We consider the initial-boundary value problem for the perturbed viscous Cahn-Hilliard equation in space dimension n ≤ 3. Applying semigroup theory, we formulate this problem as an abstract evolutionary equation with a sectorial operator in the main part. We show that the semigroup generated by this problem admits a global attractor in the phase space (H²(Ω)∩ H¹₀(Ω)) × L²(Ω) and characterize its structure.

We deal with a class on nonlinear Schrödinger equations (NLS) with potentials $V\left(x\right)\sim {\left|x\right|}^{-\alpha }$, $0<\alpha <2$, and $K\left(x\right)\sim {\left|x\right|}^{-\beta }$, $\beta >0$. Working in weighted Sobolev spaces, the existence of ground states $v_{\epsilon }$ belonging to $W^{1,2}\left({ℝ}^N\right)$ is proved under the assumption that $\sigma <p<(N+2)/\left(N-2\right)$ for some $\sigma ={\sigma }_{N,\alpha ,\beta }$. Furthermore, it is shown that $v_{\epsilon }$ are spikes concentrating at a minimum point of $𝒜=V^{\theta }{K}^{-2/\left(p-1\right)}$, where $\theta =(p+1)/\left(p-1\right)-1/2$.

We study the first eigenpair of a Dirichlet spectral problem for singularly perturbed convection-diffusion operators with oscillating locally periodic coefficients. It follows from the results of [A. Piatnitski and V. Rybalko, On the first eigenpair of singularly perturbed operators with oscillating coefficients. Preprint www.arxiv.org, arXiv:1206.3754] that the first eigenvalue remains bounded only if the integral curves of the so-called effective drift have a nonempty ω-limit set. Here we consider...

We derive a posteriori error estimates for singularly perturbed reaction–diffusion problems which yield a guaranteed upper bound on the discretization error and are fully and easily computable. Moreover, they are also locally efficient and robust in the sense that they represent local lower bounds for the actual error, up to a generic constant independent in particular of the reaction coefficient. We present our results in the framework of the vertex-centered finite volume method but their nature...

A brief survey is given to show that harmonic averages enter in a natural way in the numerical solution of various variable coefficient problems, such as in elliptic and transport equations, also of singular perturbation types. Local Green’s functions used as test functions in the Petrov-Galerkin finite element method combined with harmonic averages can be very efficient and are related to exact difference schemes.

This paper deals with the numerical study of a nonlinear, strongly anisotropic heat equation. The use of standard schemes in this situation leads to poor results, due to the high anisotropy. An Asymptotic-Preserving method is introduced in this paper, which is second-order accurate in both, temporal and spacial variables. The discretization in time is done using an L-stable Runge−Kutta scheme. The convergence of the method is shown to be independent of the anisotropy parameter , and this for fixed...

We study the asymptotic behaviour of the following nonlinear problem: $$\{\begin{array}{c}-\mathrm{div}\left(a\left(D{u}_h\right)\right)+{\left|u_h\right|}^{p-2}{u}_h=f\text{in}{\Omega }_h,a\left(D{u}_h\right)·\nu =0\text{on}\partial {\Omega }_h,\hfill \end{array}.$$ in a domain Ωh of ${ℝ}^n$ whose boundary ∂Ωh contains an oscillating part with respect to h when h tends to ∞. The oscillating boundary is defined by a set of cylinders with axis 0xn that are h-1-periodically distributed. We prove that the limit problem in the domain corresponding to the oscillating boundary identifies with a diffusion operator with respect to xn coupled with an algebraic problem for the limit fluxes.

The incompressible limit of the weak solutions to a fluid-particle interaction model is studied in this paper. By using the relative entropy method and refined energy analysis, we show that, for well-prepared initial data, the weak solutions of the compressible fluid-particle interaction model converge to the strong solution of the incompressible Navier-Stokes equations as long as the Mach number goes to zero. Furthermore, the desired convergence rates are also obtained.

On étudie un opérateur de la forme $-\Delta +V$ sur ${ℝ}^d$, où $V$ est un potentiel admettant plusieurs pôles en $a/r^2$. Plus précisément, on démontre l’estimation de résolvante tronquée à hautes fréquences, classique dans les cas non-captifs, et qui implique l’effet régularisant standard pour l’équation de Schrödinger correspondante. La preuve est basée sur l’introduction d’une mesure de défaut micro-locale semi-classique. On démontre également, dans le même contexte, des inégalités de Strichartz pour l’équation de Schrödinger....

We consider the linearized elasticity system in a multidomain of ${𝐑}^3$. This multidomain is the union of a horizontal plate with fixed cross section and small thickness ε, and of a vertical beam with fixed height and small cross section of radius $r^{\epsilon }$. The lateral boundary of the plate and the top of the beam are assumed to be clamped. When ε and $r^{\epsilon }$ tend to zero simultaneously, with $r^{\epsilon }\gg {\epsilon }^2$, we identify the limit problem. This limit problem involves six junction conditions.

Consider a one dimensional nonlinear reaction-diffusion equation (KPP equation) with non-homogeneous second order term, discontinuous initial condition and small parameter. For points ahead of the Freidlin-KPP front, the solution tends to 0 and we obtain sharp asymptotics (i.e. non logarithmic). Our study follows the work of Ben Arous and Rouault who solved this problem in the homogeneous case. Our proof is probabilistic, and is based on the Feynman-Kac formula and the large deviation principle...