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# Publications

Below you will find the list of my publications, arranged according to research topic.
The PDF file is not the published version, but it contains all the most important ideas and computations, and often differs from the final version only by minor corrections.

## Measure Theory

These papers concern the description of the range of a vector valued measure (a Zonoid by definition), extending the well known Theorem of Lyapunov (often used in Control Problems).

Collaborators: Raphaël Cerf, Carlo Mariconda.

1. S. Bianchini, C. Mariconda, R. Cerf, TWO DIMENSIONAL ZONOIDS AND CHEBYSHEV MEASURES, J. Math. Anal. Appl. 211-2 (1997), pag. 512-526.
A very simple proof that every convex compact subset of $\R^2$ containing $0$ is the range of a vector valued measure.

2. S. Bianchini, C. Mariconda, R. Cerf, CHEBYSHEV MEASURES AND THE VECTOR MEASURES WHOSE RANGE IS STRICTLY CONVEX, Atti Sem. Mat. Fis. Univ. Modena 46 (1998), pag. 525-534.

3. S. Bianchini, C. Mariconda, THE VECTOR MEASURES WHOSE RANGE IS STRICTLY CONVEX: SOME CHARACTERIZATIONS AND APPLICATIONS , J. Math. Anal. Appl. 232 (1999), pag. 1-19.
A simple characterization of measures with strictly convex range.

4. S. Bianchini, EXTREMAL FACES OF THE RANGE OF A VECTOR MEASURE AND A THEOREM OF LYAPUNOV, J. Math. Anal. Appl. 231 (1999), pag. 301-318.
A complete description of Zonoids.

## Conservation Laws

I have worked on several topics on conservation laws: existence of solutions, stability, singular approximations, regularity.
Recently I am interested in the application of tools from Geometric Measure Theory to describe the fine structure of the solutions.

Collaborators: Fabio Ancona, Alberto Bressan, Laura Caravenna, Rinaldo M. Colombo, Bernard Hanouzet, Elio Marconi, Stefano Modena, Francesca Monti, Roberto Natalini, Laura Spinolo, Lei Yu, Stefano Modena, Elio Marconi.

1. S. Bianchini, THE SEMIGROUP GENERATED BY A TEMPLE CLASS SYSTEM WITH NON CONVEX FLUX FUNCTION, Diff. Integr. Equat. 13 (10-12), pag. 1529-1550.
An extension of P. Baiti & A. Bressan, The semigroup generated by a Temple class system with large data.

2. S. Bianchini, A. Bressan, BV SOLUTIONS FOR A CLASS OF VISCOUS HYPERBOLIC SYSTEMS, Indiana Univ. Math. J. 49 (2000), pag. 1673-1713.
The vanishing viscosity approach for a special system of conservation laws.

3. S. Bianchini, ON THE SHIFT DIFFERENTIABILITY OF THE FLOW GENERATED BY A HYPERBOLIC SYSTEM OF CONSERVATION LAWS, Discr. Cont. Dyn. Systems 6-2 (2000), pag. 329-350.
A notion of differentiability w.r.t. initial data for the semigroup generated by conservation laws.

4. S. Bianchini, A GLIMM TYPE FUNCTIONAL FOR A SPECIAL JIN-XIN RELAXATION MODEL, Ann. Inst. H. Poincare' Anal. Nonlinear 18-1 (2001), pag. 19-42.

5. S. Bianchini, STABILITY OF $L^\INFTY$ SOLUTIONS FOR HYPERBOLIC SYSTEMS WITH COINCIDING SHOCKS AND RAREFACTIONS, SIAM J. Math. Anal. 33-4 (2001), pag. 959-981.
The construction of a continuous semigroup for large data in Temple type systems

6. S. Bianchini, A. Bressan, ON A LYAPUNOV FUNCTIONAL RELATING SHORTENING CURVES AND VISCOUS CONSERVATION LAWS, Nonlinear Analysis: Theory, Methods & Applications 51(4) (2002), pag. 649-662
A very simple presentation of the Glimm-Liu interaction functional for scalar conservation laws with viscosity.

