., Rend. Sem. Mat. Univ. Padova 110 pp 103– (2003)
[4] Bressan A., Rend. Sem. Mat. Univ. Padova 110 pp 97– (2003)
[5] DOI: 10.1007/BF01393835 · Zbl 0696.34049
[6] Lions P. L., Oxford Lect. Ser. Math. Appl. pp 3– (1996)
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