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 - On the convergence of viscous approximations after shock interactions JF - Discrete Contin. Dyn. Syst. 23 (2009) 29-48 Y1 - 2009 A1 - Alberto Bressan A1 - Carlotta Donadello AB - We consider a piecewise smooth solution to a scalar conservation law, with possibly interacting shocks. We show that, after the interactions have taken place, vanishing viscosity approximations can still be represented by a regular expansion on smooth regions and by a singular perturbation expansion near the shocks, in terms of powers of the viscosity coefficient. PB - American Institute of Mathematical Sciences UR - http://hdl.handle.net/1963/3412 U1 - 923 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Asymptotic variational wave equations JF - Arch. Ration. Mech. Anal. 183 (2007) 163-185 Y1 - 2007 A1 - Alberto Bressan A1 - Zhang Ping A1 - Zheng Yuxi AB - We investigate the equation $(u_t + (f(u))_x)_x = f\\\'\\\'(u) (u_x)^2/2$ where $f(u)$ is a given smooth function. Typically $f(u)= u^2/2$ or $u^3/3$. This equation models unidirectional and weakly nonlinear waves for the variational wave equation $u_{tt} - c(u) (c(u)u_x)_x =0$ which models some liquid crystals with a natural sinusoidal $c$. The equation itself is also the Euler-Lagrange equation of a variational problem. Two natural classes of solutions can be associated with this equation. A conservative solution will preserve its energy in time, while a dissipative weak solution loses energy at the time when singularities appear. Conservative solutions are globally defined, forward and backward in time, and preserve interesting geometric features, such as the Hamiltonian structure. On the other hand, dissipative solutions appear to be more natural from the physical point of view.\\nWe establish the well-posedness of the Cauchy problem within the class of conservative solutions, for initial data having finite energy and assuming that the flux function $f$ has Lipschitz continuous second-order derivative. In the case where $f$ is convex, the Cauchy problem is well-posed also within the class of dissipative solutions. However, when $f$ is not convex, we show that the dissipative solutions do not depend continuously on the initial data. UR - http://hdl.handle.net/1963/2182 U1 - 2062 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - RPRT T1 - BV instability for the Lax-Friedrichs scheme Y1 - 2007 A1 - Paolo Baiti A1 - Alberto Bressan A1 - Helge Kristian Jenssen AB - It is proved that discrete shock profiles (DSPs) for the Lax-Friedrichs scheme for a system of conservation laws do not necessarily depend continuously in BV on their speed. We construct examples of $2 \\\\times 2$-systems for which there are sequences of DSPs with speeds converging to a rational number. Due to a resonance phenomenon, the difference between the limiting DSP and any DSP in the sequence will contain an order-one amount of variation. UR - http://hdl.handle.net/1963/2335 U1 - 1681 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Nearly time optimal stabilizing patchy feedbacks JF - Ann. Inst. H. Poincare Anal. Non Lineaire 24 (2007) 279-310 Y1 - 2007 A1 - Fabio Ancona A1 - Alberto Bressan AB - We consider the time optimal stabilization problem for a nonlinear control system $\\\\dot x=f(x,u)$. Let $\\\\tau(y)$ be the minimum time needed to steer the system from the state $y\\\\in\\\\R^n$ to the origin, and call $\\\\A(T)$ the set of initial states that can be steered to the origin in time $\\\\tau(y)\\\\leq T$. Given any $\\\\ve>0$, in this paper we construct a patchy feedback $u=U(x)$ such that every solution of $\\\\dot x=f(x, U(x))$, $x(0)=y\\\\in \\\\A(T)$ reaches an $\\\\ve$-neighborhood of the origin within time $\\\\tau(y)+\\\\ve$. UR - http://hdl.handle.net/1963/2185 U1 - 2059 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Conservative Solutions to a Nonlinear Variational Wave Equation JF - Comm. Math. Phys. 266 (2006) 471-497 Y1 - 2006 A1 - Alberto Bressan A1 - Zheng Yuxi AB - We establish the existence of a conservative weak solution to the Cauchy problem for the nonlinear variational wave equation $u_{tt} - c(u)(c(u)u_x)_x=0$, for initial data of finite energy. Here $c(\\\\cdot)$ is any smooth function with uniformly positive bounded values. UR - http://hdl.handle.net/1963/2184 U1 - 2060 