7. S. Bianchini, R.M. Colombo, ON THE STABILITY OF THE STANDARD RIEMANN SEMIGROUP, Proc. AMS 130 (2002), pag. 1961-1973.

8. S. Bianchini, A NOTE ON SINGULAR LIMITS TO HYPERBOLIC SYSTEMS , Comm. Pure Appl. Anal. 2-1 (2003), pag. 51-64.

9. S. Bianchini, A. Bressan, A CASE STUDY IN VANISHING VISCOSITY, Discr. Cont. Dyn. Systems 7-3 (2001), pag. 449-476.

10. S. Bianchini, A. Bressan, A CENTER MANIFOLD TECHNIQUE FOR TRACING VISCOUS WAVES, Comm. Pure Appl. Anal. 1-2 (2002), pag. 161-190.
The idea of the center manifold of traveling profiles in the most simple case.

11. 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.

12. S. Bianchini, A NOTE ON THE RIEMANN PROBLEM FOR NONCONSERVATIVE HYPERBOLIC SYSTEMS , Archive for Rational Mechanics and Analysis 166 (2003), pag. 1-26.
Study of the compatibility of the solution of the Riemann problem w.r.t. various singular approximations: viscosity, relaxation, numerical schemes.

13. S. Bianchini, BV SOLUTIONS OF THE SEMIDISCRETE UPWIND SCHEME, Archive for Rational Mechanics and Analysis 167 (2003), pag. 1-81.
The convergence of the semidiscrete scheme to the solution of hyperbolic conservation laws in one space dimension.

14. S. Bianchini, INTERACTION ESTIMATES AND GLIMM FUNCTIONAL FOR GENERAL HYPERBOLIC SYSTEMS, Discrete and Continuous Dynamical Systems 9 (2003), pag. 133-166.
Extension of classical interaction estimates to general quasilinear hyperbolic systems.

15. S. Bianchini, RELAXATION LIMIT OF THE JIN-XIN RELAXATION MODEL, Comm. Pure Appl. Math. 59-5 (2006), pages 688-753.
Convergence of the Jin-Xin relaxation scheme to the solution of hyperbolic conservation laws in one space dimension.

16. S. Bianchini, GLIMM INTERACTION FUNCTIONAL FOR BGK SCHEMES, in "Hyperbolic Problems: Theory, Numerics and Applications - I", F. Asakura, S. Kawashima, A Matsumura, S. Nishibata, K. Nishihara editors.

17. S. Bianchini, B. Hanouzet, R. Natalini, ASYMPTOTIC BEHAVIOR OF SMOOTH SOLUTIONS FOR PARTIALLY DISSIPATIVE HYPERBOLIC SYSTEMS WITH A CONVEX ENTROPY, Comm. Pure Appl. Math. 60 (2007), p. 1559-1622.
Classical results on stability of constant states, this time for systems of balance laws.

18. S. Bianchini, L. V. Spinolo, THE BOUNDARY RIEMANN SOLVER COMING FROM THE REAL VANISHING VISCOSITY APPROXIMATION, Arch. Ration. Mech. Anal. 191-1 (2009), p. 1-96.
Construction of self similar solutions to hyperbolic systems with boundary, respecting the compatibility with the viscous approximation.

19. S. Bianchini, R.M. Colombo, F. Monti, 2x2 SYSTEMS OF CONSERVATION LAWS WITH L^infty DATA, JDE 249 (2010), p. 3466-3488.
A simplification and slight extension of the result of J. Glimm & P. D. Lax, Decay of solutions of systems of nonlinear hyperbolic conservation laws, Memoirs AMS 101 (1970).

20. S. Bianchini, L. V. Spinolo, INVARIANT MANIFOLDS FOR A SINGULAR ORDINARY DIFFERENTIAL EQUATION, J. Differential Equations. 250-4 (2011), p. 1788-1827.
Variation of standard singular ODE analysis, suitable for application to characteristic boundaries to hyperbolic conservation laws.

21. S. Bianchini, L. V. Spinolo, A CONNECTION BETWEEN VISCOUS PROFILES AND SINGULAR ODEs, Rend. Istit. Mat. Univ. Trieste 41 (2009), p. 35-41.

22. S. Bianchini, SBV REGULARITY OF SYSTEMS OF CONSERVATION LAWS AND HAMILTON-JACOBI EQUATION, In: 13th conference on Hyperbolic Problems: Theory, Numerics and Applications, Beijing (2010).

23. 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.

24. S. Bianchini, L. Yu, GLOBAL STRUCTURE OF SOLUTIONS TO PIECEWISE GENUINELY NONLINEAR HYPERBOLIC CONSERVATION LAWS IN ONE SPACE DIMENSION , Comm. PDE 39, p. 244-273.
Study of the fine structure of BV solutions.

