Mathematical Models and Methods in Applied Sciences

This paper is concerned with entropy methods for linear drift-diffusion equations with explicitly time-dependent or degenerate coefficients. Our goal is to establish a list of various qualitative properties of the solutions. The motivation for this study comes from a model for molecular motors, the so-called Brownian ratchet, and from a nonlinear equation arising in traffic flow models, for which complex long time dynamics occurs. General results are out of the scope of this paper, but we deal with several examples corresponding to most of the expected behaviors of the solutions.

We first prove a contraction property for general entropies which is a useful tool for uniqueness and for the convergence to some large time asymptotic solutions which may depend on time. Then we focus on power law and logarithmic relative entropies. When the diffusion term is of the type ∇(|x|^α∇·), we prove that the inequality relating the entropy with the entropy production term is a Hardy–Poincaré type inequality, that we establish. Here we assume that α ∈ (0,2] and the limit case α = 2 appears as a threshold for the method. As a consequence, we obtain an exponential decay of the relative entropies. In the case of time-periodic coefficients, we prove the existence of a unique time-periodic solution which attracts all other solutions.

The case of a degenerate diffusion coefficient taking the form |x|^α with α > 2 is also studied. The Gibbs state exhibits a non-integrable singularity. In this case concentration phenomena may occur, but we conjecture that an additional time-dependence restores the smoothness of the asymptotic solution.

The existence of minimizers to a geometrically exact Cosserat planar shell model with microstructure is proven. The membrane energy is a quadratic, uniformly Legendre–Hadamard elliptic energy in contrast to traditional membrane energies. The bending contribution is augmented by a curvature term representing the interaction of the rotational microstructure in the Cosserat theory. The model includes non-classical size effects, transverse shear resistance, drilling degrees of freedom and accounts implicitly for thickness extension and asymmetric shift of the midsurface. Upon linearization with zero Cosserat couple modulus μ_c = 0, one recovers the infinitesimal-displacement Reissner–Mindlin model. It is shown that the Cosserat shell formulation admits minimizers even for μ_c = 0, in which case the drill-energy is absent. The midsurface deformation m is found in H¹(ω, ℝ³). Since the existence of energy minimizers rather than equilibrium solutions is established, the proposed analysis includes the large deformation/large rotation buckling behaviour of thin shells.

In this paper we present a new approach to describe the behaviour of a pollutant slick at the sea surface. To this end, we consider that the pollutant and the water are immiscible and we propose a two-layer model where the lower layer corresponds to the water and the upper layer represents the pollutant. Since the dimension of the pollutant slick is generally much smaller than the domain occupied by the sea, we propose to compute the motion of the pollutant with a shallow water model with free boundary only in the domain occupied by the pollutant. To discretize in time the problem with free boundary, we use an ALE formulation coupled with the characteristic method. Then, to solve the space discretized problem, we approximate the pollutant velocity by using a Galerkin method with a special basis which verifies the boundary conditions and simplifies significantly the resolution. Finally we test this work in a real situation: the dam of Calacuccia (Corsica).

In this paper we consider the hyperbolic relaxation of the Cahn–Hilliard equation ruling the evolution of the relative concentration u of one component of a binary alloy system located in a bounded and regular domain Ω of ℝ³. This equation is characterized by the presence of the additional inertial term ∊u_tt that accounts for the relaxation of the diffusion flux. For this equation we address the problem of the long time stability from the point of view of global attractors. The main difficulty in dealing with this system is the low regularity of its weak solutions, which prevents us from proving a uniqueness result and a proper energy identity for the solutions. We overcome this difficulty by using a density argument based on a Faedo–Galerkin approximation scheme and the recent J. M. Ball's theory of generalized semiflows. Moreover, we address the problem of the approximation of the attractor of the continuous problem with the one of Faedo–Galerkin scheme. Finally, we show that the same type of results hold also for the damped semilinear wave equation when the nonlinearity ϕ is not Lipschitz continuous and has a super critical growth.

We present a stabilized finite element method for Stokes equations with piecewise continuous bilinear approximations for both velocity and pressure variables. The velocity field is enriched with piecewise polynomial bubble functions with null average at element edges. These functions are statically condensed at the element level and therefore they can be viewed as a continuous Q1–Q1 stabilized finite element method. The enriched velocity-pressure pair satisfies optimal inf–sup conditions and approximation properties. Numerical experiments show that the proposed discretization outperforms the Galerkin least-squares method.

We consider a chiral medium in a bounded domain, enclosed in a perfectly conducting material. We solve the transient Maxwell equations in this domain, when the medium is modeled by the Drude–Born–Fedorov constitutive equations. The input data is located on the boundary, in the form of given surface current and surface charge densities. It is proved that, except for a countable set of chirality admittance values, the problem is mathematically well-posed. This result holds for domains with non-smooth boundaries.