On the local limit for nonlocal conservation laws
- Speaker: Prof. Dr. Gianluca Crippa
- Place: Online (zoom)
- Time: 29.4.2021, 14:00 - 29.4.2021, 15:00
- Invited by: Die Dozenten des Schwerpunktes Partielle Differentialgleichungen
Consider a family of continuity equations where the velocity field is given by the convolution of the solution with a regular kernel. In the singular limit where the convolution kernel is replaced by a Dirac delta, one formally recovers a conservation law: can we rigorously justify this formal limit? This question was posed by P. Amorim, R. Colombo and A. Teixeira and a positive answer was suggested by numerical simulations. In the talk we will exhibit counterexamples showing that in general convergence of the solutions does not hold. We will also show that the answer to the above question is positive if we consider viscous perturbations of the nonlocal equations. In this case, in the singular local limit the solutions converge to the solution of the viscous conservation law. We will also discuss the possible role of numerical viscosity in numerical simulations, as well as some more recent results dealing with the case of an anisotropic convolution kernel, a more realistic case for applications to traffic modelling.
The talk will be based on some joint works with Maria Colombo, Marie Graff, Elio Marconi, and Laura Spinolo.