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1.
Eric Lombardi 《Archive for Rational Mechanics and Analysis》1997,137(3):227-304
In this paper a class of reversible analytic vector fields is investigated near an equilibrium. For these vector fields,
the part of the spectrum of the differential at the equilibrium which lies near the imaginary axis comes from the perturbation
of a double eigenvalue 0 and two simple eigenvalues , .
In the first part of this paper, we study the 4-dimensional problem. The existence of a family of solutions homoclinic to
periodic orbits of size less than μN for any fixed N, where μ is the bifurcation parameter, is known for vector fields. Using the analyticity of the vector field, we prove here the existence of solutions homoclinic to a periodic
orbit the size of which is exponentially small ( of order . This result receives its significance from the still unsolved question of whether there exist solutions that are homoclinic
to the equilibrium or whether the amplitudes of the oscillations at infinity have a positive infimum.
In the second part of this paper we prove that the exponential estimates still hold in infinite dimensions. This result cannot
be simply obtained from the study of the 4-dimensional analysis by a center-manifold reduction since this result is based
on analyticity of the vector field.
One example of such a vector field in infinite dimensions occurs when describing the irrotational flow of an inviscid fluid
layer under the influence of gravity and small surface tension (Bond number ) for a Froude number F close to 1. In this context a homoclinic solution to a periodic orbit is called a generalized solitary wave. Our work shows
that there exist generalized solitary waves with exponentially small oscillations at infinity. More precisely, we prove that
for each F close enough to 1, there exist two reversible solutions homoclinic to a periodic orbit, the size of which is less than , l being any number between 0 and π and satisfying .
(Accepted October 2, 1995) 相似文献
2.
Classical capillarity theory is based on a hypothesis that virtual motions of fluid particles distinct from those on a surface
interface have no effect on the form of the interface. That hypothesis cannot be supported for a compressible fluid. A heuristic
reasoning suggests that even small amounts of compressibility could have significant effect on surface behavior. In an earlier
work, Finn took a partial account of compressibility, and formulated a variant of the classical capillarity equation for fluid
surface height in a vertical capillary tube; he was led to a necessary condition for existence of a solution with prescribed
mass in a tube closed at the bottom. For a circular tube, he proved that the condition also suffices, and that solutions are
uniquely determined for any contact angle γ.
Later Finn took more complete account of compressibility and obtained a new equation of highly nonlinear character but for
which the same necessary condition holds. In the present work we consider that equation for circular tubes. We prove that
the necessary condition again suffices for existence when 0 ≤ γ < π, and we establish uniqueness when 0 ≤ γ ≤ π/2.
Our result is put into relief by the observation that for the unconstrained problem of a tube dipped into an infinite liquid
bath, solutions do not in general exist when γ > π/2. Presumably an actual fluid would in that case descend to the bottom
of the tube. This kind of singular behavior does not occur for the equation previously considered, nor does it occur in the
present case under the presence of a mass constraint. 相似文献
3.
The paper deals with a scalar wave equation of the form where is a Prandtl–Ishlinskii operator and are given functions. This equation describes longitudinal vibrations of an elastoplastic rod. The mass density and the Prandtl–Ishlinskii distribution function are allowed to depend on the space variable x. We prove existence, uniqueness and regularity of solution to a corresponding initial-boundary value problem. The system
is then homogenized by considering a sequence of equations of the above type with spatially periodic data and , where the spatial period tends to 0. We identify the homogenized limits and and prove the convergence of solutions to the solution of the homogenized equation.
Received June 17, 1999 相似文献
4.
V. S. Potapkin 《Fluid Dynamics》1980,15(4):545-553
Composite waves on the surface of the stationary flow of a heavy ideal incompressible liquid are steady forced waves of finite amplitude which do not disappear when the pressure on the free surface becomes constant but rather are transformed into free nonlinear waves [1]. It will be shown that such waves correspond to the case of nonlinear resonance, and mathematically to the bifurcation of the solution of the fundamental integral equation describing these waves. In [2], a study is made of the problem of composite waves in a flow of finite depth generated by a variable pressure with periodic distribution along the surface of the flow. In [3], such waves are considered for a flow with a wavy bottom. In this case, composite waves are defined as steady forced waves of finite amplitude that, when the pressure becomes constant and the bottom is straightened, do not disappear but are transformed into free nonlinear waves over a flat horizontal bottom. However, an existence and uniqueness theorem was not proved for this case. The aim of the present paper is to fill this gap and investigate the conditions under which such waves can arise.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 88–98, July–August, 1980. 相似文献
5.
Yanguang Li 《Journal of Dynamics and Differential Equations》2003,15(4):699-730
For finite-dimensional maps and periodic systems, Palmer rigorously proved Smale horseshoe theorem using shadowing lemma in 1988 [20]. For infinite-dimensional maps and periodic systems, such a proof was completed by Steinlein and Walther in 1990 [30], and Henry in 1994 [9]. For finite-dimensional autonomous systems, such a proof was accomplished by Palmer in 1996 [17]. For infinite-dimensional autonomous systems, the current article offers such a proof. First we prove an Inclination Lemma to set up a coordinate system around a pseudo-orbit. Then we utilize graph transform and the concept of persistence of invariant manifold, to prove the existence of a shadowing orbit. 相似文献
6.
