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.

}, doi = {10.22363/2413-3639-2017-63-3-418-436}, url = {http://www.mathnet.ru/php/archive.phtml?wshow=paper\&jrnid=cmfd\&paperid=327\&option_lang=eng}, author = {Stefano Bianchini and Paolo Bonicatto and Elio Marconi} } @article {2017, title = {A uniqueness result for the decomposition of vector fields in Rd}, number = {SISSA;15/2017/MATE}, year = {2017}, institution = {SISSA}, abstract = {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.

}, url = {http://preprints.sissa.it/handle/1963/35274}, author = {Stefano Bianchini and Paolo Bonicatto} } @article {1937-1632_2016_1_73, title = {On the concentration of entropy for scalar conservation laws}, journal = {Discrete \& Continuous Dynamical Systems - S}, volume = {9}, number = {1937-1632_2016_1_7}, year = {2016}, pages = {73}, abstract = {We prove that the entropy for an $L^$\infty$$-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.

}, keywords = {concentration, Conservation laws, entropy solutions, Lagrangian representation, shocks}, issn = {1937-1632}, doi = {10.3934/dcdss.2016.9.73}, url = {http://aimsciences.org//article/id/ce4eb91e-9553-4e8d-8c4c-868f07a315ae}, author = {Stefano Bianchini and Elio Marconi} } @article {2016, title = {Eulerian, Lagrangian and Broad continuous solutions to a balance law with non convex flux II}, number = {SISSA;32/2016/MATE}, year = {2016}, url = {http://urania.sissa.it/xmlui/handle/1963/35197}, author = {Giovanni Alberti and Stefano Bianchini and Laura Caravenna} } @article {2016, title = {Eulerian, Lagrangian and Broad continuous solutions to a balance law with non-convex flux I}, journal = {Journal of Differential Equations, vol. 261, issue 8 (2016): 4298-4337}, number = {Journal of Differential Equations;261}, year = {2016}, publisher = {Elsevier}, doi = {10.1016/j.jde.2016.06.026}, url = {http://urania.sissa.it/xmlui/handle/1963/35207}, author = {Giovanni Alberti and Stefano Bianchini and Laura Caravenna} } @article {doi:10.1137/15M1007380, title = {Renormalization for Autonomous Nearly Incompressible BV Vector Fields in Two Dimensions}, journal = {SIAM Journal on Mathematical Analysis}, volume = {48}, number = {1}, year = {2016}, pages = {1-33}, abstract = {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{\textquoteright}s superposition principle [4].

}, doi = {10.1137/15M1007380}, url = {https://doi.org/10.1137/15M1007380}, author = {Stefano Bianchini and Paolo Bonicatto and N.A. Gusev} } @article {2016, title = {On the structure of $L^\infty$-entropy solutions to scalar conservation laws in one-space dimension}, year = {2016}, institution = {SISSA}, abstract = {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 {modena2015convergence, title = {Convergence rate of the Glimm scheme}, journal = {Bulletin of the Institute of Mathematics of Academia Sinica (New Series)}, year = {2015}, author = {Stefano Modena and Stefano Bianchini} } @article {bianchini2015quadratic, title = {Quadratic Interaction Functional for General Systems of Conservation Laws}, journal = {Communications in Mathematical Physics}, volume = {338}, year = {2015}, pages = {1075{\textendash}1152}, abstract = {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;

}, doi = {10.1007/s00220-015-2372-2}, author = {Stefano Bianchini and Stefano Modena} } @article {2014, title = {The decomposition of optimal transportation problems with convex cost}, number = {SISSA;45/2014/MATE}, year = {2014}, institution = {SISSA}, url = {http://hdl.handle.net/1963/7433}, author = {Stefano Bianchini and Mauro Bardelloni} } @article {2014, title = {Existence and uniqueness of the gradient flow of the Entropy in the space of probability measures}, number = {Rendiconti dell{\textquoteright}Istituto di Matematica dell{\textquoteright}Universita di Trieste;Volume 46; issue 1; pp. 43-70;}, year = {2014}, note = {This paper resumes the main part of the Bachelor thesis of the second author, discussed in 2013 at the University of Trieste.}, publisher = {EUT Edizioni Universita di Trieste}, abstract = {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.}, url = {http://urania.sissa.it/xmlui/handle/1963/34693}, author = {Stefano Bianchini and Alexander Dabrowski} } @article {2014, title = {Global Structure of Admissible BV Solutions to Piecewise Genuinely Nonlinear, Strictly Hyperbolic Conservation Laws in One Space Dimension}, number = {Communications in Partial Differential Equations;Volume 39; issue 2; pp. 244-273;}, year = {2014}, publisher = {Taylor \& Francis}, abstract = {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.

