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Stefano Bianchini

Curriculum Vitae

My research interests is analysis and nonlinear PDEs, in particular the application of measure theoretic techniques to hyperbolic and transport equations. To have a brief overview of my research, these are in my opinion the most important papers:
1. S. Bianchini, A. Bressan, VANISHING VISCOSITY SOLUTIONS OF NONLINEAR HYPERBOLIC SYSTEMS, Annals of Mathematics 161 (2005), pag. 223-342.
Proof of the convergence of the vanishing viscosity approximations to the hyperbolic solution for system of conservation laws. The viscosity is the identity matrix.

2. S. Bianchini, L. Caravenna, SBV REGULARITY FOR GENUINELY NONLINEAR, STRICTLY HYPERBOLIC SYSTEMS OF CONSERVATION LAWS IN ONE SPACE DIMENSION, Comm. Math. Phys. 313 (2012), 1-33.
Solution to the SBV regularity problem for solutions to genuinely nonlinear hyperbolic systems.

3. S. Bianchini, S. Modena, QUADRATIC INTERACTION FUNCTIONAL FOR GENERAL SYSTEMS OF CONSERVATION LAWS , Communications in Mathematical Physics 338, p. 1075-1152.
The quadratic interaction estimate for general strictly hyperbolic systems of conservation laws in one space dimension.

4. S. Bianchini, E. Marconi, ON THE STRUCTURE OF $L^\INFTY$-ENTROPY SOLUTIONS TO SCALAR CONSERVATION LAWS IN ONE-SPACE DIMENSION , Archive for Rational Mechanics and Analysis 226 (1), p. 441-493.
Here the structure of $L^\infty$-entropy solutions to conservation laws is described in full details, yielding in particular the proof of the conjecture on the concentration of the entropy dissipation.

5. S. Bianchini, L. Caravenna, ON THE EXTREMALITY, UNIQUENESS AND OPTIMALITY OF TRANSFERENCE PLANS, Bullettin of the Institute of Mathematics, Academia Sinica Taipei 4 (2009), p. 353-454.
Title is self explanatory: the key argument is a uniqueness results for plans concentrated on and Borel linear preorder.

6. S. Bianchini, S. Daneri, ON SUDAKOV'S TYPE DECOMPOSITION OF TRANSFERENCE PLANS WITH NORM COSTS, Memoirs AMS 251, n. 1197.
The solution of the Monge transportation problem with convex norm using the Sudakov approach.

7. G. Alberti, S. Bianchini, G. Crippa, A UNIQUENESS RESULT FOR THE CONTINUITY EQUATION IN TWO DIMENSIONS, Journal EMS 16, p. 201-234.
Complete description of weak solutions to 2d autonomous divergence free linear transport.

8. S. Bianchini, P. Bonicatto, A UNIQUENESS RESULT FOR THE DECOMPOSITION OF VECTOR FIELDS IN R^d , preprint SISSA 2017.
Solution to the Bressan's compactness conjecture