Search Results for « asymptotic behavior »

We consider the Helmholtz equation in an angular sector partially covered by a homogeneous layer of small thickness, denoted $\varepsilon$. We propose in this work an asymptotic expansion of the solution with respect to $\varepsilon$ at any order. This is done using matched asymptotic expansion, which consists here in introducing different asymptotic expansions of the solution in three subdomains: the vicinity of the corner, the layer and the rest of the domain. These expansions are linked through matching conditions. The presence of the corner makes these matching conditions delicate to derive because the fields have singular behaviors. Our approach is to reformulate these matching conditions purely algebraically by writing all asymptotic expansions as formal series. By using algebraic calculus we reduce the matching conditions to scalar relations linking the singular behaviors of the fields. These relations have a convolutive structure and involve some coefficients that can be computed analytically. Our asymptotic expansion is justified rigorously with error estimates.

This paper focuses on the asymptotic behavior of the eigenvalues of the classical Steklov problem in a thin multidomain as it becomes increasingly narrow. The thin multidomain consists of two vertical cylinders, one positioned atop the other. We show that the eigenvalues tend to zero as the domain becomes thinner. In this case, the limiting eigenvalue problem in the upper cylinder corresponds to a one-dimensional eigenvalue problem, while the limiting problem in the lower cylinder is defined in an ( n − 1 ) -dimensional ball. We conclude our investigation by considering the case in which the thinness of the upper and lower cylinders is equal.

An asymptotic formula for the number of states of Boson gas whose Hamiltonian is given by a positive elliptic pseudo-differential operator of order one on a compact manifold is given under a integrality assumption on the spectrum of the Hamiltonian. This is regarded as an analogue of the Meinardus theorem on asymptotics of the number of partitions of a positive integer.

We analyze the asymptotic behavior of the eigenvalues of nonlinear elliptic problems under Dirichlet boundary conditions and mixed (Dirichlet, Neumann) boundary conditions on domains becoming unbounded. We make intensive use of Picone identity to overcome nonlinearity complications. Altogether the use of Picone identity makes the proof easier with respect to the known proof in the linear case. Surprisingly the asymptotic behavior under mixed boundary conditions critically differs from the case of pure Dirichlet boundary conditions for some class of problems.

In this article, we are devoted to studying the following nonlocal Navier–Stokes equations involving the time–space fractional operators { ∂ t β u + ( − Δ ) α u + u ⋅ ∇ u = ∇ p + λ | u | q − 2 u + g ( x , t ) in Ω × R + , div u = 0 in Ω × R + , u ( x , t ) = 0 in ( R N ∖ Ω ) × R + , u ( x , 0 ) = u 0 ( x ) in Ω , where Ω ⊂ R N is a bounded domain with Lipschitz boundary, ( − Δ ) α is the fractional Laplacian with 0 < α < 1 , ∂ t β is the Riemann–Liouville time fractional derivative with 0 < β < 1 , λ is a parameter, and g ∈ ( L ∞ ( 0 , ∞ ; L 2 ( Ω ) ) ) N . Due to the nonlocal nature of the fractional Laplacian, the classical weak solution frameworks proposed by J. L. Lions and E. Hopf are inapplicable to this problem. To address this challenge, we first construct a regularized problem and establish the existence of its solutions via the Galerkin method combined with fractional calculus techniques. Subsequently, we prove the uniqueness of the global weak solution to the regularized problem. Furthermore, under appropriate assumptions, a decay estimate characterizing the long-time behavior of solutions is derived. Finally, the existence of solutions to the original problem is obtained by analyzing the limiting behavior of the regularized solutions as the regularization parameter vanishes. The main contributions of this work are twofold: (1) Our problem simultaneously incorporates the Riemann–Liouville time fractional derivative and the fractional Laplacian, extending classical Navier–Stokes theory to a nonlocal time–space framework. (2) We develop a new analytical approach to estimate the nonlinear convection term u ⋅ ∇ u in the presence of fractional operators, overcoming challenges posed by the loss of localization.

We continue the study of the operator of generalized Maxwell equations and completely discover the behavior of the solutions of the time-harmonic equations as the frequency tends to zero. Thereby, we identify degenerate operators in terms of special 'polynomially growing' solutions of a corresponding static problem, which must be added to the 'usual' Neumann series in order to describe the low frequency asymptotic adequately.

In this paper the asymptotic behavior of elliptic nonlocal equations in cylindrical domains is studied. These equations are associated with symmetric pure jump Lévy stochastic processes and are generalisation of equations involving the fractional Laplacian operator. First we consider the existence and uniqueness of weak solutions in appropriate Hilbert spaces associated with the Lévy measure and then study the asymptotic behavior in cylinders.

In this note we elaborate on the asymptotic behavior of the spectral gap of a class of discrete Schrödinger operators defined on a path graph in the limit of infinite volume. We confirm recent results and generalize them to a larger class of potentials using entirely different methods. Notably, we also resolve a conjecture previously proposed in this context. This then yields new insights into the rate at which the spectral gap tends to zero as the volume increases.