}, doi = {10.1080/03605302.2013.775153}, url = {http://urania.sissa.it/xmlui/handle/1963/34694}, author = {Stefano Bianchini and Lei Yu} } @article {2014, title = {On the Lp-differentiability of certain classes of functions}, number = {Revista Matematica Iberoamericana;Volume 30; issue 1; pp. 349-367;}, year = {2014}, publisher = {European Mathematical Society}, abstract = {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.}, doi = {10.4171/rmi/782}, url = {http://urania.sissa.it/xmlui/handle/1963/34695}, author = {Giovanni Alberti and Stefano Bianchini and Gianluca Crippa} } @article {2014, title = {On a quadratic functional for scalar conservation laws}, journal = {Journal of Hyperbolic Differential Equations}, volume = {11}, number = {Journal of Hyperbolic Differential Equations;Volume 11; issue 2; pp. 355-435;}, year = {2014}, pages = {355-435}, publisher = {World Scientific Publishing}, abstract = {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.

}, doi = {10.1142/S0219891614500118}, url = {http://arxiv.org/abs/1311.2929}, author = {Stefano Bianchini and Stefano Modena} } @article {bianchini2013quadratic, title = {Quadratic interaction functional for systems of conservation laws: a case study}, journal = {Bulletin of the Institute of Mathematics of Academia Sinica (New Series)}, volume = {9}, year = {2014}, pages = {487-546}, url = {https://w3.math.sinica.edu.tw/bulletin_ns/20143/2014308.pdf}, author = {Stefano Bianchini and Stefano Modena} } @inbook {2013, title = {Reduction on characteristics for continuous of a scalar balance law}, booktitle = {AIMS Series on Applied Mathematics, vol. 8 (2014): 399 - 406}, number = {AIMS Series on Applied Mathematics}, year = {2014}, publisher = {SISSA}, organization = {SISSA}, keywords = {Method of characteristics}, url = {http://hdl.handle.net/1963/6562}, author = {Giovanni Alberti and Stefano Bianchini and Laura Caravenna} } @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 {2014, title = {A uniqueness result for the continuity equation in two dimensions: dedicated to constantine dafermos on the occasion of his 70th birthday}, number = {Journal of the European Mathematical Society;Volume 16; issue 2; pp. 201-234;}, year = {2014}, publisher = {European Mathematical Society; Springer Verlag}, abstract = {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.}, doi = {10.4171/JEMS/431}, url = {http://urania.sissa.it/xmlui/handle/1963/34692}, author = {Giovanni Alberti and Stefano Bianchini and Gianluca Crippa} } @article {Bianchini2013, title = {The Monge Problem for Distance Cost in Geodesic Spaces}, journal = {Communications in Mathematical Physics}, volume = {318}, number = {3}, year = {2013}, month = {Mar}, pages = {615{\textendash}673}, abstract = {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.

}, issn = {1432-0916}, doi = {10.1007/s00220-013-1663-8}, url = {https://doi.org/10.1007/s00220-013-1663-8}, author = {Stefano Bianchini and Fabio Cavalletti} } @article {modena2013, title = {A New Quadratic Potential for Scalar Conservation Laws}, journal = {Oberwolfach Reports}, volume = {29}, year = {2013}, author = {Stefano Bianchini and Stefano Modena} } @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 = {A Decomposition Theorem for BV functions}, journal = {Communications on Pure and Applied Analysis}, volume = {10}, number = {SISSA;20/2009/M}, year = {2011}, pages = {1549-1566}, publisher = {American Institute of Mathematical Sciences}, doi = {10.3934/cpaa.2011.10.1549}, url = {http://hdl.handle.net/20.500.11767/14599}, author = {Stefano Bianchini and Daniela Tonon} } @article {2011, title = {An Estimate on the Flow Generated by Monotone Operators}, journal = {Communications in Partial Differential Equations 36 (2011) 777-796}, number = {SISSA;29/2009/M}, year = {2011}, publisher = {Taylor \& Francis}, doi = {10.1080/03605302.2010.534224}, url = {http://hdl.handle.net/1963/3646}, author = {Stefano Bianchini and Matteo Gloyer} } @article {2011, title = {Invariant manifolds for a singular ordinary differential equation}, journal = {Journal of Differential Equations 250 (2011) 1788-1827}, number = {SISSA;04/2008/M}, year = {2011}, publisher = {Elsevier}, doi = {10.1016/j.jde.2010.11.010}, url = {http://hdl.handle.net/1963/2554}, author = {Stefano Bianchini and Laura Spinolo} } @conference {10.1007/978-1-4419-9554-4_10, title = {The Monge Problem in Geodesic Spaces}, booktitle = {Nonlinear Conservation Laws and Applications}, year = {2011}, pages = {217{\textendash}233}, publisher = {Springer US}, organization = {Springer US}, address = {Boston, MA}, abstract = {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.

