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A computational method, based on l1-norm minimization, is proposed for the problem traffic flow correction. Without extra information, the problem is generally ill-posed when a large portion of the link sensors are unhealthy. It is possible, however, to correct the corruptions accurately if there are only a few bad sensors which are located at certain links by the proposed method. We mathematically quantify these links that are robust to miscounts and relate them to the geometric structure of the traffic network. In a more realistic setting, besides the unhealthy link sensors, if small measure noise are present at the other sensors, our method guarantees to give an estimated traffic flow fairly close to the ground-truth. Both toy and real-world examples are provided to demonstrate the effectiveness of the proposed method.

In this paper, we discuss the waiting time phenomena for a class of non-Newtonian polytropic filtration equation with convection. According to the influence of the convection on the waiting time property, the convection is classified into three classes, that is, the strong convection, the mild convection and the weak convection. For different cases, the sufficient and necessary conditions on the initial data are given for the existence of waiting time respectively.

We study linear and nonlinear Schrodinger equations defined by fractal measures. Under the assumption that the Laplacian has compact resolvent, we prove that there exists a uniqueness weak solution for a linear Schrodinger equation, and then use it to obtain the existence and uniqueness of weak solution of a nonlinear Schrodinger equation. We prove that for a class of self-similar measures on R with overlaps, the Schrodinger equations can be discretized so that the finite element method can be applied to obtain approximate solutions. We also prove that the numerical solutions converge to the actual solution and obtain the rate of convergence.

The zero dissipation limit of the compressible heat-conducting NavierStokes equations in the presence of the shock is investigated. It is shown that when the heat conduction coefficient and the viscosity coefficient satisfy = O (), = O (), c> 0, as 0 (see (1.3)), if the solution of the corresponding Euler equations is piecewise smooth with shock wave satisfying the Lax entropy condition, then there exists a smooth solution to the NavierStokes equations, which converges to the piecewise smooth shock solution of the Euler equations away from the shock discontinuity at a rate of . The proof is given by a combination of the energy estimates and the matched asymptotic analysis introduced in [3].