Coclite, Giuseppe Maria and Colombo, Maria and Crippa, Gianluca and De Nitti, Nicola and Keimer, Alexander and Marconi, Elio and Pflug, Lukas and Spinolo, Laura V.. (2023) Oleı̆niktype estimates for nonlocal conservation laws and applications to the nonlocaltolocal limit. Preprints Fachbereich Mathematik, 2023 (02).

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Abstract
We consider a class of nonlocal conservation laws with exponential kernel and prove that quantities involving the nonlocal term $W:=\mathbb{1}_{(\infty,0]}(\cdot)\exp(\cdot) \ast \rho$ satisfy an Oleı̆niktype entropy condition. More precisely, under different sets of assumptions on the velocity function $V$, we prove that $W$ satisfies a onesided Lipschitz condition and that $V'(W) W \partial_x W$ satisfies a onesided bound, respectively. As a byproduct, we deduce that, as the exponential kernel is rescaled to converge to a Dirac delta distribution, the weak solution of the nonlocal problem converges to the unique entropyadmissible solution of the corresponding local conservation law, under the only assumption that the initial datum is essentially bounded and not necessarily of bounded variation.
Faculties and Departments:  05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Analysis (Crippa) 12 Special Collections > Preprints Fachbereich Mathematik 

UniBasel Contributors:  Crippa, Gianluca 
Item Type:  Preprint 
Publisher:  Universität Basel 
Language:  English 
edoc DOI:  
Last Modified:  13 Apr 2023 08:48 
Deposited On:  13 Apr 2023 08:48 
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