Mathematical Models and Methods in Applied Sciences

This paper follows the work of Colin–Gallice–Laurioux⁶ in which a new model generalizing the Schrödinger (NLS) model of the diffractive optics is derived for the laser propagation in nonlinear media. In particular, it provides good approximate solutions of the Maxwell–Lorentz system for highly oscillating initial data with broad spectrum. In real situations one is given boundary data. We propose to derive a similar evolution model but in the variable associated to the direction of propagation. However, since the space directions for the Maxwell equations are not hyperbolic, the boundary problem is ill-posed and one needs to apply a cutoff defined in the Fourier space, selecting those frequencies for which the operator is hyperbolic. The model we obtain is nearly L² conservative on its domain of validity.

We then give a justification of the derivation. For this purpose we introduce a related well-posed initial boundary value problem. Finally, we perform numerical computations on the example of Maxwell with Kerr nonlinearity in some cases of short or spectrally chirped data where our model outperforms the Schrödinger one.

We investigate the derivation and the mathematical properties of a Saint-Venant model with an energy equation and with temperature-dependent transport coefficients. These equations model shallow water flows as well as thin viscous sheets over fluid substrates like oil slicks, atlantic waters in the Strait of Gilbraltar or float glasses. We exhibit an entropy function for the system of partial differential equations and by using the corresponding entropic variable, we derive a symmetric conservative formulation of the system. The symmetrized Saint-Venant quasilinear system of partial differential equations is then shown to satisfy the nullspace invariance property and is recast into a normal form. Upon establishing the local dissipative structure of the linearized normal form, global existence results and asymptotic stability of equilibrium states are obtained. We finally derive the Saint-Venant equations with an energy equation taking into account the temperature-dependence of transport coefficients from an asymptotic limit of a three-dimensional model.

This paper deals with the analysis of the asymptotic limit for BGK model to the linearized Navier–Stokes equations when the Knudsen number ε tends to zero. The uniform (in ε) existence of global strong solutions and uniqueness theorems are proved for regular initial fluctuations. As ε tends to zero, the solution of BGK model converges strongly to the solution of the linearized Navier–Stokes systems. The validity of the BGK model is critically analyzed.

We study electromagnetic wave propagation in a periodic and frequency dependent material characterized by a space- and frequency-dependent complex-valued permittivity. The spectral parameter relates to the time-frequency, leading to spectral analysis of a holomorphic operator-valued function. We apply the Floquet transform and show for a fixed quasi-momentum that the resulting family of spectral problems has a spectrum consisting of at most countably many isolated eigenvalues of finite multiplicity. These eigenvalues depend continuously on the quasi-momentum and no nonzero real eigenvalue exists when the material is absorptive. Moreover, we reformulate the special case of a rational operator-valued function in terms of a polynomial operator pencil and study two-component dispersive and absorptive crystals in detail.

We introduce an interior penalty discontinuous Galerkin finite element method for the Reissner–Mindlin plate model that, as the plate's half-thickness ϵ tends to zero, recovers a hp interior penalty discontinuous Galerkin finite element methods for biharmonic equation. Our method does not introduce shear as an extra unknown, and does not need reduced integration techniques. We develop the a priori error analysis of these methods and prove error bounds that are optimal in h and uniform in ϵ. Numerical tests, that confirm our predictions, are provided.

P. Galenko et al. proposed a Cahn–Hilliard model with inertial term in order to model spinodal decomposition caused by deep supercooling in certain glasses. Here we analyze a finite element space semidiscretization of their model, based on a scheme introduced by C. M. Elliott et al. for the Cahn–Hilliard equation. We prove that the semidiscrete solution converges weakly to the continuous solution as the discretization parameter tends to 0. We obtain optimal a priori error estimates in energy norm and related norms, assuming enough regularity on the solution. We also show that the semidiscrete solution converges to an equilibrium as time goes to infinity and we give a simple finite difference version of the scheme.