Systems of conservation laws are a class of nonlinear partial differential equations with several applications coming from both physics and engineering. In particular, the Euler equations of fluid dynamics are systems of conservation laws. More recent applications come from models of pedestrian and vehicular traffic. From both the numerical and the theoretic viewpoint, the understanding of this class of equations is still incomplete, owing to the presence of highly nonlinear phenomena. In particular, two of the main challenges are the finite time breakdown of regular solutions and the non uniqueness of distributional solutions. This research line aims at investigating the convergence of second order viscous approximations and the qualitative properties of physically relevant solutions. This analysis is pivotal to the design of accurate numerical schemes, which is also investigated, and to other applications. This research line is also concerned with the analysis of related topics like i) transport equations with low regularity coefficients; ii) eigenvalue problems for local and nonlocal elliptic operators.  

The main ongoing collaborations are those with people from Kobe University, SISSA (Trieste), the University of Bari, the University of Basel, the University of Milano Bicocca, the University of Padova and the University of Pavia.

References:

Siddhartha Mishra and Laura V. Spinolo
Accurate numerical schemes for approximating initial-boundary value problems for systems of conservation laws, J. Hyperbolic Differ. Equ. To appear

Gianluca Crippa, Carlotta Donadello and Laura V. Spinolo
Initial-boundary value problems for continuity equations with BV coefficients J. Math. Pures Appl. (9) 102 (2014), no. 1, 79-98.

Antoine Lemenant, Emmanouil Milakis and Laura V. Spinolo
Spectral stability estimates for the Dirichlet and Neumann Laplacian in rough domains J. Funct. Anal. 264 (2013), no. 9, 2097-2135


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