01264nas 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/1963/307400387nas 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/1963/1238