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - RPRT T1 - Infinite Horizon Noncooperative Differential Games Y1 - 2006 A1 - Alberto Bressan A1 - Fabio Simone Priuli AB - For a non-cooperative differential game, the value functions of the various players satisfy a system of Hamilton-Jacobi equations. In the present paper, we consider a class of infinite-horizon games with nonlinear costs exponentially discounted in time. By the analysis of the value\\nfunctions, we establish the existence of Nash equilibrium solutions in feedback form and provide results and counterexamples on their uniqueness and stability. JF - J. Differential Equations 227 (2006) 230-257 UR - http://hdl.handle.net/1963/1720 U1 - 2431 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - An instability of the Godunov scheme JF - Comm. Pure Appl. Math. 59 (2006) 1604-1638 Y1 - 2006 A1 - Alberto Bressan A1 - Helge Kristian Jenssen A1 - Paolo Baiti AB - We construct a solution to a $2\\\\times 2$ strictly hyperbolic system of conservation laws, showing that the Godunov scheme \\\\cite{Godunov59} can produce an arbitrarily large amount of oscillations. This happens when the speed of a shock is close to rational, inducing a resonance with the grid. Differently from the Glimm scheme or the vanishing viscosity method, for systems of conservation laws our counterexample indicates that no a priori BV bounds or $L^1$ stability estimates can in general be valid for finite difference schemes. UR - http://hdl.handle.net/1963/2183 U1 - 2061 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - On the Blow-up for a Discrete Boltzmann Equation in the Plane JF - Discrete Contin. Dyn. Syst. 13 (2005) 1-12 Y1 - 2005 A1 - Alberto Bressan A1 - Massimo Fonte AB - We study the possibility of finite-time blow-up for a two dimensional Broadwell model. In a set of rescaled variables, we prove that no self-similar blow-up solution exists, and derive some a priori bounds on the blow-up rate. In the final section, a possible blow-up scenario is discussed. UR - http://hdl.handle.net/1963/2244 U1 - 2000 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Global solutions of the Hunter-Saxton equation JF - SIAM J. Math. Anal. 37 (2005) 996-1026 Y1 - 2005 A1 - Alberto Bressan A1 - Adrian Constantin AB - We construct a continuous semigroup of weak, dissipative solutions to a nonlinear partial differential equations modeling nematic liquid crystals. A new distance functional, determined by a problem of optimal transportation, yields sharp estimates on the continuity of solutions with respect to the initial data. UR - http://hdl.handle.net/1963/2256 U1 - 1991 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - RPRT T1 - An Optimal Transportation Metric for Solutions of the Camassa-Holm Equation Y1 - 2005 A1 - Alberto Bressan A1 - Massimo Fonte AB - In this paper we construct a global, continuous flow of solutions to the Camassa-Holm equation on the entire space H1. Our solutions are conservative, in the sense that the total energy int[(u2 + u2x) dx] remains a.e. constant in time. Our new approach is based on a distance functional J(u, v), defined in terms of an optimal transportation problem, which satisfies d dtJ(u(t), v(t)) ≤ κ · J(u(t), v(t)) for every couple of solutions. Using this new distance functional, we can construct arbitrary solutions as the uniform limit of multi-peakon solutions, and prove a general uniqueness result. JF - Methods Appl. Anal. 12 (2005) 191-219 UR - http://hdl.handle.net/1963/1719 U1 - 2432 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 - On the convergence rate of vanishing viscosity approximations JF - Comm. Pure Appl. Math. 57 (2004) 1075-1109 Y1 - 2004 A1 - Alberto Bressan A1 - Tong Yang AB - Given a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound $\\\\big\\\\|u(t,\\\\cdot)-u^\\\\ve(t,\\\\cdot)\\\\big\\\\|_{\\\\L^1}= \\\\O(1)(1+t)\\\\cdot \\\\sqrt\\\\ve|\\\\ln\\\\ve|$ on the distance between an exact BV solution $u$ and a viscous approximation $u^\\\\ve$, letting the viscosity coefficient $\\\\ve\\\\to 0$. In the proof, starting from $u$ we construct an approximation of the viscous solution $u^\\\\ve$ by taking a mollification $u*\\\\phi_{\\\\strut \\\\sqrt\\\\ve}$ and inserting viscous shock profiles at the locations of finitely many large shocks, for each fixed $\\\\ve$. Error estimates are then obtained by introducing new Lyapunov