Asymptotic Analysis - Volume 64, issue 3-4

Authors: Wong, R. | Zhang, L.

Article Type: Research Article

Abstract: In this paper, we study the asymptotics of polynomials orthogonal with respect to the varying quartic weight ω(x)=e−nV(x) , where V(x)=Vt (x)=x4 /4+x2 t/2. We focus on the critical case t=−2, in the sense that for t≥−2, the support of the associated equilibrium measure is a single interval, while for t<−2, the support consists of two intervals. Globally uniform asymptotic expansions are obtained for z in three unbounded regions. These regions together cover the whole complex z-plane. In particular, in the region containing the origin, the expansion involves the Ψ function affiliated with the Hastings–McLeod solution of the second Painlevé …equation. Our approach is based on a modified version of the steepest-descent method for Riemann–Hilbert problems introduced by Deift and Zhou (Ann. Math. 137 (1993), 295–370). Show more

Keywords: orthogonal polynomials, global asymptotics, Riemann–Hilbert problems, Airy functions, the second Painlevé transcendent

DOI: 10.3233/ASY-2008-0937

Citation: Asymptotic Analysis, vol. 64, no. 3-4, pp. 125-154, 2009

Authors: Bostan, Mihai

Article Type: Research Article

Abstract: The Nordström–Vlasov system describes the evolution of self-gravitating collisionless particles. We prove the existence of stationary solutions in one dimension. We show also the propagation of impulsion moments and perform an asymptotic analysis.

Keywords: Nordström equation, Vlasov equation, mild solution

DOI: 10.3233/ASY-2009-0939

Citation: Asymptotic Analysis, vol. 64, no. 3-4, pp. 155-183, 2009

Authors: Bayada, Guy | Chupin, Laurent | Grec, Bérénice

Article Type: Research Article

Abstract: The present paper deals with non-Newtonian viscoelastic flows of Oldroyd-B type in thin domains. Such geometries arise for example in the context of lubrication. More precisely, we justify rigorously the asymptotic model obtained heuristically by proving the mathematical convergence of the Navier–Stokes/Oldroyd-B system towards the asymptotic model.

Keywords: viscoelastic fluids, thin film, Oldroyd model, lubrication flow, asymptotic analysis

DOI: 10.3233/ASY-2009-0940

Citation: Asymptotic Analysis, vol. 64, no. 3-4, pp. 185-211, 2009

Authors: Shubov, Marianna A.

Article Type: Research Article

Abstract: The paper is the second in a series of works devoted to the solvability of the Possio singular integral equation. This equation relates the pressure distribution over a typical section of a slender wing in subsonic compressible air flow to the normal velocity of the points of a wing (downwash). In spite of the importance of the Possio equation, the question of the existence of its solution has not been settled yet. In the first paper “Reduction of the boundary-value problem to Possio integral equation in theoretical aeroelasticity”, we reformulated the initial boundary-value problem, involving a partial differential equation for …the velocity potential and highly nonstandard boundary conditions, in the form of a singular integral equation, the Possio equation. In our derivation, we have used the ideas and techniques totally different from the ones used by C. Possio in his original work. In the present paper, we show that the integral equation can be split up into two parts in such a way that the first part is not small with respect to a complex parameter entering the equation, while the second part is asymptotically small and tends to zero as the above parameter tends to infinity. In this paper, we justify such a splitting and in the next paper, we will prove solvability of the nonvanishing part of the Possio equation and then prove that asymptotically small terms cannot destroy the aforementioned unique solvability. Show more

Keywords: compressible flow, velocity potential, Mikhlin multipliers, Hilbert integral transform

DOI: 10.3233/ASY-2009-0941

Citation: Asymptotic Analysis, vol. 64, no. 3-4, pp. 213-238, 2009

Authors: Boccardo, Lucio | Casado-Diaz, Juan

Article Type: Research Article

Abstract: We study the H-convergence of boundary value problems of the type  −div(Mε (x)∇uε )=−div(uε Eε (x))+f(x), inΩ,  uε =0 on∂Ω if Eε ∈(L2 (Ω))N .

Keywords: homogenization, H-convergence, entropic solutions

DOI: 10.3233/ASY-2009-0943

Citation: Asymptotic Analysis, vol. 64, no. 3-4, pp. 239-249, 2009

Article Type: Other

Citation: Asymptotic Analysis, vol. 64, no. 3-4, pp. 251-251, 2009