BV solutions to some nonlocal transport equations in 1d Debora Amadori University of L’Aquila (Italy) [email protected] In this talk we deal with transport equations in one space dimension, that contain integral terms depending on the unknown variable. We introduce two different models: the first one arises in the study of two-layers granular flow under a certain limit (so called slow erosion limit); the second one is the kinetic Kuramoto model (KKM) and originates from the analysis of collective synchronized phenomena in a mean-field limit, as the number of discrete oscillators tends to infinity ([2]). Both equations are analyzed by means of a wave-front tracking algorithm which is suitable for the analysis of scalar conservation laws with nonlocal flux. Such an approach, when compared to a classical iteration procedure based on recomputing the nonlocal term at each time step, leads to a possibly simpler analysis in obtaining rigorous estimates on approximate solutions. We will present results on bounded weak solutions for these equations: existence of BV solutions and their stability. For the KKM equation, the phenomena of solutions whose L∞ -norm may grow to ∞ as t → ∞ is discussed. Talk based on [1] and on joint work with: Seung-Yeal Ha (SNU, Seoul, Korea), Jinyeong Park (SNU, Seoul, Korea). References [1] Amadori, D. and Shen, W.: Front tracking approximations for slow erosion. Discrete Contin. Dyn. Syst. 32 (2012), 1481–1502 [2] Lancellotti, C.: On the Vlasov limit for systems of nonlinearly coupled oscillators without noise. Transport theory and statistical physics 34, 523– 535 (2005)