We prove that if $u$ is the entropy solution to a scalar conservation law in one space dimension, then the entropy dissipation is a measure concentrated on countably many Lipschitz curves. This result is a consequence of a detailed analysis of the structure of the characteristics. In particular the characteristic curves are segments outside a countably 1-rectifiable set and the left and right traces of the solution exist in a $C^0$-sense up to the degeneracy due to the segments where $f{\textquoteright}{\textquoteright}=0$. We prove also that the initial data is taken in a suitably strong sense and we give some counterexamples which show that these results are sharp.

}, url = {http://urania.sissa.it/xmlui/handle/1963/35209}, author = {Stefano Bianchini and Elio Marconi} } @article {2014, title = {SBV Regularity of Systems of Conservation Laws and Hamilton{\textendash}Jacobi Equations}, number = {Journal of Mathematical Sciences;Volume 201; issue 6; pp. 733-745;}, year = {2014}, publisher = {Springer}, abstract = {We review the SBV regularity for solutions to hyperbolic systems of conservation laws and Hamilton-Jacobi equations. We give an overview of the techniques involved in the proof, and a collection of related problems concludes the paper.}, doi = {10.1007/s10958-014-2022-9}, url = {http://urania.sissa.it/xmlui/handle/1963/34691}, author = {Stefano Bianchini} } @article {2014, title = {Steady nearly incompressible vector elds in 2D: chain rule and renormalization}, year = {2014}, institution = {SISSA}, author = {Stefano Bianchini and N.A. Gusev} } @article {2014, title = {Structure of entropy solutions to general scalar conservation laws in one space dimension}, journal = {Journal of Mathematical Analysis and Applications}, volume = {428}, number = {SISSA;11/2014/MATE}, year = {2014}, month = {08/2015}, pages = {356-386}, publisher = {SISSA}, chapter = {356}, doi = {https://doi.org/10.1016/j.jmaa.2015.03.006}, url = {https://www.sciencedirect.com/science/article/pii/S0022247X15002218}, author = {Stefano Bianchini and Lei Yu} } @article {2013, title = {On Sudakov{\textquoteright}s type decomposition of transference plans with norm costs}, number = {SISSA;51/2013/MATE}, year = {2013}, institution = {SISSA}, url = {http://hdl.handle.net/1963/7206}, author = {Stefano Bianchini and Sara Daneri} } @article {2012, title = {SBV regularity for genuinely nonlinear, strictly hyperbolic systems of conservation laws in one space dimension}, journal = {Communications in Mathematical Physics 313 (2012) 1-33}, number = {SISSA;71/2010/M}, year = {2012}, publisher = {Springer}, doi = {10.1007/s00220-012-1480-5}, url = {http://hdl.handle.net/1963/4091}, author = {Stefano Bianchini and Laura Caravenna} } @article {2011, title = {SBV regularity for Hamilton-Jacobi equations with Hamiltonian depending on (t,x)}, journal = {Siam Journal on Mathematical Analysis}, volume = {44}, number = {SISSA;13/2011/M}, year = {2012}, pages = {2179-2203}, publisher = {SISSA}, doi = {10.1137/110827272}, url = {http://hdl.handle.net/20.500.11767/14066}, author = {Stefano Bianchini and Daniela Tonon} } @article {2012, title = {SBV regularity of genuinely nonlinear hyperbolic systems of conservation laws in one space dimension}, journal = {Acta Mathematica Scientia, Volume 32, Issue 1, January 2012, Pages 380-388}, year = {2012}, publisher = {Elsevier}, abstract = {The problem of the presence of Cantor part in the derivative of a solution to a hyperbolic system of conservation laws is considered. An overview of the techniques involved in the proof is given, and a collection of related problems concludes the paper. Key words hyperbolic systems; conservation laws; SBV; regularity}, keywords = {Hyperbolic systems}, doi = {10.1016/S0252-9602(12)60024-1}, url = {http://hdl.handle.net/1963/6535}, author = {Stefano Bianchini} } @article {bianchini2012sbv, title = {SBV-like regularity for general hyperbolic systems of conservation laws in one space dimension}, journal = {Rend. Istit. Mat. Univ. Trieste}, volume = {44}, year = {2012}, pages = {439{\textendash}472}, author = {Stefano Bianchini and Lei Yu} } @article {2012, title = {SBV-like regularity for Hamilton-Jacobi equations with a convex Hamiltonian}, journal = {Journal of Mathematical Analysis and Applications}, volume = {391}, number = {SISSA;45/2011/M}, year = {2012}, pages = {190-208}, publisher = {SISSA}, doi = {10.1016/j.jmaa.2012.02.017}, url = {http://hdl.handle.net/20.500.11767/13909}, author = {Stefano Bianchini and Daniela Tonon} } @article {2011, title = {SBV regularity for Hamilton-Jacobi equations in R^n}, journal = {Arch. Rational Mech. Anal. 200 (2011) 1003-1021}, number = {arXiv:1002.4087;}, year = {2011}, publisher = {Springer}, abstract = {In this paper we study the regularity of viscosity solutions to the following Hamilton-Jacobi equations $$ \partial_t u + H(D_{x} u)=0 \qquad \textrm{in}\quad \Omega\subset \mathbb{R}\times \mathbb{R}^{n} . $$ In particular, under the assumption that the Hamiltonian $H\in C^2(\mathbb{R}^n)$ is uniformly convex, we prove that $D_{x}u$ and $\partial_t u$ belong to the class $SBV_{loc}(\Omega)$.