}, isbn = {978-1-4419-9554-4}, author = {Stefano Bianchini and Fabio Cavalletti}, editor = {Alberto Bressan and Chen, Gui-Qiang G. and Marta Lewicka and Wang, Dehua} } @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 {2011, title = {A uniqueness result for the continuity equation in two dimensions}, number = {SISSA;52/2011/M}, year = {2011}, institution = {SISSA}, url = {http://hdl.handle.net/1963/4663}, author = {Giovanni Alberti and Stefano Bianchini and Gianluca Crippa} } @article {2010, title = {Estimates on path functionals over Wasserstein Spaces}, journal = {SIAM J. Math. Anal. 42 (2010) 1179-1217}, number = {SISSA;11/2009/M}, year = {2010}, abstract = {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.}, doi = {10.1137/100782693}, url = {http://hdl.handle.net/1963/3583}, author = {Stefano Bianchini and Alessio Brancolini} } @article {2010, title = {On the Euler-Lagrange equation for a variational problem : the general case II}, journal = {Math. Z. 265 (2010) 889-923}, number = {SISSA;75/2007/M}, year = {2010}, doi = {10.1007/s00209-009-0547-2}, url = {http://hdl.handle.net/1963/2551}, author = {Stefano Bianchini and Matteo Gloyer} } @article {2010, title = {On optimality of c-cyclically monotone transference plans}, journal = {Comptes Rendus Mathematique 348 (2010) 613-618}, year = {2010}, publisher = {Elsevier}, abstract = {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.}, doi = {10.1016/j.crma.2010.03.022}, url = {http://hdl.handle.net/1963/4023}, author = {Stefano Bianchini and Laura Caravenna} } @article {2009, title = {The boundary Riemann solver coming from the real vanishing viscosity approximation}, journal = {Arch. Ration. Mech. Anal. 191 (2009) 1-96}, number = {SISSA;24/2006/M}, year = {2009}, abstract = {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.}, doi = {10.1007/s00205-008-0177-6}, url = {http://hdl.handle.net/1963/1831}, author = {Stefano Bianchini and Laura Spinolo} } @article {2009, title = {A connection between viscous profiles and singular ODEs}, journal = {Rend. Istit. Mat. Univ. Trieste 41 (2009) 35-41}, number = {SISSA;05/2008/M}, year = {2009}, url = {http://hdl.handle.net/1963/2555}, author = {Stefano Bianchini and Laura Spinolo} } @article {2009, title = {On the extremality, uniqueness and optimality of transference plans}, journal = {Bull. Inst. Math. Acad. Sin. (N.S.) 4 (2009) 353-458}, number = {SISSA;46/2009/M}, year = {2009}, abstract = {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.}, url = {http://hdl.handle.net/1963/3692}, author = {Stefano Bianchini and Laura Caravenna} } @article {2008, title = {Invariant Manifolds for Viscous Profiles of a Class of Mixed Hyperbolic-Parabolic Systems}, number = {SISSA;83/2008/M}, year = {2008}, url = {http://hdl.handle.net/1963/3400}, author = {Stefano Bianchini and Laura Spinolo} } @inbook {2008, title = {Transport Rays and Applications to Hamilton{\textendash}Jacobi Equations}, booktitle = {Nonlinear PDE{\textquoteright}s and Applications : C.I.M.E. Summer School, Cetraro, Italy 2008 / Stefano Bianchini, Eric A. Carlen, Alexander Mielke, C{\'e}dric Villani. Eds. Luigi Ambrosio, Giuseppe Savar{\'e}. - Berlin : Springer, 2011. - (Lecture Notes in Mathematics ; 20}, year = {2008}, note = {This volume collects the notes of the CIME course Nonlinear PDE{\textquoteright}s and\\r\\napplications held in Cetraro (Italy) on June 23{\textendash}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).}, publisher = {Springer}, organization = {Springer}, abstract = {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).}, isbn = {978-3-642-21718-0}, doi = {10.1007/978-3-642-21861-3_1}, url = {http://hdl.handle.net/1963/5463}, author = {Stefano Bianchini and Matteo Gloyer} } @article {2007, title = {Asymptotic behaviour of smooth solutions for partially dissipative hyperbolic systems with a convex entropy}, journal = {Comm. Pure Appl. Math. 60 (2007) 1559-1622}, number = {SISSA;83/2005/M}, year = {2007}, abstract = {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.