introduced. In the TASEP/LK, for example, we have $S(\rho)=\Omega_a(1-\rho)-\Omega_d\rho$. In the continuum limit we neglect terms of $\mathcal{O}(1/L)$ so that (\[dJdx\]) becomes an ordinary first order differential equation in the continuous variable $x$. The system, however, has at least two initial conditions (e.g. the boundary conditions in the homogeneous case), thus it is overdetermined. Each initial condition at a point $x_0$ is associated with one solution of the differential equation (\[dJdx\]) $J_{x_0}(x)$ for the current and $\rho_{x_0}(x)$ for density, respectively. We call the mathematical solutions to single initial conditions $J_{x_0}(x)$ and $\rho_{x_0}(x)$ *local current/density profiles*. Physically these solutions are not necessarily realized.

For the TASEP/LK with a single defect it was shown by Pierobon et al. [@pff_1def] that the finite transport capacity at the defect site, corresponding to a local upper bound of the current, can be regarded as an additional condition on the current profile. They argued that the local solution of (\[dJdx\]) with the initial condition $J(x_d)=J^*(x_d)$ becomes relevant if the local current profiles of the boundary conditions exceed $J^*$ at the defect site. Here we want to justify this approach and generalize it to a larger class of driven lattice gases with many defects, including randomly disordered 