functionals which control shock interactions, interactions between waves of different families and by using sharp decay estimates for positive nonlinear waves. PB - Wiley UR - http://hdl.handle.net/1963/2915 U1 - 1785 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Semi-cooperative strategies for differential games JF - Internat. J. Game Theory 32 (2004) 561-593 Y1 - 2004 A1 - Alberto Bressan A1 - Wen Shen AB - The paper is concerned with a non-cooperative differential game for two players. We first consider Nash equilibrium solutions in feedback form. In this case, we show that the Cauchy problem for the value functions is generically ill-posed. Looking at vanishing viscosity approximations, one can construct special solutions in the form of chattering controls, but these also appear to be unstable. In the second part of the paper we propose an alternative \\\"semi-cooperative\\\" pair of strategies for the two players, seeking a Pareto optimum instead of a Nash equilibrium. In this case, we prove that the corresponding Hamiltonian system for the value functions is always weakly hyperbolic. PB - Springer UR - http://hdl.handle.net/1963/2893 U1 - 1807 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A sharp decay estimate for positive nonlinear waves JF - SIAM J. Math. Anal. 36 (2004) 659-677 Y1 - 2004 A1 - Alberto Bressan A1 - Tong Yang AB - We consider a strictly hyperbolic, genuinely nonlinear system of conservation laws in one space dimension. A sharp decay estimate is proved for the positive waves in an entropy weak solution. The result is stated in terms of a partial ordering among positive measures, using symmetric rearrangements and a comparison with a solution of Burgers\\\' equation with impulsive sources. PB - SIAM UR - http://hdl.handle.net/1963/2916 U1 - 1784 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Small BV solutions of hyperbolic noncooperative differential games JF - SIAM J. Control Optim. 43 (2004) 194-215 Y1 - 2004 A1 - Alberto Bressan A1 - Wen Shen AB - The paper is concerned with an n-persons differential game in one space dimension. We state conditions for which the system of Hamilton-Jacobi equations for the value functions is strictly hyperbolic. In the positive case, we show that the weak solution of a corresponding system of conservation laws determines an n-tuple of feedback strategies. These yield a Nash equilibrium solution to the non-cooperative differential game. PB - SIAM UR - http://hdl.handle.net/1963/2917 U1 - 1783 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Some remarks on multidimensional systems of conservation laws JF - Rend. Mat. Acc. Lincei, s. 9, v. 15 (2004) 3-4, pp. 225 - 233 Y1 - 2004 A1 - Alberto Bressan AB - This note is concerned with the Cauchy problem for hyperbolic systems of conservation\\nlaws in several space dimensions. We first discuss an example of ill-posedness, for a special system\\nhaving a radial symmetry property. Some conjectures are formulated, on the compactness of the set of\\nflow maps generated by vector fields with bounded variation. PB - Accademia Nazionale dei Lincei UR - http://hdl.handle.net/1963/3642 U1 - 662 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Stability rates for patchy vector fields JF - ESAIM COCV 10 (2004) 168-200 Y1 - 2004 A1 - Fabio Ancona A1 - Alberto Bressan AB - This paper is concerned with the stability of the set of trajectories of a patchy vector field, in the presence of impulsive perturbations. Patchy vector fields are discontinuous, piecewise smooth vector fields that were introduced in Ancona and Bressan (1999) to study feedback stabilization problems. For patchy vector fields in the plane, with polygonal patches in generic position, we show that the distance between a perturbed trajectory and an unperturbed one is of the same order of magnitude as the impulsive forcing term. PB - EDP Sciences UR - http://hdl.handle.net/1963/2959 U1 - 1741 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - An ill posed Cauchy problem for a hyperbolic system in two space dimensions Y1 - 2003 A1 - Alberto Bressan AB - The theory of weak solutions for nonlinear conservation laws is now well developed in the case of scalar equations [3] and for one-dimensional hyperbolic systems [1, 2]. For systems in several space dimensions, however, even the global existence of solutions to the Cauchy