My research, on the one hand, falls within the general framework of non-linear evolution equations. In particular, these partial differential equations are motivated by Fluid Mechanics. 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.Our main focus is the study of non-local regularizations defined by fractional derivatives.
The 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.
On the other hand, I collaborate in the field of social economics, providing statistical support and contributing to data analysis. The first study was motivated by the economic resilience of cooperative employment and the evolution of women's employment rate within cooperatives, particularly in the Autonomous Community of the Basque Country. However, the general picture of the Spanish provinces was also considered.
Achleitner, F., Cuesta, C. M. and Diez-Izagirre, X. (2025). Existence of undercompressive travelling waves of a non-local generalised Korteweg-de Vries-Burgers equation. Electronic Journal of Differential Equations, vol. 2025, 2025. Journal link, Preprint: arXiv:2412.03209 (2024).
Etxezarreta-Etxarri, A., Díez-Izagirre, X., Erauskin-Tolosa, A., Barinaga-Rementeria, I. eta Izagirre-Olaizola, J. (2024). Kooperatibak, erresilientzia ekonomikoa eta emakumeen enplegua Euskal Autonomia Erkidegoan. GIZAEKOA-Revista Vasca de Economía Social, (21). Aldizkariaren esteka.
Cuesta, C. M. and Diez-Izagirre, X. (2023). Large time behaviour of a conservation law regularised by a Riesz-Feller operator: the sub-critical case. Czech Math 73(4): 1057-1080. Journal link, Preprint: arXiv:2302.03981, (2023).
Cuesta, C. M. and Diez-Izagirre, X. (2020). Vanishing viscosity limit of a conservation law regularised by a Riesz-Feller operator. Monatshefte für Mathematik 192(3): 513-550. Journal link, Preprint: arXiv:1909.00685, (2019).
TBU