In this note we present a unifying approach for two classes of first order partial differential equations: we introduce the notion of Lagrangian representation in the settings of continuity equation and scalar conservation laws. This yields, on the one hand, the uniqueness of weak solutions to transport equation driven by a two dimensional BV nearly incompressible vector field. On the other hand, it is proved that the entropy dissipation measure for scalar conservation laws in one space dimension is concentrated on countably many Lipschitz curves.

PB - Peoples' Friendship University of Russia VL - 63 UR - http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=cmfd&paperid=327&option_lang=eng ER - TY - RPRT T1 - A uniqueness result for the decomposition of vector fields in Rd Y1 - 2017 A1 - Stefano Bianchini A1 - Paolo Bonicatto AB -Given a vector field $\rho (1,\b) \in L^1_\loc(\R^+\times \R^{d},\R^{d+1})$ such that $\dive_{t,x} (\rho (1,\b))$ is a measure, we consider the problem of uniqueness of the representation $\eta$ of $\rho (1,\b) \mathcal L^{d+1}$ as a superposition of characteristics $\gamma : (t^-_\gamma,t^+_\gamma) \to \R^d$, $\dot \gamma (t)= \b(t,\gamma(t))$. We give conditions in terms of a local structure of the representation $\eta$ on suitable sets in order to prove that there is a partition of $\R^{d+1}$ into disjoint trajectories $\wp_\a$, $\a \in \A$, such that the PDE \begin{equation*} \dive_{t,x} \big( u \rho (1,\b) \big) \in \mathcal M(\R^{d+1}), \qquad u \in L^\infty(\R^+\times \R^{d}), \end{equation*} can be disintegrated into a family of ODEs along $\wp_\a$ with measure r.h.s.. The decomposition $\wp_\a$ is essentially unique. We finally show that $\b \in L^1_t(\BV_x)_\loc$ satisfies this local structural assumption and this yields, in particular, the renormalization property for nearly incompressible $\BV$ vector fields.

PB - SISSA UR - http://preprints.sissa.it/handle/1963/35274 U1 - 35581 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - On the concentration of entropy for scalar conservation laws JF - Discrete & Continuous Dynamical Systems - S Y1 - 2016 A1 - Stefano Bianchini A1 - Elio Marconi KW - concentration KW - Conservation laws KW - entropy solutions KW - Lagrangian representation KW - shocks AB -We prove that the entropy for an $L^∞$-solution to a scalar conservation laws with continuous initial data is concentrated on a countably $1$-rectifiable set. To prove this result we introduce the notion of Lagrangian representation of the solution and give regularity estimates on the solution.

VL - 9 UR - http://aimsciences.org//article/id/ce4eb91e-9553-4e8d-8c4c-868f07a315ae ER - TY - JOUR T1 - Eulerian, Lagrangian and Broad continuous solutions to a balance law with non-convex flux I JF - Journal of Differential Equations, vol. 261, issue 8 (2016): 4298-4337 Y1 - 2016 A1 - Giovanni Alberti A1 - Stefano Bianchini A1 - Laura Caravenna PB - Elsevier UR - http://urania.sissa.it/xmlui/handle/1963/35207 U1 - 35507 U2 - Mathematics ER - TY - RPRT T1 - Eulerian, Lagrangian and Broad continuous solutions to a balance law with non convex flux II Y1 - 2016 A1 - Giovanni Alberti A1 - Stefano Bianchini A1 - Laura Caravenna UR - http://urania.sissa.it/xmlui/handle/1963/35197 U1 - 35494 U2 - Mathematics ER - TY - JOUR T1 - Renormalization for Autonomous Nearly Incompressible BV Vector Fields in Two Dimensions JF - SIAM Journal on Mathematical Analysis Y1 - 2016 A1 - Stefano Bianchini A1 - Paolo Bonicatto A1 - N.A. Gusev AB -Given a bounded autonomous vector field $b \colon \mathbb{R}^d \to \mathbb{R}^d$, we study the uniqueness of bounded solutions to the initial value problem for the related transport equation \begin{equation*} \partial_t u + b \cdot \nabla u= 0. \end{equation*} We are interested in the case where $b$ is of class BV and it is nearly incompressible. Assuming that the ambient space has dimension $d=2$, we prove uniqueness of weak solutions to the transport equation. The starting point of the present work is the result which has been obtained in [7] (where the steady case is treated). Our proof is based on splitting the equation onto a suitable partition of the plane: this technique was introduced in [3], using the results on the structure of level sets of Lipschitz maps obtained in [1]. Furthermore, in order to construct the partition, we use Ambrosio's superposition principle [4].

