Mathematical Models and Methods in Applied Sciences

A fundamental solution of the Cauchy problem is constructed for a pseudo-differential equation generalizing some Schrödinger equations. Then well-posedness of the Cauchy problem is proved in some Gevrey spaces whose indices depend on the lower order term of the operator.

This paper gives the asymptotic decay rate toward the planar rarefaction waves of the solutions for the scalar viscous conservation laws in two or more space dimensions. This is proved by a result on the decay rate of solutions for one-dimensional scalar viscous conservation laws and by using an L²-energy method with a weight of time.

We analyze the convergence of a mixed finite element method introduced by Zienkiewicz, Taylor, Papadopoulos and Oñate for the Reissner–Mindlin plate model. In order to do this, we compare it with a method which is known to be convergent with optimal order uniformly in the plate thickness. We show that the difference between the solutions of both methods is of higher order than the error. In particular the method does not present locking and is optimal order convergent. We also present several numerical experiments which confirm the similar behavior of both methods.

A new approach is suggested to the construction of conservative difference schemes for multidimensional second-order parabolic equations containing divergent first-order differential forms. These schemes hold the property of having fixed sign for non-negative solutions. The convergence of schemes is proved in the grid norm L₂ at the rate O(τ + h²).

We describe in this paper the stability of the interface in a free-boundary problem for a self-gravitating gas in one space dimension, with an external pressure P, and a Fourier coefficient λ, for the thermal flux.

In the first part, we consider the pure gaseous limit (perfect gas), and we find different possible asymptotic behaviors: if λ > 0, the gas tends to collapse; if λ = 0, we show that, when P > 0, the solution converges, for large time to the isothermal solution of the corresponding stationary problem, while for P = 0, the system is unstable, and the gas tends to fill all the space.

In the second part, we consider the same problem for the pure radiative limit (photon gas), and we show that analogous asymptotics hold.

The effect of an orthogonal applied magnetic field on an electrically conducting thermomicropolar fluid layer which is heated from below, is investigated. The linear stability and the nonlinear stability of the stationary state are considered when the fluid layer lies between two nonconducting rigid boundaries. The nonlinear stability study is performed for different cases of the Alfven No. A which depends on the strength of the magnetic field. A generalized energy is used for the general case A different from 1. From a variational and numerical analysis the nonlinear stability threshold is obtained and compared with the linear stability results and the nonconducting case. Both micro-rotation and magnetic field tend to stabilize the fluid layer.

We analyze the dynamics of a disparate mass binary gas or of a plasma in the homogeneous case, at various time scales, in the framework of the Boltzmann or Fokker–Planck equation. We intend to provide a rigorous foundation to the epochal relaxation phenomenon first pointed out by Grad. From general basic physical hypotheses, we derive the scaling of the equations as a function of the mass ratio of the particles, and we expand the collision operators in powers of this mass ratio. Then, Hilbert or Chapman–Enskog type expansions of the distribution functions allow us to investigate the dynamics of the mixture at various time scales, and we verify that the behavior of the obtained models is coherent with Grad's hypothesis.