Analysis, control, and singular limits for hyperbolic conservation laws

Analysis, control, and singular limits for hyperbolic conservation laws

Author: Nicola de Nitti
Date: Monday July 24, 2023

PhD Thesis: Analysis, control, and singular limits for hyperbolic conservation laws (July 24, 2023)

Abstract. The main focus of this thesis is on the study of singular limits related to scalar conservation
laws. These are first-order partial differential equations that describe how the amount of a physical
quantity in a given region of space changes over time, solely determined by the flux of that quantity
across the boundary of the region.

The first part of this manuscript deals with nonlocal regularizations of scalar conservation laws,
where the flux function depends on the solution through the convolution with a given kernel. These
models are widely used to describe vehicular traffic, where each car adjusts its velocity based on
a weighted average of the traffic density ahead. First, we establish the existence, uniqueness, and
maximum principle for solutions of the nonlocal problem under mild assumptions on the kernel and
flux function. We then investigate the convergence of the solution to that of the corresponding local
conservation law when the nonlocality is shrunk to a local evaluation (i.e., when the kernel tends to
a Dirac delta distribution). For kernels of exponential type, we analyze this singular limit for initial
data of bounded variation as well as for merely bounded ones, using Ole˘ınik-type estimates. We
also demonstrate how the techniques developed in this analysis can be used to study the long-time
behavior of a nonlocal regularization of the Burgers equation and to show that the asymptotic
profile is given by the N-wave entropy admissible solution. We also investigate the role played by
artificial viscosity in the nonlocal–to–local singular limit process. Finally, we study the boundary
controllability problem for nonlocal traffic models.

In the second part of this thesis, we address the controllability of scalar conservation laws
on networks and its relationship to the vanishing viscosity singular limit. Our main analysis is
carried out in the linear case: for a linear advection-diffusion equation, we show that the cost of
controllability blows up exponentially as the viscosity parameter vanishes for small times and decays
exponentially for a sufficiently long time-horizon. Finally, for nonlinear conservation laws, we prove
a controllability result for entropy solutions using a Lyapunov approach and highlight the stability
of this result when a small viscosity is added.

Supervisor: Prof. Dr. DhC. Enrique Zuazua

Board of examiners
• Prof. Dr. Emil Wiedemann
• Prof. Dr. Enrique Zuazua
• Prof. Dr. Giuseppe M. Coclite

Mon. July 24, 2023 at 10:30H

Room H13. Johann-Radon-Hörsaal
Felix-Klein Building, Cauerstr. 11, Erlangen
Friedrich-Alexander-Universität Erlangen-Nürnberg

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