VL - 48 UR - https://doi.org/10.1137/15M1007380 ER - TY - RPRT T1 - On the structure of $L^\infty$-entropy solutions to scalar conservation laws in one-space dimension Y1 - 2016 A1 - Stefano Bianchini A1 - Elio Marconi AB -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 - Convergence rate of the Glimm scheme JF - Bulletin of the Institute of Mathematics of Academia Sinica (New Series) Y1 - 2015 A1 - Stefano Modena A1 - Stefano Bianchini ER - TY - JOUR T1 - Quadratic Interaction Functional for General Systems of Conservation Laws JF - Communications in Mathematical Physics Y1 - 2015 A1 - Stefano Bianchini A1 - Stefano Modena AB -For the Glimm scheme approximation to the solution of the system of conservation laws in one space dimension with initial data u 0 with small total variation, we prove a quadratic (w.r.t. Tot. Var. ( u 0)) interaction estimate, which has been used in the literature for stability and convergence results. No assumptions on the structure of the flux f are made (apart from smoothness), and this estimate is the natural extension of the Glimm type interaction estimate for genuinely nonlinear systems. More precisely, we obtain the following results: a new analysis of the interaction estimates of simple waves;

VL - 338 ER - TY - RPRT T1 - The decomposition of optimal transportation problems with convex cost Y1 - 2014 A1 - Stefano Bianchini A1 - Mauro Bardelloni PB - SISSA UR - http://hdl.handle.net/1963/7433 U1 - 7527 ER - TY - JOUR T1 - Existence and uniqueness of the gradient flow of the Entropy in the space of probability measures Y1 - 2014 A1 - Stefano Bianchini A1 - Alexander Dabrowski AB - After a brief introduction on gradient flows in metric spaces and on geodesically convex functionals, we give an account of the proof (following the outline of [3, 7]) of the existence and uniqueness of the gradient flow of the Entropy in the space of Borel probability measures over a compact geodesic metric space with Ricci curvature bounded from below. PB - EUT Edizioni Universita di Trieste UR - http://urania.sissa.it/xmlui/handle/1963/34693 N1 - This paper resumes the main part of the Bachelor thesis of the second author, discussed in 2013 at the University of Trieste. U1 - 34907 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Global Structure of Admissible BV Solutions to Piecewise Genuinely Nonlinear, Strictly Hyperbolic Conservation Laws in One Space Dimension Y1 - 2014 A1 - Stefano Bianchini A1 - Lei Yu AB -The paper describes the qualitative structure of an admissible BV solution to a strictly hyperbolic system of conservation laws whose characteristic families are piecewise genuinely nonlinear. More precisely, we prove that there are a countable set of points Θ and a countable family of Lipschitz curves T{script} such that outside T{script} ∪ Θ the solution is continuous, and for all points in T{script}{set minus}Θ the solution has left and right limit. This extends the corresponding structural result in [7] for genuinely nonlinear systems. An application of this result is the stability of the wave structure of solution w.r.t. -convergence. The proof is based on the introduction of subdiscontinuities of a shock, whose behavior is qualitatively analogous to the discontinuities of the solution to genuinely nonlinear systems.

