Asymptotic Analysis - Volume 33, issue 3-4

Authors: Helffer, Bernard | Lafitte, Olivier

Article Type: Research Article

Abstract: In this paper, we study the spectrum of the Rayleigh equation, which models the Rayleigh–Taylor instability in a fluid of variable density ρ(x), where ρ(x) goes to ρ− at −∞ and ρ+ at +∞: -h^{2}{\frac{\mathrm{d}}{\mathrm{d}x}}\bigl(\rho(x){\frac{\mathrm{d}u}{\mathrm{d}x}}\bigr)+(\rho(x)+\delta \rho'(x))u=0,\quad u\in L^{2}(\mathbb{R} ). The behavior of the smallest value δ(h) of δ>0 for which there exists a nontrivial solution u is investigated in terms of h for the two regimes h→0 (called the semi‐classical regime) and h→+∞. The so‐called growth rate of the Rayleigh–Taylor instability is $\sqrt{h/\delta(h)}$ . When ρ−ρ− ∈L2 ($\mathbb{R} $ − ) and …ρ−ρ+ ∈L2 ($\mathbb{R} $ + ), we prove that δ(h)/h→(ρ+ +ρ− )/(ρ− −ρ+ ), as h→+∞, generalizing the result of Lord Rayleigh [23]. We then investigate the expansion of δ(h)/h in terms of 1/h and properties of ρ. In particular, we identify the number of terms n of the expansion in function of the behavior of ρ(x)−ρ± at ±∞, extending the results of Cherfils, Lafitte and Raviart [6]. Show more

Citation: Asymptotic Analysis, vol. 33, no. 3-4, pp. 189-235, 2003

Authors: Bresch, D. | Guillén‐González, F. | Masmoudi, N. | Rodríguez‐Bellido, M.A.

Article Type: Research Article

Abstract: This paper is devoted to the establishment of a friction boundary condition associated with the hydrostatic Navier–Stokes equations (also called the primitive equations). Usually the Navier boundary condition is used, as a wall law, for a fluid governed by the Navier–Stokes equations when we want to modelize roughness. We consider here an anisotropic Navier boundary condition corresponding to the anisotropic Navier–Stokes equations. By an asymptotic analysis with respect to the aspect ratio of the domain, we obtain a friction boundary condition for the limit system, that is the primitive equations. This asymptotic boundary condition only concerns the trace of …the horizontal components of the velocity field and the trace of its vertical derivative. Show more

Keywords: Navier boundary conditions, primitive equations, asymptotic analysis

Citation: Asymptotic Analysis, vol. 33, no. 3-4, pp. 237-259, 2003

Authors: Grasselli, Maurizio | Pata, Vittorino | Vegni, Federico Mario

Article Type: Research Article

Abstract: We consider a phase‐field model based on hereditary constitutive equations for the internal energy and the heat flux and on the assumption that the spatial average of the order parameter χ is conserved. This model consists of a parabolic integrodifferential equation for the (relative) temperature ϑ coupled with a nonlinear fourth‐order evolution equation for χ. We first show that the obtained system is indeed a nonautonomous dynamical system, provided that the phase space accounts for the past history of ϑ and appropriate boundary conditions are given. Then we establish the existence of an absorbing set, uniformly with respect to a …certain class of heat source terms. Finally, we prove that, under suitable assumptions, our dissipative dynamical system possesses a uniform attractor of finite Hausdorff and fractal dimensions. Show more

Keywords: conserved phase‐field models, materials with memory, existence and uniqueness, continuous dependence, nonautonomous dynamical systems, uniformly absorbing sets, uniform attractors, Hausdorff and fractal dimensions

Citation: Asymptotic Analysis, vol. 33, no. 3-4, pp. 261-320, 2003

Authors: Fröhlich, Andreas

Article Type: Research Article

Abstract: Using a characterisation of maximal Lp ‐regularity by ℛ‐bounded operator families we prove global in time estimates in $L^{p}({\mathbb{R}}_{+};L^{q}(\varOmega))$ , 1<p,q<∞, for solutions of the instationary Stokes system in an aperture domain $\varOmega\subset{\mathbb{R}} ^{n}$ , n≥3, with $\curpartial\varOmega\in C^{1,1}.$ The results are applied to obtain new global in time estimates for weak solutions of the Navier–Stokes equations with nonvanishing flux through the aperture.

Citation: Asymptotic Analysis, vol. 33, no. 3-4, pp. 321-335, 2003

Authors: Mellet, A.

Article Type: Research Article

Abstract: This paper follows [7], by N. Ben Abdallah, P. Degond, F. Poupaud and the author, in which a diffusion model for semiconductor superlattices was derived. At the starting point of our study, the device consists of a periodic array of localized scatters (the heterojunction between two semiconductor materials), and electron motion is described by the Boltzmann equation. Assuming that both the collisions and the scattering processes preserve the energy of the particles, we prove the convergence of the electron distribution function to the solution of the SHE model equation (for Spherical Harmonics Expansion). Since the device has microscopic periodicity, the …proof relies on the recently developed tool of two‐scale convergence. Show more

Keywords: Boltzmann equation, Spherical Harmonics Expansion model, semiconductor superlattices, diffusion approximation, homogenization, interface operators, two‐scale limit

Citation: Asymptotic Analysis, vol. 33, no. 3-4, pp. 337-361, 2003

Article Type: Other

Citation: Asymptotic Analysis, vol. 33, no. 3-4, pp. 363-364, 2003