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.

1 aBianchini, Stefano1 aCavalletti, Fabio1 aBressan, Alberto1 aChen, Gui-Qiang, G.1 aLewicka, Marta1 aWang, Dehua uhttps://www.math.sissa.it/publication/monge-problem-geodesic-spaces00806nas a2200121 4500008004300000245007400043210006700117260004800184520037100232100002100603700002400624856003600648 2009 en_Ud 00aOn the convergence of viscous approximations after shock interactions0 aconvergence of viscous approximations after shock interactions bAmerican Institute of Mathematical Sciences3 aWe 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.1 aBressan, Alberto1 aDonadello, Carlotta uhttp://hdl.handle.net/1963/341201700nas a2200121 4500008004300000245004200043210004200085520136200127100002101489700001601510700001601526856003601542 2007 en_Ud 00aAsymptotic variational wave equations0 aAsymptotic variational wave equations3 aWe 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.1 aBressan, Alberto1 aPing, Zhang1 aYuxi, Zheng uhttp://hdl.handle.net/1963/218200820nas a2200121 4500008004300000245004900043210004800092520045600140100001700596700002100613700002800634856003600662 2007 en_Ud 00aBV instability for the Lax-Friedrichs scheme0 aBV instability for the LaxFriedrichs scheme3 aIt 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.1 aBaiti, Paolo1 aBressan, Alberto1 aJenssen, Helge Kristian uhttp://hdl.handle.net/1963/233500894nas a2200109 4500008004300000245005300043210005300096520056000149100001800709700002100727856003600748 2007 en_Ud 00aNearly time optimal stabilizing patchy feedbacks0 aNearly time optimal stabilizing patchy feedbacks3 aWe 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 $\\\\tau(y)+\\\\ve$.1 aAncona, Fabio1 aBressan, Alberto uhttp://hdl.handle.net/1963/218500638nas a2200109 4500008004300000245006800043210006800111520027600179100002100455700001600476856003600492 2006 en_Ud 00aConservative Solutions to a Nonlinear Variational Wave Equation0 aConservative Solutions to a Nonlinear Variational Wave Equation3 aWe 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.1 aBressan, Alberto1 aYuxi, Zheng uhttp://hdl.handle.net/1963/218400794nas a2200109 4500008004300000245005500043210005500098520044900153100002100602700002500623856003600648 2006 en_Ud 00aInfinite Horizon Noncooperative Differential Games0 aInfinite Horizon Noncooperative Differential Games3 aFor 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.1 aBressan, Alberto1 aPriuli, Fabio Simone uhttp://hdl.handle.net/1963/172000891nas a2200121 4500008004300000245004100043210003800084520054500122100002100667700002800688700001700716856003600733 2006 en_Ud 00aAn instability of the Godunov scheme0 ainstability of the Godunov scheme3 aWe 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.1 aBressan, Alberto1 aJenssen, Helge Kristian1 aBaiti, Paolo uhttp://hdl.handle.net/1963/218300648nas a2200109 4500008004300000245006600043210005800109520029500167100002100462700001900483856003600502 2005 en_Ud 00aOn the Blow-up for a Discrete Boltzmann Equation in the Plane0 aBlowup for a Discrete Boltzmann Equation in the Plane3 aWe 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.1 aBressan, Alberto1 aFonte, Massimo uhttp://hdl.handle.net/1963/224400651nas a2200109 4500008004300000245005100043210005000094520031700144100002100461700002300482856003600505 2005 en_Ud 00aGlobal solutions of the Hunter-Saxton equation0 aGlobal solutions of the HunterSaxton equation3 aWe 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.1 aBressan, Alberto1 aConstantin, Adrian uhttp://hdl.handle.net/1963/225600982nas a2200109 4500008004300000245008000043210006900123520060400192100002100796700001900817856003600836 2005 en_Ud 00aAn Optimal Transportation