A PDE model for unidirectional flows: stationary profiles and asymptotic behaviour (Q2075906)

The authors study a continuum model for unidirectional movement of pedestrians in a corridor-like domain with an entrance and an exit at each end. By assuming symmetry of the solution and averaging over the cross section the dynamics is reduced to partial differential equation in one space dimension and by looking at stationary profiles, i.e. time-independent solutions one arrives at a singularly perturbed ODE boundary value problem \[ \begin{aligned} j'(x) &= -g(\xi) j\\ \xi'(x) & = 1 \\ \varepsilon \rho'(x) & = \rho(1-\rho)-j \end{aligned} \] on the interval \(x\in [0,1]\) with the slightly unusual boundary conditions \[ \begin{aligned} j(0) &= \alpha(1-\rho(0)),\\ j(1) &= \beta \rho(1). \end{aligned} \] Here \(\alpha\) and \(\beta\) describe the inflow resp. outflow and \(g(\xi)=\frac{k'(\xi)}{k(x)}\) with \(k(x)>0\) and \(k'(x)<0\) is linked to the cross sectional area. Using geometric singular perturbation theory the (slow) reduced system on the critical manifold \(\mathcal{C}_0\) and the (fast) layer system are constructed in the limit \(\varepsilon=0\). Starting from the line \(\mathcal{L}\) that corresponds to the boundary condition at \(x=0\) singular solutions are constructed as a concatenation of solutions of the reduced system and of the layer system such that at \(x=1\) the line \(\mathcal{R}\) corresponding to the right boundary condition is reached. Depending on the mutual position of \(\mathcal{L}\), \(\mathcal{R}\) and the critical manifold \(\mathcal{C}_0\) (which in turn depends on the choice of \(\alpha\) and \(\beta\)) there are six different regions in the \(\alpha\)-\(\beta\) parameter space with six different tpyes of singular solutions. The main result of the article states that for \(\varepsilon>0\) sufficiently small there exists a unique solution of the singularly perturbed boundary value problem which is close to one of the singular solutions. The proof is based on a shooting argument where the line \(\mathcal{L}\) corresponding to the left boundary condition at \(x=0\) is transported by the flow to \(x=1\) along invariant manifolds and intersects the line \(\mathcal{R}\) transversally. The theoretical part is complemented by some interesting numerical simulations. A comparison between the averaged 1-D model and the full 2-D model shows that in some regions the agreeement is good but that there exist also large discrepancies. In the 1-D model the six different solution types from the theorem can be found for small \(\varepsilon\) and \(k(x)=2-x\). However, if the assumption that \(k\) is monotone is dropped and \(g\) may change its sign the simulations show a more complicated picture involving canard solutions and a very sensitive dependence on the parameters \(\alpha\) and \(\beta\).