problem remains a challenging open question. In this note we construct a conterexample showing that, even for a simple class of hyperbolic systems, in two space dimensions the Cauchy problem can be ill posed. PB - Università di Padova UR - http://hdl.handle.net/1963/2913 U1 - 1787 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A lemma and a conjecture on the cost of rearrangements JF - Rend. Sem. Mat. Univ. Padova 110 (2003) 97-102 Y1 - 2003 A1 - Alberto Bressan AB - Consider a stack of books, containing both white and black books. Suppose that we want to sort them out, putting the white books on the right, and the black books on the left (fig.~1). This will be done by a finite sequence of elementary transpositions. In other words, if we have a stack of all black books of length $a$ followed by a stack of all white books of length $b$, we are allowed to reverse their order at the cost of $a+b$. We are interested in a lower bound on the total cost of the rearrangement. PB - Università di Padova UR - http://hdl.handle.net/1963/2914 U1 - 1786 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Some results on the boundary control of systems of conservation laws JF - SIAM J.Control Optim. 41 (2003),no.2, 607 Y1 - 2003 A1 - Alberto Bressan A1 - Fabio Ancona A1 - Giuseppe Maria Coclite PB - SISSA Library UR - http://hdl.handle.net/1963/1615 U1 - 2503 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - On the Boundary Control of Systems of Conservation Laws JF - SIAM J. Control Optim. 41 (2002) 607-622 Y1 - 2002 A1 - Alberto Bressan A1 - Giuseppe Maria Coclite AB - The paper is concerned with the boundary controllability of entropy weak solutions to hyperbolic systems of conservation laws. We prove a general result on the asymptotic stabilization of a system near a constant state. On the other hand, we give an example showing that exact controllability in finite time cannot be achieved, in general. PB - SIAM UR - http://hdl.handle.net/1963/3070 U1 - 1263 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 - Flow Stability of Patchy Vector Fields and Robust Feedback Stabilization JF - SIAM J. Control Optim. 41 (2002) 1455-1476 Y1 - 2002 A1 - Fabio Ancona A1 - Alberto Bressan AB - The paper is concerned with patchy vector fields, a class of discontinuous, piecewise smooth vector fields that were introduced in AB to study feedback stabilization problems. We prove the stability of the corresponding solution set w.r.t. a wide class of impulsive perturbations. These results yield the robusteness of patchy feedback controls in the presence of measurement errors and external disturbances. PB - SIAM UR - http://hdl.handle.net/1963/3073 U1 - 1260 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 - 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 - BV estimates for multicomponent chromatography with relaxation JF - Discrete Contin. Dynam. Systems 6 (2000) 21-38 Y1 - 2000 A1 - Alberto Bressan A1 - Wen Shen AB - We consider the Cauchy problem for a system of $2n$ balance laws which arises from the modelling of multi-component chromatography: $$\\\\left\\\\{ \\\\eqalign{u_t+u_x&=-{1\\\\over\\\\ve}\\\\big( F(u)-v\\\\big),\\\\cr v_t&={1\\\\over\\\\ve}\\\\big( F(u)-v\\\\big),\\\\cr}\\\\right. \\\\eqno(1)$$ This model describes a liquid flowing with unit speed over a solid bed. Several chemical substances are partly dissolved in the liquid, partly deposited on the solid bed. Their concentrations are represented respectively by the vectors $u=(u_1,\\\\ldots,u_n)$ and $v=(v_1,\\\\ldots,v_n)$. We show that, if the initial data have small total variation, then the solution of (1) remains with small variation for all times $t\\\\geq 0$. Moreover, using the $\\\\L^1$ distance, this solution depends Lipschitz continuously on the initial data, with a Lipschitz constant uniform w.r.t.~$\\\\ve$. Finally we prove that as $\\\\ve\\\\to 0$, the solutions of (1) converge to a limit described by the system $$\\\\big(u+F(u)\\\\big)_t+u_x=0,\\\\qquad\\\\qquad v=F(u).