Metric for Solutions of the Camassa-Holm Equation0 aOptimal Transportation Metric for Solutions of the CamassaHolm E3 aIn 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.1 aBressan, Alberto1 aFonte, Massimo uhttp://hdl.handle.net/1963/171901264nas a2200121 4500008004300000245006600043210006600109260002600175520086100201100002301062700002101085856003601106 2005 en_Ud 00aVanishing viscosity solutions of nonlinear hyperbolic systems0 aVanishing viscosity solutions of nonlinear hyperbolic systems bAnnals of Mathematics3 aWe 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 $u_t+f(u)_x=0$.1 aBianchini, Stefano1 aBressan, Alberto uhttp://hdl.handle.net/1963/307401240nas a2200121 4500008004300000245006600043210005900109260001000168520086800178100002101046700001501067856003601082 2004 en_Ud 00aOn the convergence rate of vanishing viscosity approximations0 aconvergence rate of vanishing viscosity approximations bWiley3 aGiven 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.1 aBressan, Alberto1 aYang, Tong uhttp://hdl.handle.net/1963/291501056nas a2200121 4500008004300000245005500043210005400098260001300152520069800165100002100863700001400884856003600898 2004 en_Ud 00aSemi-cooperative strategies for differential games0 aSemicooperative strategies for differential games bSpringer3 aThe 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.1 aBressan, Alberto1 aShen, Wen uhttp://hdl.handle.net/1963/289300741nas a2200121 4500008004300000245005600043210005400099260000900153520038500162100002100547700001500568856003600583 2004 en_Ud 00aA sharp decay estimate for positive nonlinear waves0 asharp decay estimate for positive nonlinear waves bSIAM3 aWe 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.1 aBressan, Alberto1 aYang, Tong uhttp://hdl.handle.net/1963/291600818nas a2200121 4500008004300000245007100043210006900114260000900183520043300192100002100625700001400646856003600660 2004 en_Ud 00aSmall BV solutions of hyperbolic noncooperative differential games0 aSmall BV solutions of hyperbolic noncooperative differential gam bSIAM3 aThe 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.1 aBressan, Alberto1 aShen, Wen uhttp://hdl.handle.net/1963/291700736nas a2200109 4500008004300000245006600043210006600109260003500175520035900210100002100569856003600590 2004 en_Ud 00aSome remarks on multidimensional systems of conservation laws0 aSome remarks on multidimensional systems of conservation laws bAccademia Nazionale dei Lincei3 aThis 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.1 aBressan, Alberto uhttp://hdl.handle.net/1963/364200882nas a2200121 4500008004300000245004500043210004500088260001700133520053500150100001800685700002100703856003600724 2004 en_Ud 00aStability rates for patchy vector fields0 aStability rates for patchy vector fields bEDP Sciences3 aThis 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.1 aAncona, Fabio1 aBressan, Alberto uhttp://hdl.handle.net/1963/295900871nas a2200109 4500008004300000245008000043210006900123260002600192520048600218100002100704856003600725 2003 en_Ud 00aAn ill posed Cauchy problem for a hyperbolic system in two space dimensions0 aill posed Cauchy problem for a hyperbolic system in two space di bUniversità di Padova3 aThe 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.1 aBressan, Alberto uhttp://hdl.handle.net/1963/291300867nas a2200109 4500008004300000245005900043210005700102260002600159520051500185100002100700856003600721 2003 en_Ud 00aA lemma and a conjecture on the cost of rearrangements0 alemma and a conjecture on the cost of rearrangements bUniversità di Padova3 aConsider 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.1 aBressan, Alberto uhttp://hdl.handle.net/1963/291400426nas