We study the detailed process of bifurcation in the flow’s topological structure for a two-dimensional (2-D) incompressible flow subject to no-slip boundary conditions and its connection with boundary-layer separation. The boundary-layer separation theory of M. Ghil, T. Ma and S. Wang, based on the structural-bifurcation concept, is translated into vorticity form. The vorticity formulation of the theory shows that structural bifurcation occurs whenever a degenerate singular point for the vorticity appears on the boundary; this singular point is characterized by nonzero tangential second-order derivative and nonzero time derivative of the vorticity. Furthermore, we prove the presence of an adverse pressure gradient at the critical point, due to reversal in the direction of the pressure force with respect to the basic shear flow at this point. A numerical example of 2-D driven-cavity flow, governed by the Navier Stokes equations, is presented; boundary-layer separation occurs, the bifurcation criterion is satisfied, and an adverse pressure gradient is shown to be present.

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.

We obtain the existence and uniqueness of solutions for a class of retarded parabolic equations with superlinear perturbations. The asymptotic behavior result is studied by using the pullback attractor framework.

Although the decay in time estimates of the semi-group generated by the linearized Boltzmann operator without forcing have been well established, there is no corresponding result for the case with general external force. This paper is mainly concerned with the optimal decay estimates on the solution operator in some weighted Sobolev spaces for the linearized Boltzmann equation with a time dependent external force. No time decay assumption is made on the force. The proof is based on both the energy method through the macro-micro decomposition and the L p-L q estimates from the spectral analysis. The decay estimates thus obtained are applied to the study on the global existence of the Cauchy problem to the nonlinear Boltzmann equation with time dependent external force and source. Precisely, for space dimension n>= 3, the global existence and decay rates of solutions to the Cauchy problem are

This work is devoted to study the global behavior of viscous flows contained in a symmetric domain with complete slip boundary. In such scenario the boundary no longer provides friction and therefore the perturbation of angular velocity lacks decaying structure. In fact, we show the existence of uniformly rotating solutions as steady states for the compressible Navier-Stokes equations. By manipulating the conservation law of angular momentum, we establish a suitable Korn's type inequality to control the perturbation and show the asymptotic stability of the uniformly rotating solutions with small angular velocity. In particular, the initial perturbation which preserves the angular momentum would decay exponentially in time and the solution to the Navier-Stokes equations converges to the steady state as time grows up.