Transport Equations, Mixing, and Fluid dynamics
Final Report Abstract
This research project was very successful in terms of its main objectives and it opens new perspectives and approaches for future research. It focussed on the analysis of linear advection-diffusion equations in the setting of low-regularity velocity fields, which are of relevance in turbulent fluid dynamics and which are the object of major research activities in recent years. The first achievement in the research project is an optimal stability estimate that controls the difference of two solutions in terms of the differences in their respective velocity fields, diffusion coefficients and initial data. The involved metric is a Kantorovich–Rubinstein distance with logarithmic cost function. A consequence of the stability estimate is a new uniqueness result for the advection-diffusion equation in the setting of velocity fields with merely integrable vorticity distributions. Stability estimates are crucial in the quantitative analysis of numerical discretization schemes for the advection-diffusion equation. They are exploited in the second main result of the research project, which consists of sharp estimates on the numerical error of finite-volume methods, thus providing optimal convergence rates for the discretization scheme. A somewhat dual line of research investigates the optimal regularity that is propagated by advection and advection-diffusion equations. This project examines regularity using Littlewood–Paley theory and optimal estimates which control a derivative of logarithmic order are established. The novelty in the estimates is their uniformity in the diffusivity constants. The new results can be exploited to derive sharp estimates on mixing, enhanced dissipation, and the zero-diffusivity limit. A fourth problem addressed in this project is the geometric ergodicity of passive scalars driven by random vortices. The problem is approached using Villani’s hypocoercivity method, thus constructing a Lyapunov functional for the dynamics with the help of iterated commutators. The new result provides exponential decay rates for the expectation of the solution to the associated linear advection equation with randomly varying velocity field. The new approach has potential applications in the theory of mixing of passive scalars.
Publications
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Optimal stability estimates and a new uniqueness result for advection-diffusion equations. Pure and Applied Analysis, 4(3), 571-596.
Navarro-Fernández, Víctor; Schlichting, André & Seis, Christian
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Error estimates for a finite volume scheme for advection–diffusion equations with rough coefficients. ESAIM: Mathematical Modelling and Numerical Analysis, 57(4), 2131-2158.
Navarro-Fernández, Víctor & Schlichting, André
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Stability results for advection-diffusion equations with deterministic and random vector fields. Dissertation thesis, Universität Münster, 2023
Víctor Navarro Fernández
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Propagation of regularity for transport equations: a Littlewood-Paley approach. Indiana University Mathematics Journal, 73(2), 445-473.
Meyer, David & Seis, Christian
