Let $u_t+f(u)_x=0$ 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.

}, url = {http://hdl.handle.net/1963/3256}, author = {Alberto Bressan and Paola Goatin} } @article {1999, title = {Structural stability and regularity of entropy solutions to hyperbolic systems of conservation laws}, journal = {Indiana Univ. Math. J. 48 (1999), no. 1, 43--84}, number = {SISSA;96/97/M}, year = {1999}, publisher = {Indiana University}, abstract = {The 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.}, doi = {10.1512/iumj.1999.48.1524}, url = {http://hdl.handle.net/1963/3374}, author = {Alberto Bressan and Philippe G. LeFloch} } @article {1997, title = {The semigroup generated by a temple class system with large data}, journal = {Differential Integral Equations 10 (1997), no. 3, 401-418}, number = {SISSA;121/95/M}, year = {1997}, publisher = {SISSA Library}, abstract = {We 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.}, url = {http://hdl.handle.net/1963/1023}, author = {Paolo Baiti and Alberto Bressan} } @article {1997, title = {Shift-differentiability of the flow generated by a conservation law}, journal = {Discrete Contin. Dynam. Systems 3 (1997), no. 1, 35--58.}, number = {SISSA;131/95/M}, year = {1997}, publisher = {SISSA Library}, abstract = {The 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.}, url = {http://hdl.handle.net/1963/1033}, author = {Alberto Bressan and Graziano Guerra} } @article {1997, title = {Structural stability for time-optimal planar sytheses}, journal = {Dynam. Contin. Discrete Impuls. Systems 3 (1997), no. 3, 335--371}, number = {SISSA;95/95/M}, year = {1997}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/997}, author = {Alberto Bressan and Benedetto Piccoli} } @article {1996, title = {The semigroup approach to systems of conservation laws}, journal = {Mat. Contemp. 10 (1996) 21-74}, number = {SISSA;135/95/M}, year = {1996}, publisher = {Sociedade Brasileira de Matematica}, url = {http://hdl.handle.net/1963/1037}, author = {Alberto Bressan} }