PB - Taylor & Francis UR - http://urania.sissa.it/xmlui/handle/1963/34694 U1 - 34908 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - JOUR T1 - On the Lp-differentiability of certain classes of functions Y1 - 2014 A1 - Giovanni Alberti A1 - Stefano Bianchini A1 - Gianluca Crippa AB - We prove the Lp-differentiability at almost every point for convolution products on ℝd of the form K*μ, where μ is bounded measure and K is a homogeneous kernel of degree 1-d. From this result we derive the Lp-differentiability for vector fields on R d whose curl and divergence are measures, and also for vector fields with bounded deformation. PB - European Mathematical Society UR - http://urania.sissa.it/xmlui/handle/1963/34695 U1 - 34909 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - JOUR T1 - On a quadratic functional for scalar conservation laws JF - Journal of Hyperbolic Differential Equations Y1 - 2014 A1 - Stefano Bianchini A1 - Stefano Modena AB -We prove a quadratic interaction estimate for approximate solutions to scalar conservation laws obtained by the wavefront tracking approximation or the Glimm scheme. This quadratic estimate has been used in the literature to prove the convergence rate of the Glimm scheme.

PB - World Scientific Publishing VL - 11 UR - http://arxiv.org/abs/1311.2929 IS - 2 U1 - 34903 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Quadratic interaction functional for systems of conservation laws: a case study JF - Bulletin of the Institute of Mathematics of Academia Sinica (New Series) Y1 - 2014 A1 - Stefano Bianchini A1 - Stefano Modena VL - 9 UR - https://w3.math.sinica.edu.tw/bulletin_ns/20143/2014308.pdf ER - TY - CHAP T1 - Reduction on characteristics for continuous of a scalar balance law T2 - AIMS Series on Applied Mathematics, vol. 8 (2014): 399 - 406 Y1 - 2014 A1 - Giovanni Alberti A1 - Stefano Bianchini A1 - Laura Caravenna KW - Method of characteristics JF - AIMS Series on Applied Mathematics, vol. 8 (2014): 399 - 406 PB - SISSA UR - http://hdl.handle.net/1963/6562 U1 - 6516 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - SBV Regularity of Systems of Conservation Laws and Hamilton–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 - JOUR T1 - A uniqueness result for the continuity equation in two dimensions: dedicated to constantine dafermos on the occasion of his 70th birthday Y1 - 2014 A1 - Giovanni Alberti A1 - Stefano Bianchini A1 - Gianluca Crippa AB - We characterize the autonomous, divergence-free vector fields b on the plane such that the Cauchy problem for the continuity equation ∂tu +div(bu) = 0 admits a unique bounded solution (in the weak sense) for every bounded initial datum; the characterization is given in terms of a property of Sard type for the potential f associated to b. As a corollary we obtain uniqueness under the assumption that the curl of b is a measure. This result can be extended to certain nonautonomous vector fields b with bounded divergence. PB - European Mathematical Society; Springer Verlag UR - http://urania.sissa.it/xmlui/handle/1963/34692 U1 - 34906 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - The Monge Problem for Distance Cost in Geodesic Spaces JF - Communications in Mathematical Physics Y1 - 2013 A1 - Stefano Bianchini A1 - Fabio Cavalletti AB -We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and dLis a geodesic Borel distance which makes (X, dL) a non branching geodesic space. We show that under the assumption that geodesics are d-continuous and locally compact, we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce two assumptions on the transport problem π which imply that the conditional probabilities of the first marginal on each geodesic are continuous or absolutely continuous w.r.t. the 1-dimensional Hausdorff distance induced by dL. It is known that this regularity is sufficient for the construction of a transport map. We study also the dynamics of transport along the geodesic, the stability of our conditions and show that in this setting dL-cyclical monotonicity is not sufficient for optimality.