25. S. Bianchini, S. Modena, ON A QUADRATIC FUNCTIONAL FOR SCALAR CONSERVATION LAWS , JHDE 11 (2), p. 355-435.
Extension of the original Glimm functional to general flux for scalar equations.

26. S. Bianchini, S. Modena, QUADRATIC INTERACTION FUNCTIONAL FOR SYSTEMS OF CONSERVATION LAWS: A CASE STUDY , Bullettin of the Institute of Mathematics, Academia Sinica Taipei 9, p. 487-546.
Extension of the quadratic interaction functional to a special system.

27. S. Bianchini, L. Yu, STRUCTURE OF ENTROPY SOLUTIONS TO GENERAL SCALAR CONSERVATION LAWS IN ONE SPACE DIMENSION , Journal of Mathematical Analysis and Applications 428, p. 356-386.
Method of characteristics for BV solutions to general scalar conservation laws.

28. 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.

29. G. Alberti, S. Bianchini, L. Caravenna, EULERIAN, LAGRANGIAN AND BROAD CONTINUOUS SOLUTIONS TO A BALANCE LAW WITH NON CONVEX FLUX I , J. Differential Equations 261 (8), p. 4298-4337.

30. S. Bianchini, E. Marconi, ON THE CONCENTRATION OF ENTROPY FOR SCALAR CONSERVATION LAWS , Discrete Contin. Dyn. Syst. Ser. S 9, p. 73-88.
A first attempt to the entropy dissipation structure in a special case.

31. S. Modena, S. Bianchini, CONVERGENCE RATE OF THE GLIMM SCHEME , Bulletin of the Institute of Mathematics of Academia Sinica (New Series) 11 (1), p. 235-300.
This paper fixes an open problem on the rate of convergence of the Glimm scheme.

32. S. Bianchini, L.V. Spinolo AN OVERVIEW ON THE APPROXIMATION OF BOUNDARY RIEMANN PROBLEMS THROUGH PHYSICAL VISCOSITY , Bull. Braz. Math. Soc. 47, p. 131-142.

33. S. Bianchini, M. Colombo, G. Crippa, L.V. Spinolo, OPTIMALITY OF INTEGRABILITY ESTIMATES FOR ADVECTION-DIFFUSION EQUATIONS , NODEA 24, p. 1-19.

34. 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.

## Calculus of Variations

This is the solution to an open problem proposed by Simone Bertone & Arrigo Cellina in On the Existence of Variations, possibly with Pointwise Gradient Constraints, COCV 13-2 (2007), 331-342.
Collaborator: Matteo Gloyer.

1. S. Bianchini, ON THE EULER LAGRANGE EQUATION FOR A VARIATIONAL PROBLEM, Discrete and Continuous Dynamical Systems 17 (2007), pag. 449-480.

2. S. Bianchini, M. Gloyer, ON THE EULER-LAGRANGE EQUATION FOR A VARIATIONAL PROBLEM: THE GENERAL CASE II, Math. Zeit. 265-4 (2009), p. 889-923.

## Optimal Transportation

I worked on the optimality conditions for transference plans, and on the solution to the Monge problem with norm cost.
The approach I like is from the point of view of advanced measure theory.
Collaborators: Mauro Bardelloni, Alessio Brancolini, Laura Caravenna, Fabio Cavalletti, Sara Daneri.

1. 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.

2. S. Bianchini, L. Caravenna, ON THE OPTIMALITY OF TRANSFERENCE PLANS, C. R. Acad. Sci. Paris, Ser. I 348 (2010), p. 613-618.

3. S. Bianchini, A. Brancolini, ESTIMATES ON PATH FUNCTIONALS OVER WASSERSTEIN SPACES, SIAM Math. Anal. 42 (2010), p. 1179-1217
Study of path functionals introduced in A. Brancolini, G. Buttazzo & F. Santambrogio Path functionals over Wasserstein spaces.

4. S. Bianchini, F. Cavalletti, THE MONGE PROBLEM IN GEODESIC SPACES, in "IMA Summer program: nonlinear conservation laws and applications" (2010).

5. S. Bianchini, F. Cavalletti, THE MONGE PROBLEM IN METRIC SPACES , Comm. Math. Phys. 318 (2013), 615-673.
General approach to Monge problem in non branching metric spaces.

