@article {2000, title = {A Uniqueness Condition for Hyperbolic Systems of Conservation Laws}, journal = {Discrete Contin. Dynam. Systems 6 (2000) 673-682}, number = {SISSA;88/98/M}, year = {2000}, publisher = {American Institute of Mathematical Sciences}, abstract = {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.}, url = {http://hdl.handle.net/1963/3195}, author = {Alberto Bressan and Marta Lewicka} } @article {1998, title = {Uniqueness for discontinuous ODE and conservation laws}, journal = {Nonlinear Analysis 34 (1998) 637-652}, number = {SISSA;26/97/M}, year = {1998}, publisher = {Elsevier}, abstract = {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 $\\\\L^\\\\infty$.}, doi = {10.1016/S0362-546X(97)00590-7}, url = {http://hdl.handle.net/1963/3699}, author = {Alberto Bressan and Wen Shen} } @article {1995, title = {Unique solutions of 2x2 conservation laws with large data}, journal = {Indiana Univ. Math. J. 44 (1995), no. 3, 677-725}, number = {SISSA;73/95/M}, year = {1995}, publisher = {Indiana University Mathematics Journal}, abstract = {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.}, doi = {10.1512/iumj.1995.44.2004}, url = {http://hdl.handle.net/1963/975}, author = {Alberto Bressan and Rinaldo M. Colombo} } @article {1989, title = {Upper semicontinuous differential inclusions without convexity}, journal = {Proc. Amer. Math. Soc. 106 (1989), no. 3, 771-775}, number = {SISSA;74/88/M}, year = {1989}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/670}, author = {Alberto Bressan and Arrigo Cellina and Giovanni Colombo} }