VL - 318 UR - https://doi.org/10.1007/s00220-013-1663-8 ER - TY - JOUR T1 - A New Quadratic Potential for Scalar Conservation Laws JF - Oberwolfach Reports Y1 - 2013 A1 - Stefano Bianchini A1 - Stefano Modena VL - 29 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 - A Decomposition Theorem for BV functions JF - Communications on Pure and Applied Analysis Y1 - 2011 A1 - Stefano Bianchini A1 - Daniela Tonon PB - American Institute of Mathematical Sciences VL - 10 UR - http://hdl.handle.net/20.500.11767/14599 IS - 6 U1 - 693 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - An Estimate on the Flow Generated by Monotone Operators JF - Communications in Partial Differential Equations 36 (2011) 777-796 Y1 - 2011 A1 - Stefano Bianchini A1 - Matteo Gloyer PB - Taylor & Francis UR - http://hdl.handle.net/1963/3646 U1 - 658 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Invariant manifolds for a singular ordinary differential equation JF - Journal of Differential Equations 250 (2011) 1788-1827 Y1 - 2011 A1 - Stefano Bianchini A1 - Laura Spinolo PB - Elsevier UR - http://hdl.handle.net/1963/2554 U1 - 1565 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - CONF T1 - The Monge Problem in Geodesic Spaces T2 - Nonlinear Conservation Laws and Applications Y1 - 2011 A1 - Stefano Bianchini A1 - Fabio Cavalletti ED - Alberto Bressan ED - Chen, Gui-Qiang G. ED - Marta Lewicka ED - Wang, Dehua AB -We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish non branching geodesic space. We show that we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce an assumption on the transport problem π which implies that the conditional probabilities of the first marginal on each geodesic are continuous. It is known that this regularity is sufficient for the construction of an optimal transport map.

JF - Nonlinear Conservation Laws and Applications PB - Springer US CY - Boston, MA SN - 978-1-4419-9554-4 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 - RPRT T1 - A uniqueness result for the continuity equation in two dimensions Y1 - 2011 A1 - Giovanni Alberti A1 - Stefano Bianchini A1 - Gianluca Crippa PB - SISSA UR - http://hdl.handle.net/1963/4663 U1 - 4425 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - Estimates on path functionals over Wasserstein Spaces JF - SIAM J. Math. Anal. 42 (2010) 1179-1217 Y1 - 2010 A1 - Stefano Bianchini A1 - Alessio Brancolini AB - In this paper we consider the class a functionals (introduced in [Brancolini, Buttazzo, and Santambrogio, J. Eur. Math. Soc. (JEMS), 8 (2006), pp. 415-434] $\\\\mathcal{G}_{r,p}$ defined on Lipschitz curves $\\\\gamma$ valued in the $p$-Wasserstein space. The problem considered is the following: given a measure $\\\\mu$, give conditions in order to assure the existence of a curve $\\\\gamma$ such that $\\\\gamma(0)=\\\\mu$, $\\\\gamma(1)=\\\\delta_{x_0}$, and $\\\\mathcal{G}_{r,p}(\\\\gamma)<+\\\\infty$. To this end, new estimates on $\\\\mathcal{G}_{r,p}(\\\\mu)$ are given, and a notion of dimension of a measure (called path dimension) is introduced: the path dimension specifies the values of the parameters $(r,p)$ for which the answer to the previous reachability problem is positive. Finally, we compare the path dimension with other known dimensions. UR - http://hdl.handle.net/1963/3583 U1 - 717 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - On the Euler-Lagrange equation for a variational problem : the general case II JF - Math. Z. 265 (2010) 889-923 Y1 - 2010 A1 - Stefano Bianchini A1 - Matteo Gloyer UR - http://hdl.handle.net/1963/2551 U1 - 1568 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - On optimality of c-cyclically monotone transference plans JF - Comptes Rendus Mathematique 348 (2010) 613-618 Y1 - 2010 A1 - Stefano Bianchini A1 - Laura Caravenna AB - Abstract. This note deals with the equivalence between the optimality of a transport plan for the Monge-Kantorovich problem and the condition of c-cyclical monotonicity, as an outcome of