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Optimal stability estimates and a new uniqueness result for advection-diffusion equations

Víctor Navarro-Fernández, André Schlichting and Christian Seis

Vol. 4 (2022), No. 3, 571–596
Abstract

We present two main contributions. First, we provide optimal stability estimates for advection-diffusion equations in a setting in which the velocity field is Sobolev regular in the spatial variable. This estimate is formulated with the help of Kantorovich–Rubinstein distances with logarithmic cost functions. Second, we extend the stability estimates to the advection-diffusion equations with velocity fields whose gradients are singular integrals of L1 functions entailing a new well-posedness result.

Keywords
advection-diffusion equation, stability estimates, uniqueness, Kantorovich–Rubinstein distance
Mathematical Subject Classification
Primary: 35B30, 35B35, 35B45
Secondary: 35K15
Milestones
Received: 11 March 2022
Revised: 12 July 2022
Accepted: 6 September 2022
Published: 4 December 2022
Authors
Víctor Navarro-Fernández
Institut für Analysis und Numerik
Westfälische Wilhelms-Universität Münster
Münster
Germany
André Schlichting
Institut für Analysis und Numerik
Westfälische Wilhelms-Universität Münster
Münster
Germany
Christian Seis
Institut für Analysis und Numerik
Westfälische Wilhelms-Universität Münster
Münster
Germany