Asymptotic Analysis

We prove convergence to a steady state of bounded solutions of the abstract first order semilinear Cauchy problem u_t+Lu+g(Ψ(u))Cu=0, t∈R₊, and of the second order semilinear Cauchy problem u_tt+αu_t+Lu+g(Ψ(u))Cu=0, t∈R₊.

We apply the abstract results to semilinear parabolic and hyperbolic partial differential equations including the heat equation, the wave equation, a Kuramoto–Sivashinsky model and the Kirchhoff–Carrier equation.

We consider a diphasic system in a high shear regime when separation of phases occurs. One can observe that the two phases organize themselves into numerous bands, parallel to the flow direction. Mathematically, these are solutions of a certain system of equations depending only upon the transversal variable (1D solutions). We study the stability of these 1D solutions with respect to 2D perturbations. The mathematical model used in our analysis is a coupling between a Cahn–Hilliard equation and the Navier–Stokes equations in two dimensions. We show that a small 2D perturbation of a given 1D solution persists for significant times. We give the precise size of such a perturbation and its time of persistence. Moreover, we obtain an asymptotic expansion of the solution in the considered cases. Note that, for a mathematical model to be realistic one has to take into account the fact that in experiments the high shear regime is obtained in elongated domains (a very thin Couette cell for example). Therefore, we perform the mathematical analysis of this problem in a stretched domain.

We consider the Cauchy problem for a class of scalar linear dispersive equations with rapidly oscillating initial data. The problem of the high‐frequency asymptotics of such models is reviewed, in particular we highlight the difficulties in crossing caustics when using (time‐dependent) WKB‐methods. Using Wigner measures we present an alternative approach to such asymptotic problems. We first discuss the connection of the naive WKB solutions to transport equations of Liouville type (with mono‐kinetic solutions) in the pre‐breaking regime. Further we show how the Wigner measure approach can be used to analyze high‐frequency limits in the post‐breaking regime, in comparison with the traditional Fourier integral operator method. Finally we present some illustrating examples.