Estimates and regularity results for the DiPerna-Lions flow. (English) Zbl 1160.34004

Crippa, Gianluca; De Lellis, Camillo

J. Reine Angew. Math. 616, 15-46 (2008).

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.

Reviewer: Jozef Myjak (L’Aquila)

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

Keywords:

ODE with Sobolev coefficients; Lagrangian flow; transport equation; Lipschitz estimates; stability

References:

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