a2200121 4500008004100000245007300041210006900114260001800183100002100201700001800222700002800240856003600268 2003 en d00aSome results on the boundary control of systems of conservation laws0 aSome results on the boundary control of systems of conservation bSISSA Library1 aBressan, Alberto1 aAncona, Fabio1 aCoclite, Giuseppe Maria uhttp://hdl.handle.net/1963/161500716nas a2200121 4500008004300000245006000043210005300103260000900156520034400165100002100509700002800530856003600558 2002 en_Ud 00aOn the Boundary Control of Systems of Conservation Laws0 aBoundary Control of Systems of Conservation Laws bSIAM3 aThe 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.1 aBressan, Alberto1 aCoclite, Giuseppe Maria uhttp://hdl.handle.net/1963/307000816nas a2200121 4500008004300000245005800043210005600101260004800157520040900205100002300614700002100637856003600658 2002 en_Ud 00aA center manifold technique for tracing viscous waves0 acenter manifold technique for tracing viscous waves bAmerican Institute of Mathematical Sciences3 aIn 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.1 aBianchini, Stefano1 aBressan, Alberto uhttp://hdl.handle.net/1963/307500809nas a2200121 4500008004300000245007700043210006900120260000900189520041400198100001800612700002100630856003600651 2002 en_Ud 00aFlow Stability of Patchy Vector Fields and Robust Feedback Stabilization0 aFlow Stability of Patchy Vector Fields and Robust Feedback Stabi bSIAM3 aThe 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.1 aAncona, Fabio1 aBressan, Alberto uhttp://hdl.handle.net/1963/307300788nas a2200121 4500008004100000245008600041210006900127260001300196520037700209100002300586700002100609856003600630 2002 en d00aOn a Lyapunov functional relating shortening curves and viscous conservation laws0 aLyapunov functional relating shortening curves and viscous conse bElsevier3 aWe 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.1 aBianchini, Stefano1 aBressan, Alberto uhttp://hdl.handle.net/1963/133700359nas a2200109 4500008004300000245004000043210003800083260004800121100002300169700002100192856003600213 2001 en_Ud 00aA case study in vanishing viscosity0 acase study in vanishing viscosity bAmerican Institute of Mathematical Sciences1 aBianchini, Stefano1 aBressan, Alberto uhttp://hdl.handle.net/1963/309101967nas a2200121 4500008004100000245006700041210006700108260001800175520158100193100002101774700001401795856003601809 2000 en d00aBV estimates for multicomponent chromatography with relaxation0 aBV estimates for multicomponent chromatography with relaxation bSISSA Library3 aWe 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.1 aBressan, Alberto1 aShen, Wen uhttp://hdl.handle.net/1963/133600394nas a2200109 4500008004300000245005900043210005900102260004300161100002300204700002100227856003600248 2000 en_Ud 00aBV solutions for a class of viscous hyperbolic systems0 aBV solutions for a class of viscous hyperbolic systems bIndiana University Mathematics Journal1 aBianchini, Stefano1 aBressan, Alberto uhttp://hdl.handle.net/1963/319400395nas a2200109 4500008004100000245007400041210006700115260001800182100002100200700002800221856003600249 2000 en d00aOn the convergence of Godunov scheme for nonlinear hyperbolic systems0 aconvergence of Godunov scheme for nonlinear hyperbolic systems bSISSA Library1 aBressan, Alberto1 aJenssen, Helge Kristian uhttp://hdl.handle.net/1963/147300947nas a2200121 4500008004300000245005900043210005800102260002300160520056700183100002100750700001800771856003600789 2000 en_Ud 00aStability of L^infty Solutions of Temple Class Systems0 aStability of Linfty Solutions of Temple Class Systems bKhayyam Publishing3 aLet $u_t+f(u)_x=0$ 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.

