共查询到20条相似文献,搜索用时 15 毫秒
1.
Ovidiu Costin 《纯数学与应用数学通讯》1999,52(4):461-478
We extend the technique of asymptotic series matching to exponential asymptotics expansions (transseries) and show by using asymptotic information that the extension provides a method of finding singularities of solutions of nonlinear differential equations. This transasymptotic matching method is applied to Painlevé's first equation, P1. The solutions of P1 that are bounded in some direction towards infinity can be expressed as series of functions obtained by generalized Borel summation of formal transseries solutions; the series converge in a neighborhood of infinity. We prove (under certain restrictions) that the boundary of the region of convergence contains actual poles of the associated solution. As a consequence, the position of these exterior poles is derived from asymptotic data. In particular, we prove that the location of the outermost pole xp(C) on ℝ+ of a solution is monotonic in a parameter C describing its asymptotics on anti‐Stokes lines and obtain rigorous bounds for xp(C). We also derive the behavior of xp(C) for large C ∈ ℂ. The appendix gives a detailed classical proof that the only singularities of solutions of P1 are poles. © 1999 John Wiley & Sons, Inc. 相似文献
2.
We consider the asymptotic behaviour of positive solutions u of the conformal scalar curvature equation, , in the neighbourhood of isolated singularities in the standard Euclidean ball. Although asymptotic radial symmetry for such
solutions was proved some time ago, [2], we present a much simpler and more geometric derivation of this fact. We also discuss
a refinement, showing that any such solution is asymptotic to one of the deformed radial singular solutions. Finally we give
some applications of these refined asymptotics, first to computing the global Pohožaev invariants of solutions on the sphere
with isolated singularities, and then to the regularity of the moduli space of all such solutions.
Oblatum 26-II-1997 & 6-II-1998 / Published online: 12 November 1998 相似文献
3.
We investigate the regularization of Moore’s singularities by surface tension in the evolution of vortex sheets and its dependence
on the Weber number (which is inversely proportional to surface tension coefficient). The curvature of the vortex sheet, instead
of blowing up at finite time t
0, grows exponentially fast up to a O(We) limiting value close to t
0. We describe the analytic structure of the solutions and their self-similar features and characteristic scales in terms of
the Weber number in a O(We−1) neighborhood of the time at which, in absence of surface tension effects, Moore’s singularity would appear. Our arguments
rely on asymptotic techniques and are supported by full numerical simulations of the PDEs describing the evolution of vortex
sheets. 相似文献
4.
The detailed behavior of solutions to Stokes equations on regions with corners has been historically difficult to characterize. The solutions to Stokes equations on regions with corners are known to develop singularities in the vicinity of corners; in particular, the solutions are known to have infinite oscillations along almost every ray that meet at the corner. While the nature of singularities for the differential equation have been analyzed in great detail, very little is known about the nature of singularities for the corresponding integral equations. In this paper, we observe that, when the Stokes equation is formulated as a boundary integral equation, the solutions are representable by rapidly convergent series of the form , where t is the distance from the corner and the parameters μj, βj are real, and are determined via an explicit formula depending on the angle at the corner. In addition to being analytically perspicuous, these representations lend themselves to the construction of highly accurate and efficient numerical discretizations, significantly reducing the number of degrees of freedom required for the solution of the corresponding integral equations. The results are illustrated by several numerical examples. © 2020 Wiley Periodicals LLC 相似文献
5.
P. Karamian‐Surville J. Sanchez‐Hubert E. Sanchez Palencia 《Mathematical Methods in the Applied Sciences》2003,26(17):1451-1485
We consider problems of statics of thin elastic shells with hyperbolic middle surface subjected to boundary conditions ensuring the geometric rigidity of the surface. The asymptotic behaviour of the solutions when the relative thickness tends to zero is then given by the membrane approximation. It is a hyperbolic problem propagating singularities along the characteristics. We address here the reflection phenomena when the propagated singularities arrive to a boundary. As the boundary conditions are not the classical ones for a hyperbolic system, there are various cases of reflection. Roughly speaking, singularities provoked elsewhere are not reflected at all at a free boundary, whereas at a fixed (or clamped) boundary the reflected singularity is less singular than the incident one. Reflection of singularities provoked along a non‐characteristic curve C are also considered. Copyright © 2003 John Wiley & Sons, Ltd. 相似文献
6.
A Classification of Isolated Singularities of Elliptic Monge‐Ampére Equations in Dimension Two
下载免费PDF全文
![点击此处可从《纯数学与应用数学通讯》网站下载免费的PDF全文](/ch/ext_images/free.gif)
Let ?1 denote the space of solutions z(x,y) to an elliptic, real analytic Monge‐Ampére equation whose graphs have a non‐removable isolated singularity at the origin. We prove that ?1 is in one‐to‐one correspondence with ?2 × ?2, where ?2 is a suitable subset of the class of regular, real analytic, strictly convex Jordan curves in ?2. We also describe the asymptotic behavior of solutions of the Monge‐Ampére equation in the Ck‐smooth case, and a general existence theorem for isolated singularities of analytic solutions of the more general equation .© 2015 Wiley Periodicals, Inc. 相似文献
7.
