共查询到20条相似文献,搜索用时 31 毫秒
1.
2.
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
3.
Stephan Blazy Sergueï Nazarov Maria Specovius-Neugebauer 《Journal of Mathematical Fluid Mechanics》2007,9(1):1-33
In a three-dimensional domain Ω with J cylindrical outlets to infinity the problem is treated how solutions to the stationary Stokes and Navier–Stokes system with
pressure conditions at infinity can be approximated by solutions on bounded subdomains. The optimal artificial boundary conditions
turn out to have singular coefficients. Existence, uniqueness and asymptotically precise estimates for the truncation error
are proved for the linear problem and for the nonlinear problem with small data. The results include also estimates for the
so called “do-nothing” condition. 相似文献
4.
Jiří Neustupa 《Journal of Mathematical Fluid Mechanics》2009,11(1):22-45
We derive a sufficient condition for stability of a steady solution of the Navier–Stokes equation in a 3D exterior domain
Ω. The condition is formulated as a requirement on integrability on the time interval (0, +∞) of a semigroup generated by
the linearized problem for perturbations, applied to a finite family of certain functions. The norm of the semigroup is measured
in a bounded sub-domain of Ω. We do not use any condition on “smallness” of the basic steady solution.
相似文献
5.
Sebastian Bönisch Vincent Heuveline Peter Wittwer 《Journal of Mathematical Fluid Mechanics》2008,10(1):45-70
We consider the problem of solving numerically the stationary incompressible Navier–Stokes equations in an exterior domain
in two dimensions. For numerical purposes we truncate the domain to a finite sub-domain, which leads to the problem of finding
so called “artificial boundary conditions” to replace the boundary conditions at infinity. To solve this problem we construct
– by combining results from dynamical systems theory with matched asymptotic expansion techniques based on the old ideas of
Goldstein and Van Dyke – a smooth divergence free vector field depending explicitly on drag and lift and describing the solution
to second and dominant third order, asymptotically at large distances from the body. The resulting expression appears to be
new, even on a formal level. This improves the method introduced by the authors in a previous paper and generalizes it to
non-symmetric flows. The numerical scheme determines the boundary conditions and the forces on the body in a self-consistent
way as an integral part of the solution process. When compared with our previous paper where first order asymptotic expressions
were used on the boundary, the inclusion of second and third order asymptotic terms further reduces the computational cost
for determining lift and drag to a given precision by typically another order of magnitude.
Peter Wittwer: Supported in part by the Fonds National Suisse. 相似文献
6.
We study the boundary-value problem associated with the Oseen system in the exterior of m Lipschitz domains of an euclidean point space
We show, among other things, that there are two positive constants
and α depending on the Lipschitz character of Ω such that: (i) if the boundary datum a belongs to Lq(∂Ω), with q ∈ [2,+∞), then there exists a solution (u, p), with
and u ∈ L∞(Ω) if a ∈ L∞(∂Ω), expressed by a simple layer potential plus a linear combination of regular explicit functions; as a consequence, u tends nontangentially to a almost everywhere on ∂Ω; (ii) if a ∈ W1-1/q,q(∂Ω), with
then ∇u, p ∈ Lq(Ω) and if a ∈ C0,μ(∂Ω), with μ ∈ [0, α), then
also, natural estimates holds. 相似文献
7.
We prove that in linear periodic gravity water waves there are no closed orbits for the water particles in the fluid. Each
particle experiences per period a backward-forward motion that leads overall to a forward drift.
This paper was written while both authors participated in the program “Wave Motion” at the Mittag-Leffler Institute, Stockholm,
in the Fall of 2005. 相似文献
8.
Petr Kaplický 《Journal of Mathematical Fluid Mechanics》2008,10(1):71-88
We study the steady flow of an anisotropic generalised Newtonian fluid under Dirichlet boundary conditions in a bounded domain
. The fluid is characterised by a nonlinear dependence of the stress tensor on the symmetric gradient of the velocity vector
field. We prove the existence of a C
1,α-solution of this problem under certain assumptions on the growth of the elliptic term. The result is global: we prove the
regularity up to the boundary of the domain Ω.
This research was supported by the grant GACR 201/03/0934; partially also by the grants
MSM 0021620839 and GAUK 262/2002/B-MAT/MFF. 相似文献
9.
