Asymptotic Analysis - Volume 84, issue 1-2

Authors: Huang, Min-Hai | Zhao, Yu-Qiu

Article Type: Research Article

Abstract: We demonstrate an approach to derive uniform and point-wise asymptotic formulas based on Fokas' transform method. To this aim, we study the high-frequency uniform asymptotics for the solution of the Helmholtz equation, in a quarter plane and subject to a specific Neumann condition. The analysis is based on the integral representation of the solution derived via Fokas' transform method. In the case of piecewise constant boundary data, full point-wise asymptotic expansions of the solution are obtained by using the method of steepest descents for definite integrals. A uniform asymptotic expansion, holding in the whole quadrant, is also derived in terms …of the complementary error function. The uniform expansion exhibits smooth transitions across certain critical vertical lines, along which the point-wise asymptotic approximations have jumps. Show more

Keywords: high-frequency asymptotics, Fokas' transform method, method of steepest descents, Helmholtz equation, Neumann condition

DOI: 10.3233/ASY-121160

Citation: Asymptotic Analysis, vol. 84, no. 1-2, pp. 1-15, 2013

Authors: Gie, Gung-Min | Jung, Chang-Yeol

Article Type: Research Article

Abstract: We study the asymptotic behavior, at small viscosity ε, of the Navier–Stokes equations in a 2D curved domain. The Navier–Stokes equations are supplemented with the slip boundary condition, which is a special case of the Navier friction boundary condition where the friction coefficient is equal to two times the curvature on the boundary. We construct an artificial function, which is called a corrector, to balance the discrepancy on the boundary of the Navier–Stokes and Euler vorticities. Then, performing the error analysis on the corrected difference of the Navier–Stokes and Euler vorticity equations, we prove convergence results in the L2 …norm in space uniformly in time, and in the norm of H1 in space and L2 in time with rates of order ε3/4 and ε1/4 , respectively. In addition, using the smallness of the corrector, we obtain the convergence of the Navier–Stokes solution to the Euler solution in the H1 norm uniformly in time with rate of order ε1/4 . Show more

Keywords: boundary layers, singular perturbations, Navier–Stokes equations, Euler equations, Navier friction boundary condition, slip boundary condition

DOI: 10.3233/ASY-131164

Citation: Asymptotic Analysis, vol. 84, no. 1-2, pp. 17-33, 2013

Authors: Kopylova, E.A.

Article Type: Research Article

Abstract: We obtain a dispersion long-time decay in weighted norms for solutions of the 2D Dirac equation with generic potentials. For high energy component we develop the approach of Boussaid relying on the Mourre estimates and the minimal escape velocity estimates of Hunziker, Sigal and Soffer. For the low energy component we obtain the asymptotics of the resolvent near the thresholds in the nonsingular case and apply 2D version of Jensen–Kato lemma on “one-and-half partial integration”. The results can be applied to the study of asymptotic stability and scattering of solitons for nonlinear 2D Dirac equations.

Keywords: dispersion decay, weighted norms, Dirac equation

DOI: 10.3233/ASY-131166

Citation: Asymptotic Analysis, vol. 84, no. 1-2, pp. 35-46, 2013

Authors: Lizama, Carlos | Miana, Pedro J. | Poblete, Felipe

Article Type: Research Article

Abstract: In this article we study the uniform stability of an (a,k)-regularized family {S(t)}t≥0 generated by a closed operator A. We give sufficient conditions, on the scalar kernels a, k and the operator A, to ensure the uniform stability of the family {S(t)}t≥0 in Hilbert spaces. Our main result is a generalization of Theorem 1 in [Proc. Amer. Math. Soc. 132(1) (2004), 175–181], concerning the stability of resolvent families, and can be seen as a substantial generalization of the Gearhart–Greiner–Prüss characterization of exponential stability for strongly continuous semigroups.

Keywords: stability, strongly continuous semigroups, strongly continuous cosine families, Hilbert spaces

DOI: 10.3233/ASY-131169

Citation: Asymptotic Analysis, vol. 84, no. 1-2, pp. 47-60, 2013

Authors: Danchin, Raphaël | He, Lingbing

Article Type: Research Article

Abstract: In this paper we study the validity of the so-called Oberbeck–Boussinesq approximation for compressible viscous perfect gases in the whole three-dimensional space. Both the cases of fluids with positive heat conductivity and zero conductivity are considered. For small perturbations of a constant equilibrium, we establish the global existence of unique strong solutions in a critical regularity functional framework. Next, taking advantage of Strichartz estimates for the associated system of acoustic waves, and of uniform estimates with respect to the Mach number, we obtain all-time convergence to the Boussinesq system with a explicit decay rate.

Keywords: compressible fluids, Navier–Stokes, low Mach number, low Froude number, critical regularity

DOI: 10.3233/ASY-131170

Citation: Asymptotic Analysis, vol. 84, no. 1-2, pp. 61-102, 2013

Authors: Fabricius, John | Koroleva, Yulia | Wall, Peter

Article Type: Research Article

Abstract: We study the asymptotic behavior of solutions of the evolution Stokes equation in a thin three-dimensional domain bounded by two moving surfaces in the limit as the distance between the surfaces approaches zero. Using only a priori estimates and compactness it is rigorously verified that the limit velocity field and pressure are governed by the time-dependent Reynolds equation.

Keywords: lubrication theory, thin film, asymptotic behavior, evolution Stokes equation, time-dependent Reynolds equation

DOI: 10.3233/ASY-131165

Citation: Asymptotic Analysis, vol. 84, no. 1-2, pp. 103-121, 2013