A quantitative theory for the continuity equation

Abstract

In this work, we provide stability estimates for the continuity equation with Sobolev vector fields. The results are inferred from contraction estimates for certain logarithmic Kantorovich--Rubinstein distances. As a by-product, we obtain a new proof of uniqueness in the DiPerna--Lions setting. The novelty in the proof lies in the fact that it is not based on the theory of renormalized solutions.

Final version, includes optimality result. Accepted for publication in Annales IHP

References in corpus (1)

• A new proof of the uniqueness of the flow for ordinary differential equations with BV vector fields

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• Analysis of the implicit upwind finite volume scheme with rough coefficients
• Quantitative stability estimates for Fokker-Planck equations
• A note on the vanishing viscosity limit in the Yudovich class
• On the dynamics of point vortices for the 2D Euler equation with $L^p$ vorticity
• Error estimates for a finite volume scheme for advection-diffusion equations with rough coefficients
• Differentiability properties of the flow of 2d autonomous vector fields
• Kantorovich-Rubinstein Distance and Approximation for Non-local Fokker-Planck Equations