00841nas 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/1963/319501052nas 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/1963/369900871nas 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/1963/97500413nas 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/1963/670