In this paper we study the two-dimensional hydrostatic Euler equations in a periodic channel. We prove the local existence
and uniqueness of H
s
solutions under the local Rayleigh condition. This extends Brenier’s (Nonlinearity 12(3):495–512, 1999) existence result by removing an artificial condition and proving uniqueness. In addition, we prove weak–strong uniqueness,
mathematical justification of the formal derivation and stability of the hydrostatic Euler equations. These results are based
on weighted H
s
a priori estimates, which come from a new type of nonlinear cancellation between velocity and vorticity. 相似文献
7.
We prove the existence of Cantor families of periodic solutions for nonlinear wave equations in higher spatial dimensions
with periodic boundary conditions. We study both forced and autonomous PDEs. In the latter case our theorems generalize previous
results of Bourgain to more general nonlinearities of class C
k
and assuming weaker non-resonance conditions. Our solutions have Sobolev regularity both in time and space. The proofs are
based on a differentiable Nash–Moser iteration scheme, where it is sufficient to get estimates of interpolation-type for the
inverse linearized operators. Our approach works also in presence of very large “clusters of small divisors”. 相似文献
8.
I. B. Bakholdin 《Fluid Dynamics》2000,35(4):584-590
Jumps of the bore type arising in a fluid layer with an ice sheet are investigated. These jump structures are considered for
a determining mechanism in the form of dispersion due to the presence of an ice sheet. For this purpose a generalized Korteweg-de
Vires equation [1] is used. The structure of these jumps consists of a wave zone that expand with time. On the boundary of
the wave zone there are transitions between uniform and periodic states which can be locally considered as jumps. Among them
are jumps which can be regarded as steady in the coordinate system moving with the boundary of the wave zone. These are jumps
between a sequence of solitons and a uniform state (jumps of soliton type) on the boundary of the wave zone and jumps between
periodic and uniform states (jumps with radiation). In addition, there are jumps which are unsteady even from the standpoint
of a local analysis. In order to investigate the effect of dissipation processes on the jumps considered a system of generalized
Boussinesq equations is derived with allowance for bottom slope and bottom and ice friction. The jump damping process is investigated
numerically. This system of equations also makes it possible to investigate undamped jumps of the floodwater wave type.
Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 139–146. July–August, 2000. 相似文献
9.
葛渭高 《应用数学和力学(英文版)》1985,6(10):1007-1016
In this paper we prove some theorems on the existence of harmonic solutions of somesecond-order nonlinear equations under a periodic force.These theorems extend relevantresults in refs.[1]-[8]. 相似文献
10.
Noureddine Mehidi 《Meccanica》2007,42(4):403-407
We first approximate the solutions of the nonautonomous oscillating suspension point pendulum equation by the solutions of
a second order autonomous differential equation. Using the strict monotonicity of the periodic solutions of the approximating
equation, we prove the existence of a large number of subharmonic periodic solutions of the plane pendulum when its point
of suspension is excited parametrically. 相似文献
11.
We investigate the Andronov-Hopf bifurcation of the birth of a periodic solution from a space-homogeneous stationary solution
of the Neumann problem on a disk for a parabolic equation with a transformation of space variables in the case where this
transformation is the composition of a rotation by a constant angle and a radial contraction. Under general assumptions, we
prove a theorem on the existence of a rotating structure, deduce conditions for its orbital stability, and construct its asymptotic
form.
__________
Translated from Neliniini Kolyvannya, Vol. 9, No. 2, pp. 155–169, April–June, 2006. 相似文献
12.
The existence and linear stability problem for the Stokes periodic wavetrain on fluids of finite depth is formulated in terms
of the spatial and temporal Hamiltonian structure of the water-wave problem. A proof, within the Hamiltonian framework, of
instability of the Stokes periodic wavetrain is presented. A Hamiltonian center-manifold analysis reduces the linear stability
problem to an ordinary differential eigenvalue problem on ℝ4. A projection of the reduced stability problem onto the tangent space of the 2-manifold of periodic Stokes waves is used
to prove the existence of a dispersion relation Λ(λ,σ, I
1, I
2)=0 where λ ε ℂ is the stability exponent for the Stokes wave with amplitude I
1 and mass flux I
2 and σ is the “sideband’ or spatial exponent. A rigorous analysis of the dispersion relation proves the result, first discovered
in the 1960's, that the Stokes gravity wavetrain of sufficiently small amplitude is unstable for F ε (0,F0) where F
0 ≈ 0.8 and F is the Froude number. 相似文献
13.