We consider a problem on an ellipsoidal inhomogeneity in an infinitely extended homogeneous isotropic elastic medium. The
inhomogeneity differs from the ambient body in the elastic moduli (Poisson’s ratio ν and shear modulus μ) and in that it has intrinsic strains. We use the equivalent inclusion method to write out expressions
for the Helmholtz and Gibbs free energy of the inhomogeneity as quadratic forms in the intrinsic strains and strains at infinity.
The general expressions for the coefficients of these quadratic forms are written out as three rank four tensors characterizing
the contribution to the energy by the plastic strain (ɛ
p
2), by the strain at infinity (ɛ
02), and (only for the Gibbs energy) by the cross term ɛ
0
ɛ
p
. 相似文献
10.
Zdeněk Skalák 《Journal of Mathematical Fluid Mechanics》2007,9(4):565-587
In the paper we study the asymptotic dynamics of strong global solutions of the Navier Stokes equations. We are concerned
with the question whether or not a strong global solution w can pass through arbitrarily large fast decays. Avoiding results on higher regularity of w used in other papers we prove as the main result that for the case of homogeneous Navier–Stokes equations the answer is negative:
If [0, 1/4) and δ0 > 0, then the quotient remains bounded for all t ≥ 0 and δ∈[0, δ0]. This result is not valid for the non-homogeneous case. We present an example of a strong global solution w of the non-homogeneous Navier–Stokes equations, where the exterior force f decreases very quickly to zero for while w passes infinitely often through stages of arbitrarily large fast decays. Nevertheless, we show that for the non-homogeneous
case arbitrarily large fast decays (measured in the norm cannot occur at the time t in which the norm is greater than a given positive number.
相似文献
11.
Xinyu He 《Journal of Mathematical Fluid Mechanics》2004,6(4):389-404
A self-similar solution of the three-dimensional (3d) incompressible Euler equations is defined byu(x,t) =U(y)/t*-t)
α, y = x/(t* ~ t)β,α,β> 0, whereU(y) satisfiesζU + βy. ΔU + U. VU + VP = 0,divU = 0. For α = β = 1/2, which is the limiting case of Leray’s self-similar Navier—Stokes equations, we prove the existence of(U,P) ε H3(Ω,R3 X R) in a smooth bounded domain Ω, with the inflow boundary data of non-zero vorticity. This implies the possibility that
solutions of the Euler equations blow up at a timet = t*, t* < +∞. 相似文献
12.
H. Beirão da Veiga 《Journal of Mathematical Fluid Mechanics》2007,9(4):506-516
In reference [7] it is proved that the solution of the evolution Navier–Stokes equations in the whole of R
3 must be smooth if the direction of the vorticity is Lipschitz continuous with respect to the space variables. In reference
[5] the authors improve the above result by showing that Lipschitz continuity may be replaced by 1/2-H?lder continuity. A
central point in the proofs is to estimate the integral of the term (ω · ∇)u · ω, where u is the velocity and ω = ∇ × u is the vorticity. In reference [4] we extend the main estimates on the above integral term to solutions under the slip boundary
condition in the half-space R
+3. This allows an immediate extension to this problem of the 1/2-H?lder sufficient condition.
The aim of these notes is to show that under the non-slip boundary condition the above integral term may be estimated as well
in a similar, even simpler, way. Nevertheless, without further hypotheses, we are not able now to extend to the non slip (or
adherence) boundary condition the 1/2-H?lder sufficient condition. This is not due to the “nonlinear" term (ω · ∇)u · ω but to a boundary integral which is due to the combination of viscosity and adherence to the boundary. On the other hand,
by appealing to the properties of Green functions, we are able to consider here a regular, arbitrary open set Ω.
相似文献
13.
Yasunori Maekawa 《Journal of Mathematical Fluid Mechanics》2008,10(1):89-105
In this paper we establish spatial decay estimates for derivatives of vorticities solving the two-dimensional vorticity equations
equivalent to the Navier–Stokes equations. As an application we derive asymptotic behaviors of derivatives of vorticities
at time infinity. It is well known by now that the vorticity behaves asymptotically as the Oseen vortex provided that the
initial vorticity is integrable. We show that each derivative of the vorticity also behaves asymptotically as that of the
Oseen vortex.
相似文献
14.
