Lagrangian flows for vector fields with gradient given by a singular integral

Abstract

We prove quantitative estimates on flows of ordinary differential equations with vector field with gradient given by a singular integral of an $L^1$ function. Such estimates allow to prove existence, uniqueness, quantitative stability and compactness for the flow, going beyond the $BV$ theory. We illustrate the related well-posedness theory of Lagrangian solutions to the continuity and transport equations.

References in corpus (2)

• An Ill Posed Cauchy Problem for a Hyperbolic System in Two Space Dimensions
• A Lemma and a Conjecture on the Cost of Rearrangements

Cited by in corpus (13)

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• Renormalization for autonomous nearly incompressible BV vector fields in 2D
• Smooth approximation is not a selection principle for the transport equation with rough vector field
• The Lagrangian structure of the Vlasov-Poisson system in domains with specular reflection
• Existence of weak solutions to the two-dimensional incompressible Euler equations in the presence of sources and sinks
• Error estimates for a finite volume scheme for advection-diffusion equations with rough coefficients
• A regularity result for the Fokker-Planck equation with non-smooth drift and diffusion