Research

My research interests concern partial differential equations coming from continuum mechanics, in particular hyperbolic conservation laws, the equations of fluid dynamics, convex integration methods and transport with irregular flows. 

Papers and preprints

1. Non-uniqueness and energy dissipation for 2D Euler equations with vorticity in Hardy spaces
   preprint arXiv, with M. Buck
   Non-uniqueness in L2 for the 2D Euler equations on the full space with control on the vorticity in some real Hardy space which scales below L1 

2. Local Nonuniqueness for Stochastic Transport Equations with Deterministic Drift
   preprint arXiv, with A. Schenke
   Is transport noise enough to guarantee uniqueness to the linear transport equation when the vector field is below Lipschitz? No (in some cases). 

3. On the failure of the chain rule for the divergence of Sobolev vector fields  arXiv
   Journal of hyperbolic differential equations, 20(2), 349-385, 2023 with M. Buck
   The "static" version of convex integration applied to transport equation

4. Non-uniqueness of power-law flows
   Communications in Mathematical Physics, 388(1), 199-243, with Jan Burczak and László Székelyhidi
   Convex integration applied to non-Newtonian fluids to get nonuniqueness of Leray-Hopf solutions "below the compactness exponent"

5. Convex integration solutions to the transport equation with full dimensional concentration  arXiv
   Annales de l’Institut Henri Poincaré C, Analyse non linéaire, 37(5) (2020) 1075–1108, with Gabriel Sattig
   The proof of non-uniqueness of solutions to the transport equation in the case of an L^p density and a W^{1,q} vector field, with 1/p + 1/q > 1 + 1/d (where d is the dimension of the physical space)

6. Non renormalized solutions to the continuity equation
   Calculus of Variations and PDE, 58, 208 (2019), with László Székelyhidi
   The existence of Sobolev, continuous and incompressible vector fields, for which uniqueness of solutions to the continuity equation fails

7. Non-uniqueness for the transport equation with Sobolev vector fields
   Annals of PDE,  4(2) (2018), with László Székelyhidi
   The construction of a large class of examples of non-uniqueness for the linear transport equation in Sobolev spaces

8. Quadratic Interaction Estimate for Hyperbolic Conservation Laws, an overview 
   Journal of Mathematical Sciences, 233(6) (2018), 905-929
   A "user friendly version" of the paper "Quadratic Interaction Functional for General Systems of Conservation Laws"

9. A “forward-in-time” quadratic potential for systems of conservation laws
   Nonlinear Differential Equations and Applications, 24(5) (2017), 23 pages.
   An improved version of the quadratic interaction functional for system of conservation laws, with the property that its value at any time does not depend on future history of the solution

10. Convergence rate of the Glimm scheme
    Bulletin of the Institute of Mathematics of Academia Sinica (New Series), 11(1) (2016), 235-300, with Stefano Bianchini
    The convergence of the Glimm scheme (and its rate of convergence) for strictly hyperbolic systems of conservation laws (without convexity assumptions)

11. Quadratic Interaction Functional for General Systems of Conservation Laws
    Communications in Mathematical Physics, 338 (2015), 1075-1152, 2015, with Stefano Bianchini
    The Glimm interaction potential for strictly hyperbolic systems of conservation laws (without convexity assumptions). See also the user friendly version.

12. Quadratic interaction functional for systems of conservation laws: a case study
    Bulletin of the Institute of Mathematics of Academia Sinica (New Series), 9(3) (2014), 487-546, with Stefano Bianchini
    The Glimm interaction potential for a triangular system of conservation laws

13. On a Quadratic Functional for Scalar Conservation Laws,
    Journal of Hyperbolic Differential Equations, 11(2) (2014), 355-435, with Stefano Bianchini
    The Glimm interaction potential for scalar conservation laws with non-convex flux

14. Compact Hausdorff Pseudoradial Spaces and their Pseudoradial Order
    Extracta Mathematicae, 25(3) (2010), 309-315, with Gino Tironi
    The construction of compact Hausdorff topological spaces of any finite pseudoradial order

15. Pseudoradial Order of Pseudoradial Spaces
    Mathematica Pannonica, 21(2) (2010), 159-175, with Gino Tironi
    The construction of topological spaces of any pseudoradial order

Proceedings

16. On some recent results concerning non-uniqueness for the transport equation arXiv
    Hyperbolic problems: theory, numerics, applications. Proceedings of the 17th international conference on Hyperbolic Problems, AIMS 10 (2020), 562-568
    An overview on some recent works (with Gabriel Sattig and László Székelyhidi) about non-uniqueness for the linear transport equation in the DiPerna-Lions' setting.

17. Lagrangian representation for systems of conservation laws: an overview PDF
    Theory, Numerics and Applications of Hyperbolic Problems II. HYP 2016. Springer Proceedings in Mathematics and Statistics, 237 (2018), 335-347
    An overview on some forthcoming works (with Stefano Bianchini) in which we propose a way to describe BV solutions to hyperbolic systems of conservation laws in one space dimension from a Lagrangian point of view

18. Lagrangian structure of BV solutions for hyperbolic systems of conservation laws PDF
    Oberwolfach Reports, 30 (2016), 41-44
    A very brief introduction to the notion of "Lagrangian representation"

19. A quadratic interaction estimate for conservation laws: motivations, techniques and open problems
    Bulletin of the Brazilian Math. Society, 47(2) (2016), 589-604
    A description of the importance of a quadratic interaction estimate in the theory of conservation laws

20. A New Quadratic Potential for Scalar Conservation Laws PDF
    Oberwolfach Reports, 29 (2013), 58, with Stefano Bianchini
    A short note on the extension of the Glimm potential to general systems of conservation laws

Thesis

21. Ph.D. Thesis (2015): Interaction functionals, Glimm approximations and Lagrangian structure of BV solutions for hyperbolic systems of conservation laws

22. Master Thesis (2012): On a quadratic potential for scalar conservation laws

23. Bachelor Thesis (2009): Spazi pseudoradiali e ordine di pseudoradialità