00794nas 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/1963/218300871nas 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/1963/2913