Asymptotic representations of solutions to the boundary-value problems of elasticity theory are studied in domains with parabolic
exit at infinity (or in bounded domains with singularities like polynomial zero sharpness). The procedure of derivating a
formal asymptotic expansion looks like the algorithm of asymptotic analysis in domains. Under the Dirichlet conditions (displacements
are prescribed on the boundary of a domain), it is not hard to justify the power asymptotic series. It follows from the theorem
on the unique solvability of the problem in spaces of the type L2 containing degrees of distance r=|x| as weight multipliers. For the Neumann conditions (stresses are prescribed on the boundary
of a domain) an asymptotic expansion is justified by introducing the Eiry function Φ transforming the Lamé system to the biharmonic
equation. Due to the appearance of the Dirichlet condition on Φ, the study of the asymptotic behavior of a solution to the
last problem is simplified. The existence theorems and conditions for solvability of the “elastic” Neumann problem are presented.
These results are based on the weighted Korn inequality. Bibliography: 29 titles.
Translated fromProblemy Matematicheskogo Analiza. No. 15, 1995, pp. 162–200 相似文献
8.
We prove the global existence of weak solutions to the Navier-Stokes equations for compressible, heat-conducting flow in one space dimension with large, discontinuous initial data, and we obtain apriori estimates for these solutions which are independent of time, sufficient to determine their asymptotic behavior. In particular, we show that, as time goes to infinity, the solution tends to a constant state determined by the initial mass and the initial energy. and that the magnitudes of singularities in the solution decay to zero. 相似文献
9.
This paper deals with the basic approximation properties of the h–p version of the boundary element method (BEM) in ℝ3. We extend the results on the exponential convergence of the h–p version of the boundary element method on geometric meshes from problems in polygonal domains to problems in polyhedral domains. In 2D elliptic boundary value problems the solutions have only corner singularities whereas in 3D problems they contain additional edge and corner-edge singularities. The solutions of the corresponding boundary integral equations inherit those singularities. The detailed investigations in our analysis take care of the various types of those singularities. While edge singularities can be analysed using standard one-dimensional approximation results the corner-edge singularities demand a new analysis. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd. 相似文献
10.
J. Knežević-Miljanović 《Differential Equations》2011,47(1):149-152
We prove a number of theorems on asymptotic properties of solutions of the equation y″+x
a
y
σ
= 0, σ < 0. First, we prove the absence of solutions on (x
1, +∞) for some values of the parameters a and σ; after that, we obtain asymptotic formulas for solutions defined on (x
0, +∞). 相似文献
11.
L. J. Boele 《Journal of Difference Equations and Applications》2013,19(5):717-750
Given a nonlinear analytic difference equation of level 1 with a formal power series solution ? 0 we associate with it a stable manifold of solutions with asymptotic expansion ? 0. This manifold can be represented by means of Borel summable series. All solutions with asymptotic expansion ? 0 in some sector can be written as certain exponential series which are called transseries. Some of their properties are investigated: are resurgence properties and Stokes transition. Analogous problems for differential equations have been studied by Costin in [7] 相似文献
12.
We show that any homeomorphic solution of the Beltrami equation v from the Sobolev class W
loc1,1 is a so-called lower Q-homeomorphism with Q(z) = K
μ(z), where K
μ(z) is the dilatation ratio of this equation. On this basis, we develop the theory of boundary behavior and removing of singularities
of these solutions. 相似文献
13.
Gilbert Stengle 《Results in Mathematics》1988,13(1-2):173-187
We use an elementary method to draw analytic conclusions from divergent formal power series solutions of systems of differential equations containing a parameter and give some applications to the theory of turning points. Our main result shows that a divergent formal series transformation of one system into another in which the coefficients satisfy certain estimates is necessarily the asymptotic expansion of an actual transformation. We use it to show the following. Given a two dimensional system εPdy/dx = A(x,ε)y with A holomorphic at (x0,0), suppose that x0 is formally not a turning point in the sense that no singularities appear at x0 during the standard formal solution procedure with formal fractional power series in ε. Then the formal solution is necessarily a uniform asymptotic representation of a fundamental matrix of the system on a full neighborhood of x0. (This conclusion is known to fail under weaker hypotheses on A). We also obtain similar but less complete results for higher order systems under more specialized hypotheses. 相似文献
14.