6. S. Bianchini, A. Dabrowski, EXISTENCE AND UNIQUENESS OF THE GRADIENT FLOW OF THE ENTROPY IN THE SPACE OF PROBABILITY MEASURES , Rendiconti dell'Istituto di Matematica dell'Universita' di Trieste 46, 1-28.
A short review of the results of Ambrosio-Gigli on gradient flows.

7. 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.

8. S. Bianchini, M. Bardelloni, THE DECOMPOSITION OF OPTIMAL TRANSPORTATION PROBLEMS WITH CONVEX COST , Bulletin of Institute of Mathematics, Academia Sinica. New Series 11, 401-484.
The solution of the Monge transportation problem with convex cost.

## Linear Transport

These are some miscellaneus papers on various unrelated problems. More recently I addressed Bressan's compactness conjecture.
Collaborators: Giovanni Alberti, Paolo Bonicatto, Gianluca Crippa, Matteo Gloyer, Nikolay Gusev, Daniela Tonon.

1. S. Bianchini, ON BRESSAN CONJECTURE ON MIXING PROPERTIES OF VECTOR FIELDS, Banach Center Publications 74 (2006), 13-31.
The proof of the following Bressan Conjecture in the one dimensional case for measure preserving BV maps.

2. S. Bianchini, M. Gloyer, AN ESTIMATE ON THE FLOW GENERATED BY MONOTONE OPERATORS. Comm. PDE 36-5 (2010), p. 777-796.
A stability result for flows generated by monotone operators.

3. S. Bianchini, D. Tonon, A DECOMPOSITION THEOREM FOR BV FUNCTIONS, CPAA 10, p. 1549-1566.
Another definition of monotone functions and related decomposition.

4. 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.

5. G. Alberti, S. Bianchini, G. Crippa, STRUCTURE OF LEVEL SETS AND SARD-TYPE PROPERTIES OF LIPSCHITZ MAPS, Annali SNS 12, p. 863-902.

6. G. Alberti, S. Bianchini, G. Crippa, ON THE LP-DIFFERENTIABILITY OF CERTAIN CLASSES OF FUNCTIONS , Revista Matematica Iberoamericana 30, p. 349-367.

7. S. Bianchini, N.A. Gusev, STEADY NEARLY INCOMPRESSIBLE VECTOR FIELDS IN 2D: CHAIN RULE AND RENORMALIZATION, Archive for Rational Mechanics and Analysis 222, p. 451-505.
In this paper a fundamental example of tangent vector fields is constructed, together with the proof of renormalization.

8. S. Bianchini, P. Bonicatto, N.A. Gusev, RENORMALIZATION FOR AUTONOMOUS NEARLY INCOMPRESSIBLE BV VECTOR FIELDS IN 2D , SIAM J. Math. Anal. 48 (1), p. 1-33.
Bressan renormalization conjecture in 2d.

## Hamilton-Jacobi Equations

The topic of my research is the SBV regularity of the solution to HJ equations under various convexity assumptions on the Hamiltonian.
Collaborators: Camillo De Lellis, Matteo Gloyer, Roger Robyr, Daniela Tonon.

1. S. Bianchini, C. De Lellis, R. Robyr, SBV REGULARITY FOR HAMILTON-JACOBI EQUATIONS IN R^n, Arch. Ration. Mech. Anal. 200 (2011), p. 1003-1021.
Proof of SBV regularity for HJ equation with uniformly convex Hamiltonian.

2. S. Bianchini, M. Gloyer, TRANSPORT RAYS AND APPLICATIONS TO HAMILTON-JACOBI EQUATIONS, Lecture Notes in Mathematics 2028 (2011), p. 1-16.

3. S. Bianchini, D. Tonon, SBV-LIKE REGULARITY FOR HAMILTON-JACOBI EQUATIONS WITH A CONVEX HAMILTONIAN , J. Math. Anal. Appl. 391 (2012), p. 190-208.

4. S. Bianchini, D. Tonon, SBV REGULARITY FOR HAMILTON-JACOBI EQUATIONS WITH HAMILTONIAN DEPENDING ON (t, x) , SIAM Math. Anal. 44 (2012), p. 2179-2203.

## Some work as an editor

Part 1 of the lecture notes of the intensive trimester

### Nonlinear Hyperbolic PDEs, Dispersive and Transport Equations

Giovanni Alberti, Fabio Ancona, Stefano Bianchini, Gianluca Crippa, Camillo De Lellis, Andrea Marson, Corrado Mascia (Eds.)