1 aBressan, Alberto1 aGoatin, Paola uhttp://hdl.handle.net/1963/325600841nas a2200121 4500008004300000245007100043210006900114260004800183520041200231100002100643700001900664856003600683 2000 en_Ud 00aA Uniqueness Condition for Hyperbolic Systems of Conservation Laws0 aUniqueness Condition for Hyperbolic Systems of Conservation Laws bAmerican Institute of Mathematical Sciences3 aConsider 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.1 aBressan, Alberto1 aLewicka, Marta uhttp://hdl.handle.net/1963/319500449nam a2200121 4500008004300000245008000043210006900123260003400192100002100226700002100247700002300268856003600291 2000 en_Ud 00aWell-posedness of the Cauchy problem for n x n systems of conservation laws0 aWellposedness of the Cauchy problem for n x n systems of conserv bAmerican Mathematical Society1 aBressan, Alberto1 aCrasta, Graziano1 aPiccoli, Benedetto uhttp://hdl.handle.net/1963/349500796nas a2200097 4500008004100000245004400041210004400085520051200129100002100641856003600662 1999 en d00aHyperbolic Systems of Conservation Laws0 aHyperbolic Systems of Conservation Laws3 aThis 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.1 aBressan, Alberto uhttp://hdl.handle.net/1963/185501453nas a2200133 4500008004300000245005600043210005500099260001300154520106200167100002101229700001801250700001501268856003601283 1999 en_Ud 00aL-1 stability estimates for n x n conservation laws0 aL1 stability estimates for n x n conservation laws bSpringer3 aLet $u_t+f(u)_x=0$ 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.\\\'\\\'1 aBressan, Alberto1 aLiu, Tai-Ping1 aYang, Tong uhttp://hdl.handle.net/1963/337300990nas a2200121 4500008004300000245007000043210006900113260001300182520059800195100002100793700001800814856003600832 1999 en_Ud 00aOleinik type estimates and uniqueness for n x n conservation laws0 aOleinik type estimates and uniqueness for n x n conservation law bElsevier3 aLet $u_t+f(u)_x=0$ 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.1 aBressan, Alberto1 aGoatin, Paola uhttp://hdl.handle.net/1963/337500739nas a2200121 4500008004300000245010400043210006900147260002300216520029600239100002100535700002500556856003600581 1999 en_Ud 00aStructural stability and regularity of entropy solutions to hyperbolic systems of conservation laws0 aStructural stability and regularity of entropy solutions to hype bIndiana University3 aThe 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.1 aBressan, Alberto1 aLeFloch, Philippe G. uhttp://hdl.handle.net/1963/337400387nas a2200109 4500008004100000245006900041210006900110260001800179100002300197700002100220856003600241 1999 en d00aVanishing viscosity solutions of hyperbolic systems on manifolds0 aVanishing viscosity solutions of hyperbolic systems on manifolds bSISSA Library1 aBianchini, Stefano1 aBressan, Alberto uhttp://hdl.handle.net/1963/123800924nas a2200121 4500008004100000245006500041210006500106260001300171520054200184100001900726700002100745856003600766 1998 en d00aError bounds for a deterministic version of the Glimm scheme0 aError bounds for a deterministic version of the Glimm scheme bSpringer3 aConsider 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 $$ \\\\left\\\\Vert u^\\\\varepsilon(t,\\\\cdot)-u(t,\\\\cdot) \\\\right\\\\Vert_1=o(1)\\\\cdot\\\\sqrt{\\\\Delta x}\\\\vert\\\\ln\\\\Delta x\\\\vert. $$1 aMarson, Andrea1 aBressan, Alberto uhttp://hdl.handle.net/1963/104501087nas a2200121 4500008004100000245007400041210006900115260001800184520068400202100002100886700002300907856003500930 1998 en d00aA generic classification of time-optimal planar stabilizing feedbacks0 ageneric classification of timeoptimal planar stabilizing feedbac bSISSA Library3 aConsider 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.1 aBressan, Alberto1 aPiccoli, Benedetto uhttp://hdl.handle.net/1963/99801052nas a2200121 4500008004300000245005900043210005900102260001300161520068500174100002100859700001400880856003600894 