\\\\eqno(2)$$ The proof of the uniform BV estimates relies on the application of probabilistic techniques. It is shown that the components of the gradients $v_x,u_x$ can be interpreted as densities of random particles travelling with speed 0 or 1. The amount of coupling between different components is estimated in terms of the expected number of crossing of these random particles. This provides a first example where BV estimates are proved for general solutions to a class of $2n\\\\times 2n$ systems with relaxation. PB - SISSA Library UR - http://hdl.handle.net/1963/1336 U1 - 3119 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 - On the convergence of Godunov scheme for nonlinear hyperbolic systems JF - Chinese Ann. Math. B, 2000, 21, 269 Y1 - 2000 A1 - Alberto Bressan A1 - Helge Kristian Jenssen PB - SISSA Library UR - http://hdl.handle.net/1963/1473 U1 - 2690 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Stability of L^infty Solutions of Temple Class Systems JF - Differential Integral Equations 13 (2000) 1503-1528 Y1 - 2000 A1 - Alberto Bressan A1 - Paola Goatin AB -Let $u_t+f(u)_x=0$ be a strictly hyperbolic, genuinely nonlinear system of conservation laws of Temple class. In this paper, a continuous semigroup of solutions is constructed on a domain of $L^\infty$ functions, with possibly unbounded variation. Trajectories depend Lipschitz continuously on the initial data, in the $L^1$ distance. Moreover, we show that a weak solution of the Cauchy problem coincides with the corresponding semigroup trajectory if and only if it satisfies an entropy condition of Oleinik type, concerning the decay of positive waves.

PB - Khayyam Publishing UR - http://hdl.handle.net/1963/3256 U1 - 1445 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A Uniqueness Condition for Hyperbolic Systems of Conservation Laws JF - Discrete Contin. Dynam. Systems 6 (2000) 673-682 Y1 - 2000 A1 - Alberto Bressan A1 - Marta Lewicka AB - Consider the Cauchy problem for a hyperbolic $n\\\\times n$ system of conservation laws in one space dimension: $$u_t+f(u)_x=0, u(0,x)=\\\\bar u(x).\\\\eqno(CP)$$ Relying on the existence of a continuous semigroup of solutions, we prove that the entropy admissible solution of (CP) is unique within the class of functions $u=u(t,x)$ which have bounded variation along a suitable family of space-like curves. PB - American Institute of Mathematical Sciences UR - http://hdl.handle.net/1963/3195 U1 - 1106 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - BOOK T1 - Well-posedness of the Cauchy problem for n x n systems of conservation laws T2 - Mem. Amer. Math. Soc. 146 (2000), no. 694, 134 p. Y1 - 2000 A1 - Alberto Bressan A1 - Graziano Crasta A1 - Benedetto Piccoli JF - Mem. Amer. Math. Soc. 146 (2000), no. 694, 134 p. PB - American Mathematical Society UR - http://hdl.handle.net/1963/3495 N1 - Chapter 1 and 2 U1 - 769 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Hyperbolic Systems of Conservation Laws JF - Rev. Mat. Complut. 12 (1999) 135-200 Y1 - 1999 A1 - Alberto Bressan AB - This is a survey paper, written in the occasion of an invited talk given by the author at the Universidad Complutense in Madrid, October 1998. Its purpose is to provide an account of some recent advances in the mathematical theory of hyperbolic systems of conservation laws in one space dimension. After a brief review of basic concepts, we describe in detail the method of wave-front tracking approximation and present some of the latest results on uniqueness and stability of entropy weak solutions. UR - http://hdl.handle.net/1963/1855 U1 - 77 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - L-1 stability estimates for n x n conservation laws JF - Arch. Ration. Mech. Anal. 149 (1999), no. 1, 1--22 Y1 - 1999 A1 - Alberto Bressan A1 - Tai-Ping Liu A1 - Tong Yang AB - Let $u_t+f(u)_x=0$ be a strictly hyperbolic $n\\\\times n$ system of conservation laws, each characteristic field being linearly degenerate or genuinely nonlinear. In this paper we explicitly define a functional $\\\\Phi=\\\\Phi(u,v)$, equivalent to the $L^1$ distance, which is `almost decreasing\\\', i.e., $\\\\Phi(u(t),v(t))-\\\\Phi(u(s),v(s))\\\\leq\\\\break O (\\\\epsilon)·(t-s)$ for all $t>s\\\\geq 0$, for every pair of $\\\\epsilon$-approximate solutions $u,v$ with small total variation, generated by a wave-front-tracking algorithm. The small parameter $\\\\epsilon$ here controls the errors in the wave speeds, the maximum size of rarefaction fronts and the total strength of all non-physical waves in $u$ and in $v$. From the above estimate, it follows that front-tracking approximations converge to a unique limit solution, depending Lipschitz continuously on the initial data, in the $L^1$ norm. This provides a new proof of the existence of the standard Riemann semigroup generated by an $n\\\\times n$ system of conservation laws.