Transport phenomena in collective dynamics: from micro to social hydrodynamics

Nonlocal-to-local limit of conservation laws

Maria Colombo

ETH Zurich

We consider a nonlocal approximation of the Burgers' equation obtained by convolving the velocity field with a smooth, compactly supported convolution kernel $\rho_\varepsilon$ in some of its occurrencies $$\partial_t u^\varepsilon + \partial_x( (u^\varepsilon \ast \rho_\varepsilon) u^\varepsilon ) = 0.$$ As $\varepsilon$ goes to $0$, Zumbrun proved the convergence of $u^\varepsilon$ to a solution of the Burgers' equation holds provided that the convolution kernel is even and the limit is sufficiently smooth. Motivated by numerical simulations, in particular by Amorim, Colombo and Teixeira, we investigate the convergence when the limit solution is nonsmooth, providing several counterexamples and some positive results for the corresponding viscous problem.