Asymptotic Analysis - Volume 76, issue 2

Authors: Jiang, Ning | Levermore, C. David

Article Type: Research Article

Abstract: We construct the weakly nonlinear-dissipative approximate system for the general compressible Navier–Stokes system in a periodic domain. It was shown in Arch. Rational Mech. Anal. 201 (2011), 377–412, that because the Navier–Stokes system has an entropy structure, its approximate system will have Leray-like global weak solutions. These solutions decompose into an incompressible part governed by an incompressible Navier–Stokes system, and an acoustic part governed by a nonlocal quadratic equation which couples it to the incompressible part. We obtain regularity results for the acoustic part of the solution via a Littlewood–Paley decomposition that extend to this general setting results found by …Masmoudi [Ann. Inst. H. Poincaré 18 (2001), 199–224] and Danchin [Amer. J. Math. 124 (2002), 1153–1219] in the γ-law barotropic setting. Show more

Keywords: weakly compressible, Navier–Stokes, averaging, regularity

DOI: 10.3233/ASY-2011-1066

Citation: Asymptotic Analysis, vol. 76, no. 2, pp. 61-86, 2012

Authors: Getmanenko, Alexander

Article Type: Research Article

Abstract: The paper is devoted to some foundational questions in resurgent analysis. As a main technical result, it is shown that under appropriate conditions the infinite sum of endlessly continuable majors commutes with the Laplace transform. A similar statement is proven for compatibility of a convolution and of an infinite sum of majors. We generalize the results of Candelpergher–Nosmas–Pham and prove a theorem about substitution of a small (extended) resurgent function into a holomorphic parameter of another resurgent function. Finally, we discuss an application of these results to the question of resurgence of eigenfunctions of a one-dimensional Schrödinger operator corresponding to …a small resurgent eigenvalue. Show more

Keywords: resurgence, Laplace transform, linear ODEs

DOI: 10.3233/ASY-2011-1068

Citation: Asymptotic Analysis, vol. 76, no. 2, pp. 87-114, 2012

Authors: Allain, Geneviève | Beaulieu, Anne

Article Type: Research Article

Abstract: We consider the positive solutions u of −Δu+u−up =0 in [0,2π]×RN−1 , which are 2π-periodic in x1 and tend uniformly to 0 in the other variables. There exists a constant C such that any solution u verifies u(x1 ,x′)≤Cw0 (x′) where w0 is the ground state solution of −Δv+v−vp =0 in RN−1 . We prove that exactly the same estimate is true when the period is 2π/ε, even when ε tends to 0. We have a similar result for the gradient.

Keywords: asymptotic behavior of solutions, semilinear elliptic equations, periodic solutions

DOI: 10.3233/ASY-2011-1076

Citation: Asymptotic Analysis, vol. 76, no. 2, pp. 115-122, 2012