Xiao-Biao Lin 《Journal of Dynamics and Differential Equations》2006,18(1):1-52
The Dafermos regularization of a system of n hyperbolic conservation laws in one space dimension has, near a Riemann solution consisting of n Lax shock waves, a self-similar solution u = u
ε(X/T). In Lin and Schecter (2003, SIAM J. Math. Anal. 35, 884–921) it is shown that the linearized Dafermos operator at such a solution may have two kinds of eigenvalues: fast eigenvalues of order 1/ε and slow eigenvalues of order one. The fast eigenvalues represent motion in an initial time layer, where near the shock waves solutions quickly converge to traveling-wave-like motion. The slow eigenvalues represent motion after the initial time layer, where motion between the shock waves is dominant. In this paper we use tools from dynamical systems and singular perturbation theory to study the slow eigenvalues. We show how to construct asymptotic expansions of eigenvalue-eigenfunction pairs to any order in ε. We also prove the existence of true eigenvalue-eigenfunction pairs near the asymptotic expansions. 相似文献
15.
Hiroko Morimoto 《Journal of Mathematical Fluid Mechanics》2007,9(3):411-418
Let Ω be a 2-dimensional bounded domain, symmetric with respect to the x2-axis. The boundary has several connected components, intersecting the x2-axis. The boundary value is symmetric with respect to the x2-axis satisfying the general outflow condition. The existence of the symmetric solution to the steady Navier–Stokes equations
was established by Amick [2] and Fujita [4]. Fujita [4] proved a key lemma concerning the solenoidal extension of the boundary
value by virtual drain method. In this note, we give a different proof via elementary approach by means of the stream function. 相似文献
16.
Under assumptions on smoothness of the initial velocity and the external body force, we prove that there exists T
0 > 0, ν
0 > 0 and a unique continuous family of strong solutions u
ν
(0 ≤ ν < ν
0) of the Euler or Navier–Stokes initial-boundary value problem on the time interval (0, T
0). In addition to the condition of the zero flux, the solutions of the Navier–Stokes equation satisfy certain natural boundary
conditions imposed on curl
u
ν
and curl
2
u
ν
.
相似文献
17.
18.
Hamiltonian evolution equations which are equivariant with respect to the action of a Lie group are models for physical phenomena
such as oceanographic flows, optical fibres and atmospheric flows, and such systems often have a wide variety of solitary-wave
or front solutions. In this paper, we present a new symplectic framework for analysing the spectral problem associated with
the linearization about such solitary waves and fronts. At the heart of the analysis is a multi-symplectic formulation of
Hamiltonian partial differential equations where a distinct symplectic structure is assigned for the time and space directions,
with a third symplectic structure – with two-form denoted by Ω– associated with a coordinate frame moving at the speed of
the wave. This leads to a geometric decomposition and symplectification of the Evans function formulation for the linearization
about solitary waves and fronts.
We introduce the concept of the symplectic Evans matrix, a matrix consisting of restricted Ω-symplectic forms. By applying Hodge duality to the exterior algebra formulation of
the Evans function, we find that the zeros of the Evans function correspond to zeros of the determinant of the symplectic
Evans matrix. Based on this formulation, we prove several new properties of the Evans function. Restricting the spectral
parameter λ to the real axis, we obtain rigorous results on the derivatives of the Evans function near the origin, based solely
on the abstract geometry of the equations, and results for the large |λ| behaviour which use primarily the symplectic structure,
but also extend to the non-symplectic case.
The Lie group symmetry affects the Evans function by generating zero eigenvalues of large multiplicity in the so-called systems
at infinity. We present a new geometric theory which describes precisely how these zero eigenvalues behave under perturbation.
By combining all these results, a new rigorous sufficient condition for instability of solitary waves and fronts is obtained.
The theory applies to a large class of solitary waves and fronts including waves which are bi-asymptotic to a nonconstant
manifold of states as $|x|$ tends to infinity. To illustrate the theory, it is applied to three examples: a Boussinesq model
from oceanography, a class of nonlinear Schr?dinger equations from optics and a nonlinear Klein-Gordon equation from atmospheric
dynamics.
Accepted August 7, 2000 ?Published online January 22, 2001 相似文献
19.
20.
Gilles Carbou 《Journal of Mathematical Fluid Mechanics》2008,10(1):126-158
In this paper we study a penalization method used to compute the flow of a viscous fluid around a thin layer of porous material.
Using a BKW method, we perform an asymptotic expansion of the solution when a little parameter, measuring the thickness of
the thin layer and the inverse of the penalization coefficient, tends to zero. We compare then this numerical method with
a Brinkman model for the flow around a porous thin layer.
相似文献