The Cauchy problem for a class of \(2\times 2\) hyperbolic systems of conservation laws with unbounded propagation speed (Q1916350)

This paper deals with quasilinear systems \[ r_t+\lambda \biggl(\bigl(r(t,x), s(t,x)\bigr)\biggr) r_x=0 \quad s_t+ \mu\biggl(\bigl(r(t,x), s(t,x)\bigr)\biggr) s_x=0, \tag{\(*\)} \] where \((r,s)= (r_0(x), s_0(x))\) at \(t=0\), and \(\lambda (\rho,\sigma)\), \(\mu(\rho,\sigma)\) are real continuous functions defined on some interval \(I_r\times I_s\). After the work of P. Lax, J. Johnson, M. Yamaguti and T. Nishida, we know that for any pair of monotone increasing initial data \((r_0, s_0)\) such that \(\{(r_0(x), s_0(x)):x \in \mathbb{R}\}\) is compact in \(I_r \times I_s\), system (*) has a unique global continuous solution provided that \(\lambda,\mu\) are \(C^1\), \(\mu(\rho,\sigma) < \lambda (\rho,\sigma)\), and \(\lambda(\rho,\sigma)\), \(\mu (\rho,\sigma)\) are increasing, respectively, in \(\rho\) and in \(\sigma\). Here the authors drop the compactness assumption on the range of initial data, and consider the degenerate case wherein the functions \(\lambda, \mu\) are only continuous and a weak hyperbolicity condition \(\mu(\rho,\sigma) \leq A\leq\lambda (\rho,\sigma)\) holds for some constant \(A\). In such a situation, both existence and uniqueness may fail. However, the existence of a global continuous solution can be proved under the following additional assumptions: \(\lambda\) and \(\mu\) are decreasing in \(\sigma\) and \(\rho\) respectively; the equalities \(\lambda(\rho, \sigma)=A\) and \(\mu(\rho,\sigma) =A\) may hold only for \(\rho=\inf I_r\) and \(\sigma= \sup I_s\) respectively; \(\sup_\sigma \inf_\rho \lambda (\rho,\sigma) \in \mathbb{R}\) if \(\inf r_0(x) \notin I_r\) and \(\inf_\rho \sup_\sigma\mu (\rho,\sigma) \in \mathbb{R}\) if \(\sup s_0(x)\notin I_s\). Such a solution \((r(t,x), s(t,x))\) is a ``characteristic solution'', in the sense that \(r\) and \(s\) are constant along the right and (respectively) the left characteristic curves. In order to prove the uniqueness of the characteristic solutions, some conditions on the growth of the initial propagation speeds are needed. These results apply in particular to the conservation laws \(\partial_tu_j +\partial_x [F_j(u_1(t,x),u_2(t,x))] =0\), \(j=1,2\), with \(F_1(u)= -f(u_2)\), \(f'\) increasing, \(F_2(u)= -u_1\).