1998 en_Ud 00aUniqueness for discontinuous ODE and conservation laws0 aUniqueness for discontinuous ODE and conservation laws bElsevier3 aConsider 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 $\\\\L^\\\\infty$.1 aBressan, Alberto1 aShen, Wen uhttp://hdl.handle.net/1963/369900841nas a2200121 4500008004100000245006900041210006500110260001800175520045200193100001700645700002100662856003600683 1997 en d00aThe semigroup generated by a temple class system with large data0 asemigroup generated by a temple class system with large data bSISSA Library3 aWe 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.1 aBaiti, Paolo1 aBressan, Alberto uhttp://hdl.handle.net/1963/102300768nas a2200121 4500008004100000245007200041210006900113260001800182520036800200100002100568700002100589856003600610 1997 en d00aShift-differentiability of the flow generated by a conservation law0 aShiftdifferentiability of the flow generated by a conservation l bSISSA Library3 aThe 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.1 aBressan, Alberto1 aGuerra, Graziano uhttp://hdl.handle.net/1963/103300363nas a2200109 4500008004100000245005800041210005700099260001800156100002100174700002300195856003500218 1997 en d00aStructural stability for time-optimal planar sytheses0 aStructural stability for timeoptimal planar sytheses bSISSA Library1 aBressan, Alberto1 aPiccoli, Benedetto uhttp://hdl.handle.net/1963/99700349nas a2200097 4500008004100000245005900041210005500100260003900155100002100194856003600215 1996 en d00aThe semigroup approach to systems of conservation laws0 asemigroup approach to systems of conservation laws bSociedade Brasileira de Matematica1 aBressan, Alberto uhttp://hdl.handle.net/1963/103700871nas a2200121 4500008004100000245006200041210006200103260004300165520046100208100002100669700002400690856003500714 1995 en d00aUnique solutions of 2x2 conservation laws with large data0 aUnique solutions of 2x2 conservation laws with large data bIndiana University Mathematics Journal3 aFor 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.1 aBressan, Alberto1 aColombo, Rinaldo M. uhttp://hdl.handle.net/1963/97500404nas a2200121 4500008004100000245006200041210006000103260001800163100002100181700002000202700002500222856003500247 1991 en d00aA class of absolute retracts of dwarf spheroidal galaxies0 aclass of absolute retracts of dwarf spheroidal galaxies bSISSA Library1 aBressan, Alberto1 aCellina, Arrigo1 aFryszkowski, Andrzej uhttp://hdl.handle.net/1963/83700376nas a2200109 4500008004100000245006600041210006300107260001800170100002100188700002200209856003500231 1990 en d00aExistence and continuous dependence for discontinuous O.D.E.s0 aExistence and continuous dependence for discontinuous ODEs bSISSA Library1 aBressan, Alberto1 aColombo, Giovanni uhttp://hdl.handle.net/1963/71600413nas a2200121 4500008004100000245006700041210006700108260001800175100002100193700002000214700002200234856003500256 1989 en d00aUpper semicontinuous differential inclusions without convexity0 aUpper semicontinuous differential inclusions without convexity bSISSA Library1 aBressan, Alberto1 aCellina, Arrigo1 aColombo, Giovanni uhttp://hdl.handle.net/1963/67000375nas a2200109 4500008004100000245006700041210006200108260001800170100002100188700002100209856003500230 1988 en d00aOn differential systems with vector-valued impulsive controls.0 adifferential systems with vectorvalued impulsive controls bSISSA Library1 aBressan, Alberto1 aRampazzo, Franco uhttp://hdl.handle.net/1963/53500393nas a2200109 4500008004100000245007700041210006900118260001800187100002100205700002200226856003500248 1988 en d00aGeneralized Baire category and differential inclusions in Banach spaces.0 aGeneralized Baire category and differential inclusions in Banach bSISSA Library1 aBressan, Alberto1 aColombo, Giovanni uhttp://hdl.handle.net/1963/538