\\\'\\\' PB - Springer UR - http://hdl.handle.net/1963/3373 U1 - 957 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Oleinik type estimates and uniqueness for n x n conservation laws JF - J. Differential Equations 156 (1999), no. 1, 26--49 Y1 - 1999 A1 - Alberto Bressan A1 - Paola Goatin AB - Let $u_t+f(u)_x=0$ be a strictly hyperbolic $n\\\\times n$ system of conservation laws in one space dimension. Relying on the existence of a semigroup of solutions, we first establish the uniqueness of entropy admissible weak solutions to the Cauchy problem, under a mild assumption on the local oscillation of $u$ in a forward neighborhood of each point in the $t\\\\text{-}x$ plane. In turn, this yields the uniqueness of weak solutions which satisfy a decay estimate on positive waves of genuinely nonlinear families, thus extending a classical result proved by Oleĭnik in the scalar case. PB - Elsevier UR - http://hdl.handle.net/1963/3375 U1 - 955 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Structural stability and regularity of entropy solutions to hyperbolic systems of conservation laws JF - Indiana Univ. Math. J. 48 (1999), no. 1, 43--84 Y1 - 1999 A1 - Alberto Bressan A1 - Philippe G. LeFloch AB - The paper is concerned with the qualitative structure of entropy solutions to a strictly hyperbolic, genuinely nonlinear system of conservation laws. We first give an accurate description of the local and global wave-front structure of a BV solution, generated by a front tracking algorithm. PB - Indiana University UR - http://hdl.handle.net/1963/3374 U1 - 956 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 - Error bounds for a deterministic version of the Glimm scheme JF - Arch. Rational Mech. Anal. 142 (1998), no. 2, 155-176 Y1 - 1998 A1 - Andrea Marson A1 - Alberto Bressan AB - Consider the hyperbolic system of conservation laws $u_t F(u)_x=0. Let $u$ be the unique viscosity solution with initial condition $u(0,x)=\\\\bar u(x)$ and let $u^\\\\varepsilon$ be an approximate solution constructed by the Glimm scheme, corresponding to the mesh sizes $\\\\Delta x,\\\\Delta t=O(\\\\Delta x). With a suitable choise of the sampling sequence, we prove the estimate $$ \\\\left\\\\Vert u^\\\\varepsilon(t,\\\\cdot)-u(t,\\\\cdot) \\\\right\\\\Vert_1=o(1)\\\\cdot\\\\sqrt{\\\\Delta x}\\\\vert\\\\ln\\\\Delta x\\\\vert. $$ PB - Springer UR - http://hdl.handle.net/1963/1045 U1 - 2811 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A generic classification of time-optimal planar stabilizing feedbacks JF - SIAM J. Control Optim. 36 (1998) 12-32 Y1 - 1998 A1 - Alberto Bressan A1 - Benedetto Piccoli AB - Consider the problem of stabilization at the origin in minimum time for a planar control system affine with respect to the control. For a family of generic vector fields, a topological equivalence relation on the corresponding time-optimal feedback synthesis was introduced in a previous paper [Dynamics of Continuous, Discrete and Impulsive Systems, 3 (1997), pp. 335--371]. The set of equivalence classes can be put in a one-to-one correspondence with a discrete family of graphs. This provides a classification of the global structure of generic time-optimal stabilizing feedbacks in the plane, analogous to the classification of smooth dynamical systems developed by Peixoto. PB - SISSA Library UR - http://hdl.handle.net/1963/998 U1 - 2858 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Uniqueness for discontinuous ODE and conservation laws JF - Nonlinear Analysis 34 (1998) 637-652 Y1 - 1998 A1 - Alberto Bressan A1 - Wen Shen AB - Consider a scalar O.D.E. of the form $\\\\dot x=f(t,x),$ where $f$ is possibly discontinuous w.r.t. both variables $t,x$. Under suitable assumptions, we prove that the corresponding Cauchy problem admits a unique solution, which depends H\\\\\\\"older continuously on the initial data.\\nOur result applies in particular to the case where $f$ can be written in the form $f(t,x)\\\\doteq g\\\\big( u(t,x)\\\\big)$, for some function $g$ and some solution $u$ of a scalar conservation law, say $u_t+F(u)_x=0$. In turn, this yields the uniqueness and continuous dependence of solutions to a class of $2\\\\times 2$ strictly hyperbolic systems, with initial data in $\\\\L^\\\\infty$. PB - Elsevier UR - http://hdl.handle.net/1963/3699 U1 - 606 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - The semigroup generated by a temple class system with large data JF - Differential Integral Equations 10 (1997), no. 3, 401-418 Y1 - 1997 A1 - Paolo Baiti A1 - Alberto Bressan AB - We consider the Cauchy problem $$u_t + [F(u)]_x=0, u(0,x)=\\\\bar u(x) (*)$$ for a nonlinear $n\\\\times n$ system of conservation laws with coinciding shock and rarefaction curves. Assuming the existence of a coordinates system made of Riemann invariants, we prove the existence of a weak solution of (*) that depends in a lipschitz continuous way on the initial data, in the class of functions with arbitrarily large but bounded total variation. PB - SISSA Library UR - http://hdl.handle.net/1963/1023 U1 - 2833 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Shift-differentiability of the flow generated by a conservation law JF - Discrete Contin. Dynam. Systems 3 (1997), no. 1, 35--58. Y1 - 1997 A1 - Alberto Bressan A1 - Graziano Guerra AB - The paper introduces a notion of \\\"shift-differentials\\\" for maps with values in the space BV. These differentials describe first order variations of a given functin $u$, obtained by horizontal shifts of the points of its graph. The flow generated by a scalar conservation law is proved to be generically shift-differentiable, according to the new definition. PB - SISSA Library UR - http://hdl.handle.net/1963/1033 U1 - 2823 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Structural stability for time-optimal planar sytheses JF - Dynam. Contin. Discrete Impuls. Systems 3 (1997), no. 3, 335--371 Y1 - 1997 A1 - Alberto Bressan A1 - Benedetto Piccoli PB - SISSA Library UR - http://hdl.handle.net/1963/997 U1 - 2859 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - The semigroup approach to systems of conservation laws JF - Mat. Contemp. 10 (1996) 21-74 Y1 - 1996 A1 - Alberto Bressan PB - Sociedade Brasileira de Matematica UR - http://hdl.handle.net/1963/1037 U1 - 2819 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Unique solutions of 2x2 conservation laws with large data JF - Indiana Univ. Math. J. 44 (1995), no. 3, 677-725 Y1 - 1995 A1 - Alberto Bressan A1 - Rinaldo M. Colombo AB - For a 2x2 hyperbolic system of conservation laws, we first consider a Riemann problem with arbitrarily large data. A stability assumption is introduced, which yields the existence of a Lipschitz semigroup of solutions, defined on a domain containing all suitably small BV perturbations of the Riemann data. We then establish a uniqueness result for large BV solutions, valid within the same class of functions where a local existence theorem can be proved. PB - Indiana University Mathematics Journal UR - http://hdl.handle.net/1963/975 U1 - 3479 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A class of absolute retracts of dwarf spheroidal galaxies JF - Proc.Amer.Math.Soc. 112 (1991), no.2, 413 Y1 - 1991 A1 - Alberto Bressan A1 - Arrigo Cellina A1 - Andrzej Fryszkowski PB - SISSA Library UR - http://hdl.handle.net/1963/837 U1 - 2954 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Existence and continuous dependence for discontinuous O.D.E.s JF - Boll. Un. Mat. Ital. B (7) 4 (1990), no. 2, 295--311 Y1 - 1990 A1 - Alberto Bressan A1 - Giovanni Colombo PB - SISSA Library UR - http://hdl.handle.net/1963/716 U1 - 3210 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Upper semicontinuous differential inclusions without convexity JF - Proc. Amer. Math. Soc. 106 (1989), no. 3, 771-775 Y1 - 1989 A1 - Alberto Bressan A1 - Arrigo Cellina A1 - Giovanni Colombo PB - SISSA Library UR - http://hdl.handle.net/1963/670 U1 - 3256 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - On differential systems with vector-valued impulsive controls. JF - Boll. Un. Mat. Ital. B (7) 2 (1988), no. 3, 641-656 Y1 - 1988 A1 - Alberto Bressan A1 - Franco Rampazzo PB - SISSA Library UR - http://hdl.handle.net/1963/535 U1 - 3369 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Generalized Baire category and differential inclusions in Banach spaces. JF - J. Differential Equations 76 (1988), no. 1, 135-158. Y1 - 1988 A1 - Alberto Bressan A1 - Giovanni Colombo PB - SISSA Library UR - http://hdl.handle.net/1963/538 U1 - 3366 U2 - Mathematics U3 - Functional Analysis and Applications ER -