}, doi = {10.1002/cpa.20195}, url = {http://hdl.handle.net/1963/1780}, author = {Stefano Bianchini and Bernard Hanouzet and Roberto Natalini} } @article {2007, title = {On the Euler-Lagrange equation for a variational problem}, journal = {Discrete Contin. Dynam. Systems A 17 (2007) 449-480}, number = {SISSA;95/2005/M}, year = {2007}, url = {http://hdl.handle.net/1963/1792}, author = {Stefano Bianchini} } @article {2007, title = {Perturbation techniques applied to the real vanishing viscosity approximation of an initial boundary value problem}, year = {2007}, institution = {SISSA}, url = {http://preprints.sissa.it/handle/1963/35315}, author = {Stefano Bianchini} } @article {2006, title = {On Bressan\\\'s conjecture on mixing properties of vector fields}, journal = {Self-Similar Solutions of Nonlinear PDE / Ed. Piotr Biler and Grzegorz Karch. - Banach Center Publ. 74 (2006) 13-31}, number = {SISSA;70/2005/M}, year = {2006}, url = {http://hdl.handle.net/1963/1806}, author = {Stefano Bianchini} } @article {2006, title = {Glimm interaction functional for BGK schemes}, number = {SISSA;69/2005/M}, year = {2006}, url = {http://hdl.handle.net/1963/1770}, author = {Stefano Bianchini} } @article {2005, title = {Vanishing viscosity solutions of nonlinear hyperbolic systems}, journal = {Ann. of Math. 161 (2005) 223-342}, number = {SISSA;86/2001/M}, year = {2005}, publisher = {Annals of Mathematics}, abstract = {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 $u_t+f(u)_x=0$.}, url = {http://hdl.handle.net/1963/3074}, author = {Stefano Bianchini and Alberto Bressan} } @article {2003, title = {A note on singular limits to hyperbolic systems of conservation laws}, journal = {Commun. Pure Appl. Ana., 2003, 2, 51-64}, number = {SISSA;85/00/M}, year = {2003}, publisher = {SISSA Library}, abstract = {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.}, url = {http://hdl.handle.net/1963/1542}, author = {Stefano Bianchini} } @article {2002, title = {A center manifold technique for tracing viscous waves}, journal = {Commun. Pure Appl. Anal. 1 (2002) 161-190}, number = {SISSA;85/2001/M}, year = {2002}, publisher = {American Institute of Mathematical Sciences}, abstract = {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.}, url = {http://hdl.handle.net/1963/3075}, author = {Stefano Bianchini and Alberto Bressan} } @article {2002, title = {On a Lyapunov functional relating shortening curves and viscous conservation laws}, journal = {Nonlinear Anal. 51 (2002) 649-662}, number = {SISSA;123/99/M}, year = {2002}, publisher = {Elsevier}, abstract = {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.}, doi = {10.1016/S0362-546X(01)00848-3}, url = {http://hdl.handle.net/1963/1337}, author = {Stefano Bianchini and Alberto Bressan} } @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 = {A case study in vanishing viscosity}, journal = {Discrete Cont. Dyn. Syst. 7 (2001) 449-476}, year = {2001}, publisher = {American Institute of Mathematical Sciences}, url = {http://hdl.handle.net/1963/3091}, author = {Stefano Bianchini and Alberto Bressan} } @article {2001, title = {A Glimm type functional for a special Jin-Xin relaxation model}, journal = {Ann. Inst. H. Poincare\\\' Anal. Non Lineaire 18 (2001), no. 1, 19-42}, number = {SISSA;140/99/M}, year = {2001}, publisher = {Elsevier}, doi = {10.1016/S0294-1449(00)00124-4}, url = {http://hdl.handle.net/1963/1355}, author = {Stefano Bianchini} } @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 = {BV solutions for a class of viscous hyperbolic systems}, journal = {Indiana Univ. Math. J. 49 (2000) 1673-1714}, year = {2000}, publisher = {Indiana University Mathematics Journal}, doi = {10.1512/iumj.2000.49.1776}, url = {http://hdl.handle.net/1963/3194}, author = {Stefano Bianchini and Alberto Bressan} } @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} } @article {1999, title = {Extremal faces of the range of a vector measure and a theorem of Lyapunov}, journal = {J. Math. Anal. Appl. 231 (1999) 301-318}, number = {SISSA;4/98/M}, year = {1999}, publisher = {Elsevier}, doi = {10.1006/jmaa.1998.6260}, url = {http://hdl.handle.net/1963/3370}, author = {Stefano Bianchini} } @article {1999, title = {Vanishing viscosity solutions of hyperbolic systems on manifolds}, number = {SISSA;24/99/M}, year = {1999}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1238}, author = {Stefano Bianchini and Alberto Bressan} } @article {1999, title = {The vector measures whose range is strictly convex}, journal = {J. Math. Anal. Appl. 232 (1999) 1-19}, number = {SISSA;108/96/M}, year = {1999}, publisher = {Elsevier}, doi = {10.1006/jmaa.1998.6215}, url = {http://hdl.handle.net/1963/3546}, author = {Stefano Bianchini and Carlo Mariconda} }