Research

Our study is mainly focused on the analysis of non-local regularisations of a scalar conservation law given by a fractional derivative which can be seen as a non-local integro-differential operator. 

The major motivation for studying this kind of non-local conservation laws is that they can be used to describe the internal structure of hydraulic jumps in a shallow water model. In particular, the non-local generalised Kortewed-de Vries-Burgers equation with fractional derivative of order 4/3 and either a quadratic or cubic flux function has been derived as a model to describe hydraulic jumps in a shallow water limit.

• C. M. Cuesta, and X. Diez-Izagirre. Large time behaviour of a conservation law regularised by a Riesz-Feller operator: the sub-critical case. Czech Math J Published online on July 20, (2023) Journal link, Preprint: arXiv:2302.03981, (2023).

• C. M. Cuesta, and X. Diez-Izagirre. Vanishing viscosity limit of a conservation law regularised by a Riesz-Feller operator. Monatshefte für Mathematik 192(3): 513-550, 2020 Journal link, Preprint: arXiv:1909.00685, (2019).

• F. Achleitner, C. M. Cuesta, and X. Diez-Izagirre. Non-classical shocks in a non-local generalised Korteweg-de Vries-Burgers equation.