}, doi = {10.1007/s00205-010-0381-z}, url = {http://hdl.handle.net/1963/4911}, author = {Stefano Bianchini and Camillo De Lellis and Roger Robyr} } @article {2011, title = {Structure of level sets and Sard-type properties of Lipschitz maps}, number = {SISSA;51/2011/M}, year = {2011}, institution = {SISSA}, url = {http://hdl.handle.net/1963/4657}, author = {Giovanni Alberti and Stefano Bianchini and Gianluca Crippa} } @article {2002, title = {On the Stability of the Standard Riemann Semigroup}, journal = {P. Am. Math. Soc., 2002, 130, 1961}, number = {SISSA;71/00/M}, year = {2002}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1528}, author = {Stefano Bianchini and Rinaldo M. Colombo} } @article {2001, title = {Stability of L-infinity solutions for hyperbolic systems with coinciding shocks and rarefactions}, journal = {Siam J. Math. Anal., 2001, 33, 959}, number = {SISSA;65/00/M}, year = {2001}, publisher = {SISSA Library}, abstract = {We consider a hyperbolic system of conservation laws u_t + f(u)_x = 0 and u(0,\\\\cdot) = u_0, where each characteristic field is either linearly degenerate or genuinely nonlinear. Under the assumption of coinciding shock and rarefaction curves and the existence of a set of Riemann coordinates $w$, we prove that there exists a semigroup of solutions $u(t) = \\\\mathcal{S}_t u_0$, defined on initial data $u_0 \\\\in L^\\\\infty$. The semigroup $\\\\mathcal{S}$ is continuous w.r.t. time and the initial data $u_0$ in the $L^1_{\\\\text{loc}}$ topology. Moreover $\\\\mathcal{S}$ is unique and its trajectories are obtained as limits of wave front tracking approximations.}, doi = {10.1137/S0036141000377900}, url = {http://hdl.handle.net/1963/1523}, author = {Stefano Bianchini} } @article {2000, title = {The semigroup generated by a Temple class system with non-convex flux function}, journal = {Differential Integral Equations 13 (2000) 1529-1550}, number = {SISSA;107/98/M}, year = {2000}, publisher = {Khayyam Publishing}, abstract = {We consider the Cauchy problem for a nonlinear n {\texttimes} n system of conservation laws of Temple class, i.e. with coinciding shock and rarefaction curves and with a coordinate system made of Riemann invariants. Without any assumption on the convexity of the flux function, we prove the existence of a semigroup made of weak solutions of the equations and depending Lipschitz continuously on the initial data with bounded total variation.}, url = {http://hdl.handle.net/1963/3221}, author = {Stefano Bianchini} } @article {2000, title = {On the shift differentiability of the flow generated by a hyperbolic system of conservation laws}, journal = {Discrete Contin. Dynam. Systems 6 (2000), no. 2, 329-350}, number = {SISSA;60/99/M}, year = {2000}, publisher = {American Institute of Mathematical Sciences}, doi = {10.3934/dcds.2000.6.329}, url = {http://hdl.handle.net/1963/1274}, author = {Stefano Bianchini} }