the construction in [7]. We emphasize the measurability assumption on the hidden structure of linear preorder, applied also to extremality and uniqueness problems. Resume. Dans la presente note nous decrivons brievement la construction introduite dans [7] a propos de l\\\'equivalence entre l\\\'optimalite d\\\'un plan de transport pour le probleme de Monge-Kantorovich et la condition de monotonie c-cyclique ainsi que d\\\'autres sujets que cela nous amene a aborder. Nous souhaitons mettre en evidence l\\\'hypothese de mesurabilite sur la structure sous-jacente de pre-ordre lineaire. PB - Elsevier UR - http://hdl.handle.net/1963/4023 U1 - 379 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - The boundary Riemann solver coming from the real vanishing viscosity approximation JF - Arch. Ration. Mech. Anal. 191 (2009) 1-96 Y1 - 2009 A1 - Stefano Bianchini A1 - Laura Spinolo AB - We study the limit of the hyperbolic-parabolic approximation $$ \\\\begin{array}{lll} v_t + \\\\tilde{A} ( v, \\\\, \\\\varepsilon v_x ) v_x = \\\\varepsilon \\\\tilde{B}(v ) v_{xx} \\\\qquad v \\\\in R^N\\\\\\\\ \\\\tilde \\\\beta (v (t, \\\\, 0)) = \\\\bar g \\\\\\\\ v (0, \\\\, x) = \\\\bar v_0. \\\\\\\\ \\\\end{array} \\\\right. $$\\nThe function $\\\\tilde \\\\beta$ is defined in such a way to guarantee that the initial boundary value problem is well posed even if $\\\\tilde \\\\beta$ is not invertible.\\nThe data $\\\\bar g$ and $\\\\bar v_0$ are constant. When $\\\\tilde B$ is invertible, the previous problem takes the simpler form $$ \\\\left\\\\{ \\\\begin{array}{lll} v_t + \\\\tilde{A} \\\\big( v, \\\\, \\\\varepsilon v_x \\\\big) v_x = \\\\varepsilon \\\\tilde{B}(v ) v_{xx} \\\\qquad v \\\\in \\\\mathbb{R}^N\\\\\\\\ v (t, \\\\, 0) \\\\equiv \\\\bar v_b \\\\\\\\ v (0, \\\\, x) \\\\equiv \\\\bar{v}_0. \\\\\\\\ \\\\end{array} \\\\right. $$\\nAgain, the data $\\\\bar v_b$ and $\\\\bar v_0$ are constant. The conservative case is included in the previous formulations. It is assumed convergence of the v, smallness of the total variation and other technical hypotheses and it is provided a complete characterization of the limit. The most interesting points are the following two. First, the boundary characteristic case is considered, i.e. one eigenvalue of $\\\\tilde A$ can be 0.\\n Second, as pointed out before we take into account the possibility that $\\\\tilde B$ is not invertible. To deal with this case, we take as hypotheses conditions that were introduced by Kawashima and Shizuta relying on physically meaningful examples. We also introduce a new condition of block linear degeneracy. We prove that, if it is not satisfied, then pathological behaviours may occur. UR - http://hdl.handle.net/1963/1831 U1 - 2385 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A connection between viscous profiles and singular ODEs JF - Rend. Istit. Mat. Univ. Trieste 41 (2009) 35-41 Y1 - 2009 A1 - Stefano Bianchini A1 - Laura Spinolo UR - http://hdl.handle.net/1963/2555 U1 - 1564 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - On the extremality, uniqueness and optimality of transference plans JF - Bull. Inst. Math. Acad. Sin. (N.S.) 4 (2009) 353-458 Y1 - 2009 A1 - Stefano Bianchini A1 - Laura Caravenna AB - We consider the following standard problems appearing in optimal mass transportation theory: when a transference plan is extremal; when a transference plan is the unique transference plan concentrated on a set A,; when a transference plan is optimal. UR - http://hdl.handle.net/1963/3692 U1 - 613 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - RPRT T1 - Invariant Manifolds for Viscous Profiles of a Class of Mixed Hyperbolic-Parabolic Systems Y1 - 2008 A1 - Stefano Bianchini A1 - Laura Spinolo UR - http://hdl.handle.net/1963/3400 U1 - 932 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - CHAP T1 - Transport Rays and Applications to Hamilton–Jacobi Equations T2 - Nonlinear PDE’s and Applications : C.I.M.E. Summer School, Cetraro, Italy 2008 / Stefano Bianchini, Eric A. Carlen, Alexander Mielke, Cédric Villani. Eds. Luigi Ambrosio, Giuseppe Savaré. - Berlin : Springer, 2011. - (Lecture Notes in Mathematics ; 20 Y1 - 2008 A1 - Stefano Bianchini A1 - Matteo Gloyer AB - The aim of these notes is to introduce the readers to the use of the Disintegration Theorem for measures as an effective tool for reducing problems in transport equations to simpler ones. The basic idea is to partition Rd into one dimensional sets, on which the problem under consideration becomes one space dimensional (and thus much easier, hopefully). JF - Nonlinear PDE’s and Applications : C.I.M.E. Summer School, Cetraro, Italy 2008 / Stefano Bianchini, Eric A. Carlen, Alexander Mielke, Cédric Villani. Eds. Luigi Ambrosio, Giuseppe Savaré. - Berlin : Springer, 2011. - (Lecture Notes in Mathematics ; 20 PB - Springer SN - 978-3-642-21718-0 UR - http://hdl.handle.net/1963/5463 N1 - This volume collects the notes of the CIME course Nonlinear PDE’s and\\r\\napplications held in Cetraro (Italy) on June 23–28, 2008. The school consisted\\r\\nin 5 series of lectures, delivered by Stefano Bianchini (SISSA, Trieste), Eric A. Carlen (Rutgers University), Alexander Mielke (WIAS, Berlin), Felix Otto (Bonn University), Cedric Villani (Ecole Normale Superieure de Lyon). U1 - 5298 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - Asymptotic behaviour of smooth solutions for partially dissipative hyperbolic systems with a convex entropy JF - Comm. Pure Appl. Math. 60 (2007) 1559-1622 Y1 - 2007 A1 - Stefano Bianchini A1 - Bernard Hanouzet A1 - Roberto Natalini AB - We study the asymptotic time behavior of global smooth solutions to general entropy dissipative hyperbolic systems of balance law in m space dimensions, under the Shizuta-Kawashima condition. UR - http://hdl.handle.net/1963/1780 U1 - 2764 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - On the Euler-Lagrange equation for a variational problem JF - Discrete Contin. Dynam. Systems A 17 (2007) 449-480 Y1 - 2007 A1 - Stefano Bianchini UR - http://hdl.handle.net/1963/1792 U1 - 2752 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - RPRT T1 - Perturbation techniques applied to the real vanishing viscosity approximation of an initial boundary value problem Y1 - 2007 A1 - Stefano Bianchini PB - SISSA UR - http://preprints.sissa.it/handle/1963/35315 U1 - 35623 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - On Bressan\\\'s conjecture on mixing properties of vector fields JF - Self-Similar Solutions of Nonlinear PDE / Ed. Piotr Biler and Grzegorz Karch. - Banach Center Publ. 74 (2006) 13-31 Y1 - 2006 A1 - Stefano Bianchini UR - http://hdl.handle.net/1963/1806 U1 - 2408 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - RPRT T1 - Glimm interaction functional for BGK schemes Y1 - 2006 A1 - Stefano Bianchini UR - http://hdl.handle.net/1963/1770 U1 - 2774 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Vanishing viscosity solutions of nonlinear hyperbolic systems JF - Ann. of Math. 161 (2005) 223-342 Y1 - 2005 A1 - Stefano Bianchini A1 - Alberto Bressan AB - We consider the Cauchy problem for a strictly hyperbolic, $n\\\\times n$ system in one space dimension: $u_t+A(u)u_x=0$, assuming that the initial data has small total variation.\\nWe show that the solutions of the viscous approximations $u_t+A(u)u_x=\\\\ve u_{xx}$ are defined globally in time and satisfy uniform BV estimates, independent of $\\\\ve$. Moreover, they depend continuously on the initial data in the $\\\\L^1$ distance, with a Lipschitz constant independent of $t,\\\\ve$. Letting $\\\\ve\\\\to 0$, these viscous solutions converge to a unique limit, depending Lipschitz continuously on the initial data. In the conservative case where $A=Df$ is the Jacobian of some flux function $f:\\\\R^n\\\\mapsto\\\\R^n$, the vanishing viscosity limits are precisely the unique entropy weak solutions to the system of conservation laws $u_t+f(u)_x=0$. PB - Annals of Mathematics UR - http://hdl.handle.net/1963/3074 U1 - 1259 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A note on singular limits to hyperbolic systems of conservation laws JF - Commun. Pure Appl. Ana., 2003, 2, 51-64 Y1 - 2003 A1 - Stefano Bianchini AB - In this note we consider two different singular limits to hyperbolic system of conservation laws, namely the standard backward schemes for non linear semigroups and the semidiscrete scheme. \\nUnder the assumption that the rarefaction curve of the corresponding hyperbolic system are straight lines, we prove the stability of the solution and the convergence to the perturbed system to the unique solution of the limit system for initial data with small total variation. PB - SISSA Library UR - http://hdl.handle.net/1963/1542 U1 - 2621 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A center manifold technique for tracing viscous waves JF - Commun. Pure Appl. Anal. 1 (2002) 161-190 Y1 - 2002 A1 - Stefano Bianchini A1 - Alberto Bressan AB - In this paper we introduce a new technique for tracing viscous travelling profiles. To illustrate the method, we consider a special 2 x 2 hyperbolic system of conservation laws with viscosity, and show that any solution can be locally decomposed as the sum of 2 viscous travelling profiles. This yields the global existence, stability and uniform BV bounds for every solution with suitably small BV data. PB - American Institute of Mathematical Sciences UR - http://hdl.handle.net/1963/3075 U1 - 1258 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - On a Lyapunov functional relating shortening curves and viscous conservation laws JF - Nonlinear Anal. 51 (2002) 649-662 Y1 - 2002 A1 - Stefano Bianchini A1 - Alberto Bressan AB - We study a nonlinear functional which controls the area swept by a curve moving in the plane in the direction of curvature. In turn, this yields a priori estimates on solutions to a class of parabolic equations and of scalar viscous conservation laws. A further application provides an estimate on the \\\"change of shape\\\" of a BV solution to a scalar conservation law. PB - Elsevier UR - http://hdl.handle.net/1963/1337 U1 - 3118 U2 - Mathematics U3 - Functional Analysis and Applications 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 - A case study in vanishing viscosity JF - Discrete Cont. Dyn. Syst. 7 (2001) 449-476 Y1 - 2001 A1 - Stefano Bianchini A1 - Alberto Bressan PB - American Institute of Mathematical Sciences UR - http://hdl.handle.net/1963/3091 U1 - 1242 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A Glimm type functional for a special Jin-Xin relaxation model JF - Ann. Inst. H. Poincare\\\' Anal. Non Lineaire 18 (2001), no. 1, 19-42 Y1 - 2001 A1 - Stefano Bianchini PB - Elsevier UR - http://hdl.handle.net/1963/1355 U1 - 3100 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 - BV solutions for a class of viscous hyperbolic systems JF - Indiana Univ. Math. J. 49 (2000) 1673-1714 Y1 - 2000 A1 - Stefano Bianchini A1 - Alberto Bressan PB - Indiana University Mathematics Journal UR - http://hdl.handle.net/1963/3194 U1 - 1107 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 × 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 - TY - JOUR T1 - Extremal faces of the range of a vector measure and a theorem of Lyapunov JF - J. Math. Anal. Appl. 231 (1999) 301-318 Y1 - 1999 A1 - Stefano Bianchini PB - Elsevier UR - http://hdl.handle.net/1963/3370 U1 - 960 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Vanishing viscosity solutions of hyperbolic systems on manifolds Y1 - 1999 A1 - Stefano Bianchini A1 - Alberto Bressan PB - SISSA Library UR - http://hdl.handle.net/1963/1238 U1 - 2705 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - The vector measures whose range is strictly convex JF - J. Math. Anal. Appl. 232 (1999) 1-19 Y1 - 1999 A1 - Stefano Bianchini A1 - Carlo Mariconda PB - Elsevier UR - http://hdl.handle.net/1963/3546 U1 - 1155 U2 - Mathematics U3 - Functional Analysis and Applications ER -