TY - JOUR
T1 - A Uniqueness Condition for Hyperbolic Systems of Conservation Laws
JF - Discrete Contin. Dynam. Systems 6 (2000) 673-682
Y1 - 2000
A1 - Alberto Bressan
A1 - Marta Lewicka
AB - 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.
PB - American Institute of Mathematical Sciences
UR - http://hdl.handle.net/1963/3195
U1 - 1106
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - Uniqueness for discontinuous ODE and conservation laws
JF - Nonlinear Analysis 34 (1998) 637-652
Y1 - 1998
A1 - Alberto Bressan
A1 - Wen Shen
AB - 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 $\\\\L^\\\\infty$.
PB - Elsevier
UR - http://hdl.handle.net/1963/3699
U1 - 606
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - Unique solutions of 2x2 conservation laws with large data
JF - Indiana Univ. Math. J. 44 (1995), no. 3, 677-725
Y1 - 1995
A1 - Alberto Bressan
A1 - Rinaldo M. Colombo
AB - 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.
PB - Indiana University Mathematics Journal
UR - http://hdl.handle.net/1963/975
U1 - 3479
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - Upper semicontinuous differential inclusions without convexity
JF - Proc. Amer. Math. Soc. 106 (1989), no. 3, 771-775
Y1 - 1989
A1 - Alberto Bressan
A1 - Arrigo Cellina
A1 - Giovanni Colombo
PB - SISSA Library
UR - http://hdl.handle.net/1963/670
U1 - 3256
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -