On the Cauchy problem of transportation equations (Q1367242)

We consider the transportation system \[ \rho_t+(\rho u)_x= 0,\quad (\rho u)_t+(\rho u^2)_x= 0, \] with initial data \((\rho, u)|_{t=0}= (\rho_0(x),u_0(x))\), \(\rho_0(x)>0\) a.e., where \(\rho\) and \(u\) are density and velocity, respectively. This system is non-strictly hyperbolic with a double eigenvalue \(\lambda_1=\lambda_2= u\), both \(\lambda_1\), \(\lambda_2\) being linearly degenerate. We introduce a new notion of generalized solutions from the physical point of view and prove the global existence of such solutions.