This paper is concerned with the Cauchy problem $$(*)~~~~~~~~~~~~~~~u_t+[F(u)]_x=g(t,x,u),\quad u(0,x)=\bar{u}(x),~~~~~~~~~~~~~~~~~~$$ for a nonlinear $2\times 2$ hyperbolic system of inhomogeneous balance laws in one space dimension. As usual, we assume that the system is strictly hyperbolic and that each characteristic field is either linearly degenerate or genuinely nonlinear. \par Under suitable assumptions on $g$, we prove that there exists $T>0$ such that, for every $\bar{u}$ with sufficiently small total variation, the Cauchy problem ($*$) has a unique ``viscosity solution'', defined for $t\in [0,T]$, depending continuously on the initial data.

Viscosity solutions and uniqueness for systems of inhomogeneous balance laws / Crasta, Graziano; Benedetto, Piccoli. - In: DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. - ISSN 1078-0947. - 3:4(1997), pp. 477-502. [10.3934/dcds.1997.3.477]

Viscosity solutions and uniqueness for systems of inhomogeneous balance laws

CRASTA, Graziano;
1997

Abstract

This paper is concerned with the Cauchy problem $$(*)~~~~~~~~~~~~~~~u_t+[F(u)]_x=g(t,x,u),\quad u(0,x)=\bar{u}(x),~~~~~~~~~~~~~~~~~~$$ for a nonlinear $2\times 2$ hyperbolic system of inhomogeneous balance laws in one space dimension. As usual, we assume that the system is strictly hyperbolic and that each characteristic field is either linearly degenerate or genuinely nonlinear. \par Under suitable assumptions on $g$, we prove that there exists $T>0$ such that, for every $\bar{u}$ with sufficiently small total variation, the Cauchy problem ($*$) has a unique ``viscosity solution'', defined for $t\in [0,T]$, depending continuously on the initial data.
1997
continuous dependence; systems of conservation laws; uniqueness
01 Pubblicazione su rivista::01a Articolo in rivista
Viscosity solutions and uniqueness for systems of inhomogeneous balance laws / Crasta, Graziano; Benedetto, Piccoli. - In: DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. - ISSN 1078-0947. - 3:4(1997), pp. 477-502. [10.3934/dcds.1997.3.477]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/48747
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