Global BV solutions and relaxation limit for a system of conservation laws (Q2710417)

The authors consider the Cauchy problem for the following strictly hyperbolic genuinely nonlinear system of conservation laws with relaxation NEWLINE\[NEWLINE u_t-v_x=0, \;v_t-\sigma(u)_x=\tfrac {1}{\varepsilon} r(u,v), \quad \varepsilon>0. NEWLINE\]NEWLINE Under some restrictions on the functions \(\sigma\) and \(r\) they firstly establish existence of global entropy BV-solutions \(u^\varepsilon\), \(v^\varepsilon\) to the Cauchy problem for the system above if initial functions have small enough BV-norm. Then the authors assume that there exists an equilibrium curve \(A(u)\) such that \(r(u,A(u))=0\) and study limit behavior of the solutions \(u^\varepsilon\), \(v^\varepsilon\) as the parameter \(\varepsilon\to 0+\). It is proved that a subsequence of \(u^\varepsilon\), \(v^\varepsilon\) can be extracted which converges to some \(u\), \(v\), and these limit functions \(u\), \(v\) satisfy the equilibrium equation NEWLINE\[NEWLINE u_t-A(u)_x=0, \qquad v=A(u).NEWLINE\]