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 $f''=0$. 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.

PB - SISSA UR - http://urania.sissa.it/xmlui/handle/1963/35209 U1 - 35508 U2 - Mathematics U5 - MAT/05 ER - TY - JOUR T1 - SBV Regularity of Systems of Conservation Laws and Hamiltonâ€“Jacobi Equations Y1 - 2014 A1 - Stefano Bianchini AB - 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. PB - Springer UR - http://urania.sissa.it/xmlui/handle/1963/34691 U1 - 34904 U2 - Mathematics U4 - 1 ER - TY - RPRT T1 - Steady nearly incompressible vector elds in 2D: chain rule and renormalization Y1 - 2014 A1 - Stefano Bianchini A1 - N.A. Gusev PB - SISSA U1 - 7464 U2 - Mathematics U4 - -1 ER - TY - JOUR T1 - Structure of entropy solutions to general scalar conservation laws in one space dimension JF - Journal of Mathematical Analysis and Applications Y1 - 2014 A1 - Stefano Bianchini A1 - Lei Yu PB - SISSA VL - 428 UR - https://www.sciencedirect.com/science/article/pii/S0022247X15002218 IS - 1 U1 - 7305 U2 - Mathematics U4 - -1 ER - TY - RPRT T1 - On Sudakov's type decomposition of transference plans with norm costs Y1 - 2013 A1 - Stefano Bianchini A1 - Sara Daneri PB - SISSA UR - http://hdl.handle.net/1963/7206 U1 - 7234 U2 - Mathematics U4 - -1 ER - TY - JOUR T1 - SBV regularity for genuinely nonlinear, strictly hyperbolic systems of conservation laws in one space dimension JF - Communications in Mathematical Physics 313 (2012) 1-33 Y1 - 2012 A1 - Stefano Bianchini A1 - Laura Caravenna PB - Springer UR - http://hdl.handle.net/1963/4091 U1 - 313 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - SBV regularity for Hamilton-Jacobi equations with Hamiltonian depending on (t,x) JF - Siam Journal on Mathematical Analysis Y1 - 2012 A1 - Stefano Bianchini A1 - Daniela Tonon PB - SISSA VL - 44 UR - http://hdl.handle.net/20.500.11767/14066 IS - 3 U1 - 3890 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - SBV regularity of genuinely nonlinear hyperbolic systems of conservation laws in one space dimension JF - Acta Mathematica Scientia, Volume 32, Issue 1, January 2012, Pages 380-388 Y1 - 2012 A1 - Stefano Bianchini KW - Hyperbolic systems AB - 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 PB - Elsevier UR - http://hdl.handle.net/1963/6535 U1 - 6510 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - SBV-like regularity for general hyperbolic systems of conservation laws in one space dimension JF - Rend. Istit. Mat. Univ. Trieste Y1 - 2012 A1 - Stefano Bianchini A1 - Lei Yu VL - 44 ER - TY - JOUR T1 - SBV-like regularity for Hamilton-Jacobi equations with a convex Hamiltonian JF - Journal of Mathematical Analysis and Applications Y1 - 2012 A1 - Stefano Bianchini A1 - Daniela Tonon PB - SISSA VL - 391 UR - http://hdl.handle.net/20.500.11767/13909 IS - 1 U1 - 4352 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - SBV regularity for Hamilton-Jacobi equations in R^n JF - Arch. Rational Mech. Anal. 200 (2011) 1003-1021 Y1 - 2011 A1 - Stefano Bianchini A1 - Camillo De Lellis A1 - Roger Robyr AB -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 $SBV_{loc}(\Omega)$.

PB - Springer UR - http://hdl.handle.net/1963/4911 U1 - 4695 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - RPRT T1 - Structure of level sets and Sard-type properties of Lipschitz maps Y1 - 2011 A1 - Giovanni Alberti A1 - Stefano Bianchini A1 - Gianluca Crippa PB - SISSA UR - http://hdl.handle.net/1963/4657 U1 - 4424 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - On the Stability of the Standard Riemann Semigroup JF - P. Am. Math. Soc., 2002, 130, 1961 Y1 - 2002 A1 - Stefano Bianchini A1 - Rinaldo M. Colombo PB - SISSA Library UR - http://hdl.handle.net/1963/1528 U1 - 2635 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Stability of L-infinity solutions for hyperbolic systems with coinciding shocks and rarefactions JF - Siam J. Math. Anal., 2001, 33, 959 Y1 - 2001 A1 - Stefano Bianchini AB - 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. PB - SISSA Library UR - http://hdl.handle.net/1963/1523 U1 - 2640 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - The semigroup generated by a Temple class system with non-convex flux function JF - Differential Integral Equations 13 (2000) 1529-1550 Y1 - 2000 A1 - Stefano Bianchini AB - We consider the Cauchy problem for a nonlinear n Ă— 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. PB - Khayyam Publishing UR - http://hdl.handle.net/1963/3221 U1 - 1080 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - On the shift differentiability of the flow generated by a hyperbolic system of conservation laws JF - Discrete Contin. Dynam. Systems 6 (2000), no. 2, 329-350 Y1 - 2000 A1 - Stefano Bianchini PB - American Institute of Mathematical Sciences UR - http://hdl.handle.net/1963/1274 U1 - 3181 U2 - Mathematics U3 - Functional Analysis and Applications ER -