V. V. Pelipenko 《Journal of Mathematical Sciences》1990,51(5):2560-2564
Using the connection of solutions of the sine-Gordon equationz
xy
=sinz with the construction of a grid of asymptotic lines on surfaces of constant negative curvature inE
3 we show that those surfaces and only those that have curvature –1 and any two asymptotic lines of a single family of which are congruent correspond to solutions of stationary-wave typez=z(x+by).Translated from Ukrainskií Geometricheskií Sbornik, No. 30, 1987, pp. 81–87. 相似文献
15.
L. G. Vachnadze N. A. Kiknadze K. Sh. Turashvili A. A. Khelashvili 《Theoretical and Mathematical Physics》1994,100(1):811-818
A three-gluon vertex that depends on a gauge vectorn
and is free of kinematic and spurion (np)–1 singularities in the physical region is constructed. It is shown that in the case of integer power-law asymptotic behavior of the gluon propagator there is separation of the singular contributions of the single- and two-loop diagrams in the Dyson—Schwinger equation. Necessary conditions for realization ofp
–4 asymptotic behavior are obtained, and it is shown that an important role in their fulfillment is played by the parts of the three-gluon vertex that depend on the gauge vector. A solution of these conditions is found; when they are fulfilled, they lead to infrared asymptotic behaviors of the complete Dyson—Schwinger equation and its single-loop approximation that are mutually compatible.Tbilisi State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No. 1, pp. 14–23, July, 1994. 相似文献
16.
Ricardo Weder∗ 《偏微分方程通讯》2013,38(5):675-691
We prove that the averaged scattering solutions to the Schrödinger equation with short-range electromagnetic potentials (V, A) where V(x) = O(|x|?ρ), A(x) = O(|x|?ρ), |x| → ∞, ρ > 1, are dense in the set of all solutions to the Schrödinger equation that are in L 2(K) where K is any connected bounded open set in ? n ,n ≥ 2, with smooth boundary. We use this result to prove that if two short-range electromagnetic potentials (V 1, A 1) and (V 2, A 2) in ? n , n ≥ 3, have the same scattering matrix at a fixed positive energy and if the electric potentials V j and the magnetic fields F j : = curl A j , j = 1, 2, coincide outside of some ball they necessarily coincide everywhere. In a previous paper of Weder and Yafaev the case of electric potentials and magnetic fields that are asymptotic sums of homogeneous terms at infinity was studied. It was proven that all these terms can be uniquely reconstructed from the singularities in the forward direction of the scattering amplitude at a fixed positive energy. The combination of the new uniqueness result of this paper and the result of Weder and Yafaev implies that the scattering matrix at a fixed positive energy uniquely determines electric potentials and magnetic fields that are a finite sum of homogeneous terms at infinity, or more generally, that are asymptotic sums of homogeneous terms that actually converge, respectively, to the electric potential and to the magnetic field. 相似文献
17.
We first show asymptotic L
2 bounds for a class of equations, which includes the Burger-Sivashinsly model for odd solutions with periodic boundary conditions.
We consider the conditional stability of stationary solutions of Kuramoto-Sivashinsky equation in the periodic setting. We
establish rigorously the general structure of the spectrum of the linearized operator, in particular the linear instability
of steady states. In addition, we show conditional asymptotic stability with asymptotic phase, under a natural spectral hypothesis
for the corresponding linearized operator. For the zero solution, we have more precise results. Namely, in the non-resonant
regime L ≠
n
π, we prove conditional asymptotic stability, provided one considers only mean value zero data. If, however, L = n
0
π (but still
ò\nolimits-LL u0(x) dx=0{\int\nolimits_{-L}^L u_0(x) dx=0}), then we have conditional orbital stability. More specifically, the solutions relax to a small (but generally non-zero)
function as long as the initial data are small and lie on a center-stable manifold of codimension 2(n
0 − 1). 相似文献
18.
Thomas M. Fischer George C. Hsiao Wolfgang L. Wendland 《Journal of Mathematical Analysis and Applications》1985,110(2):583-603
In this paper the rigorous justification of the formal asymptotic expansions constructed by the method of matched inner and outer expansions is established for the three-dimensional steady flow of a viscous, incompressible fluid past an arbitrary obstacle. The justification is based on the series representation of the solution to the Navier-Stokes equations due to Finn, and it involves the reductions of various exterior boundary value problems for the Stokes and Oseen equations to boundary integral equations of the first kind from which existence as well as asymptotic error estimates for the solutions are deduced. In particular, it is shown that the force exerted on the obstacle by the fluid admits the asymptotic representation F = A0 + A1Re + O(Re2 ln Re−1) as the Reynolds number Re → 0+, where the vectors A0 and A1 can be obtained from the method of matched inner and outer expansions. 相似文献
19.
20.
Octavian G. Mustafa 《Monatshefte für Mathematik》2007,150(2):133-140
Via a special integral transformation, asymptotic integration results for ordinary differential equations are used to establish
accurate asymptotic developments for radial solutions of the elliptic equation Δu + K(|x|)e
u
= 0, |x| > x
0 > 0, in the bidimensional case. 相似文献