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RWTH Publication No: 855807 2023 
TITLE 
Conservation laws with nonlocal velocity : the singular limit problem 
AUTHORS 
Jan Friedrich, Simone Göttlich, Alexander Keimer, Lukas Pflug 
ABSTRACT 
We consider conservation laws with nonlocal velocity and show for nonlocal weights
of exponential type that the unique solutions converge in a weak or strong sense (dependent on the
regularity of the velocity) to the entropy solution of the local conservation law when the nonlocal
weight approaches a Dirac distribution. To this end, we establish first a uniform total variation
estimate on the nonlocal velocity which enables it to prove that the nonlocal solution is entropy
admissible in the limit. For the entropy solution, we use a tailored entropy flux pair which allows
the usage of only one entropy to obtain uniqueness (given some additional constraints). For general
weights, we show that monotonicity of the initial datum is preserved over time which enables it to
prove the convergence to the local entropy solution for rather general kernels and monotone initial
datum as well. This covers the archetypes of local conservation laws: Shock waves and rarefactions. It
also underlines that a “nonlocal in the velocity” approximation might be better suited to approximate
local conservation laws than a nonlocal in the solution approximation where such monotonicity does
only hold for specific velocities. 
KEYWORDS 
nonlocal conservation law, nonlocal in velocity, convergence, weak entropy solution,
monotonicity preserving, singular limit, singular limit for nonlocal in velocity conservation laws 