Susanna Terracini Andrea Venturelli 《Archive for Rational Mechanics and Analysis》2007,184(3):465-493
We consider the problem of 2N bodies of equal masses in
for the Newtonian-like weak-force potential r
−σ, and we prove the existence of a family of collision-free nonplanar and nonhomographic symmetric solutions that are periodic
modulo rotations. In addition, the rotation number with respect to the vertical axis ranges in a suitable interval. These
solutions have the hip-hop symmetry, a generalization of that introduced in [19], for the case of many bodies and taking account of a topological constraint.
The argument exploits the variational structure of the problem, and is based on the minimization of Lagrangian action on a
given class of paths. 相似文献
14.
G. Rein 《Archive for Rational Mechanics and Analysis》1997,140(4):335-351
We consider the Vlasov‐Poisson system in a cosmological setting as studied in [18] and prove nonlinear stability of homogeneous
solutions against small, spatially periodic perturbations in the L
∞‐norm of the spatial mass density. This result is connected with the question of how large scale structures such as galaxies
have evolved out of the homogeneous state of the early universe.
(Accepted June 28, 1996) 相似文献
15.
Y. -G. Oh N. Sreenath P. S. Krishnaprasad J. E. Marsden 《Journal of Dynamics and Differential Equations》1989,1(3):269-298
We give a complete bifurcation and stability analysis for the relative equilibria of the dynamics of three coupled planar rigid bodies. We also use the equivariant Weinstein-Moser theorem to show the existence of two periodic orbits distinguished by symmetry type near the stable equilibrium. Finally we prove that the dynamics is chaotic in the sense of Poincaré-Birkhoff-Smale horseshoes using the version of Melnikov's method suitable for systems with symmetry due to Holmes and Marsden. 相似文献
16.
Periodic Solutions of the Elliptic Isosceles Restricted Three-body Problem with Collision 总被引:1,自引:0,他引:1
Lúcia de Fatima Brandão Claudio Vidal 《Journal of Dynamics and Differential Equations》2008,20(2):377-423
The elliptic isosceles restricted three-body problem with collision, is a restricted three-body problem where the primaries
move having consecutive elliptic collisions and the infinitesimal mass is moving in the plane perpendicular to the primaries
motion that passes through the center of mass of the primary system. Our purpose in this paper is to prove the existence of
many families of periodic solutions using Continuation’s method, where the perturbing parameter is related with the energy
of the primaries. This work is merely analytic and uses symmetry conditions and appropriate coordinates.
Partially supported by Dirección de Investigación UBB, 064608 3/RS. 相似文献
17.
U. Kadri R.F. Mudde R.V.A. Oliemans M. Bonizzi P. Andreussi 《International Journal of Multiphase Flow》2009
In stratified gas–liquid horizontal pipe flow, growing long wavelength waves may reach the top of the pipe and form a slug flow, or evolve into roll-waves. At certain flow conditions, slugs may grow to become extremely long, e.g. 500 pipe diameter. The existence of long slugs may cause operational upsets and a reduction in the flow efficiency. Therefore, predicting the flow conditions at which the long slugs appear contributes to a better design and management of the flow to maximize the flow efficiency. 相似文献
18.
Yasushi Taniuchi Tomoya Tashiro Tsuyoshi Yoneda 《Journal of Mathematical Fluid Mechanics》2010,12(4):594-612
We consider the nonstationary Euler equations in
\mathbbR2{\mathbb{R}}^2 with almost periodic unbounded vorticity. We show that a unique solution is always spatially almost periodic at any time
when the almost periodic initial data belongs to some function space. In order to prove this, we demonstrate the continuity
with respect to initial data which do not decay at spatial infinity. The proof of the continuity with respect to initial data
is based on that of Vishik’s uniqueness theorem. 相似文献
19.
V. I. Merkulov A. P. Mikhailov S. O. Savchenko 《Journal of Applied Mechanics and Technical Physics》1984,25(4):536-541
The effect of a rigid bottom of periodic form on small periodic oscillations of the free surface of a liquid is considered with the assumption of low amplitude roughness. The methodologically most significant study in this direction, [1], will be utilized. In [1] the steady-state problem for flow over an arbitrarily rough bottom was studied. Other studies have recently appeared on small free oscillations above a rough bottom. Essentially these have considered the effect of underwater obstacles and cavities on surface waves in the shallow-water approximation (for example, [2], [3]). Liquid oscillations in a layer of arbitrary depth slowly varying with length were considered in [4]. However, these results cannot be applied to the study of resonant interaction of gravitational waves with a periodically curved bottom.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 43–48, July–August, 1984. 相似文献
20.
With the consideration of mechanism of prevention and control for the spread of viral diseases, in this paper, we propose two novel virus dynamics models where state feedback control strategies are introduced. The first model incorporates the density of infected cells (or free virus) as control threshold value; we analytically show the existence and orbit stability of positive periodic solution. Theoretical results imply that the density of infected cells (or free virus) can be controlled within an adequate level. The other model determines the control strategies by monitoring the density of uninfected cells when it reaches a risk threshold value. We analytically prove the existence and orbit stability of semi-trivial periodic solution, which show that the viral disease dies out. Numerical simulations are carried out to illustrate the main results. 相似文献