共查询到20条相似文献,搜索用时 31 毫秒
1.
Yoshikazu Giga Katsuya Inui Alex Mahalov Shin’ya Matsui Jürgen Saal 《Archive for Rational Mechanics and Analysis》2007,186(2):177-224
We prove time local existence and uniqueness of solutions to a boundary layer problem in a rotating frame around the stationary
solution called the Ekman spiral. We choose initial data in the vector-valued homogeneous Besov space for 2 < p < ∞. Here the L
p
-integrability is imposed in the normal direction, while we may have no decay in tangential components, since the Besov space
contains nondecaying functions such as almost periodic functions. A crucial ingredient is theory for vector-valued homogeneous
Besov spaces. For instance we provide and apply an operator-valued bounded H
∞-calculus for the Laplacian in for a general Banach space . 相似文献
2.
We study the global attractor of the non-autonomous 2D Navier–Stokes (N.–S.) system with singularly oscillating external force of the form . If the functions g
0(x, t) and g
1 (z, t) are translation bounded in the corresponding spaces, then it is known that the global attractor is bounded in the space H, however, its norm may be unbounded as since the magnitude of the external force is growing. Assuming that the function g
1 (z, t) has a divergence representation of the form where the functions (see Section 3), we prove that the global attractors of the N.–S. equations are uniformly bounded with respect to for all . We also consider the “limiting” 2D N.–S. system with external force g
0(x, t). We have found an estimate for the deviation of a solution of the original N.–S. system from a solution u
0(x, t) of the “limiting” N.–S. system with the same initial data. If the function g
1 (z, t) admits the divergence representation, the functions g
0(x, t) and g
1 (z, t) are translation compact in the corresponding spaces, and , then we prove that the global attractors converges to the global attractor of the “limiting” system as in the norm of H. In the last section, we present an estimate for the Hausdorff deviation of from of the form: in the case, when the global attractor is exponential (the Grashof number of the “limiting” 2D N.–S. system is small).
相似文献
3.
We study the limit of the hyperbolic–parabolic approximation
The function is defined in such a way as to guarantee that the initial boundary value problem is well posed even if is not invertible. The data and are constant. When is invertible, the previous problem takes the simpler form
Again, the data and are constant. The conservative case is included in the previous formulations. Convergence of the , smallness of the total variation and other technical hypotheses are assumed, and a complete characterization of the limit
is provided. The most interesting points are the following: First, the boundary characteristic case is considered, that is,
one eigenvalue of can be 0. Second, as pointed out before, we take into account the possibility that is not invertible. To deal with this case, we take as hypotheses conditions that were introduced by Kawashima and Shizuta
relying on physically meaningful examples. We also introduce a new condition of block linear degeneracy. We prove that, if
this condition is not satisfied, then pathological behaviors may occur. 相似文献
4.
The unsteady dynamics of the Stokes flows, where
, is shown to verify the vector potential–vorticity (
) correlation
, where the field
is the pressure-gradient vector potential defined by
. This correlation is analyzed for the Stokes eigenmodes,
, subjected to no-slip boundary conditions on any two-dimensional (2D) closed contour or three-dimensional (3D) surface. It is established that an asymptotic linear relationship appears, verified in the core part of the domain, between the vector potential and vorticity,
, where
is a constant offset field, possibly zero. 相似文献
5.
Let be the set of m × m matrices A(λ) depending analytically on a parameter λ in a closed interval . Consider one-parameter families of quasi-periodic linear differential equations: , where is analytic and sufficiently small. We prove that there is an open and dense set in , such that for each the equation can be reduced to an equation with constant coefficients by a quasi-periodic linear transformation for almost
all in Lebesgue measure sense provided that g is sufficiently small. The result gives an affirmative answer to a conjecture of Eliasson (In: Proceeding of Symposia in
Pure Mathematics).
Dedicated to Professor Zhifen Zhang on the occasion of her 80th birthday 相似文献
6.
Thierry Cazenave Flávio Dickstein Fred B. Weissler 《Journal of Dynamics and Differential Equations》2007,19(3):789-818
In this paper, we construct solutions u(t,x) of the heat equation on such that has nontrivial limit points in as t → ∞ for certain values of μ > 0 and β > 1/2. We also show the existence of solutions of this type for nonlinear heat equations.
相似文献
7.
Giuseppe Da Prato Arnaud Debussche 《Journal of Dynamics and Differential Equations》2008,20(2):301-335
We study the long time behavior of the solution X(t, s, x) of a 2D-Navier–Stokes equation subjected to a periodic time dependent forcing term. We prove in particular that as , approaches a periodic orbit independently of s and x for any continuous and bounded real function .
相似文献
8.
Joel Avrin 《Journal of Dynamics and Differential Equations》2008,20(2):479-518
We obtain attractor and inertial-manifold results for a class of 3D turbulent flow models on a periodic spatial domain in
which hyperviscous terms are added spectrally to the standard incompressible Navier–Stokes equations (NSE). Let P
m
be the projection onto the first m eigenspaces of A =−Δ, let μ and α be positive constants with α ≥3/2, and let Q
m
=I − P
m
, then we add to the NSE operators μ A
φ in a general family such that A
φ≥Q
m
A
α in the sense of quadratic forms. The models are motivated by characteristics of spectral eddy-viscosity (SEV) and spectral
vanishing viscosity (SVV) models. A distinguished class of our models adds extra hyperviscosity terms only to high wavenumbers
past a cutoff λ
m0
where m
0 ≤ m, so that for large enough m
0 the inertial-range wavenumbers see only standard NSE viscosity.
We first obtain estimates on the Hausdorff and fractal dimensions of the attractor (respectively and ). For a constant K
α on the order of unity we show if μ ≥ ν that and if μ ≤ ν that where ν is the standard viscosity coefficient, l
0 = λ1−1/2 represents characteristic macroscopic length, and is the Kolmogorov length scale, i.e. where is Kolmogorov’s mean rate of dissipation of energy in turbulent flow. All bracketed constants and K
α are dimensionless and scale-invariant. The estimate grows in m due to the term λ
m
/λ1 but at a rate lower than m
3/5, and the estimate grows in μ as the relative size of ν to μ. The exponent on is significantly less than the Landau–Lifschitz predicted value of 3. If we impose the condition , the estimates become for μ ≥ ν and for μ ≤ ν. This result holds independently of α, with K
α and c
α independent of m. In an SVV example μ ≥ ν, and for μ ≤ ν aspects of SEV theory and observation suggest setting for 1/c within α orders of magnitude of unity, giving the estimate where c
α is within an order of magnitude of unity. These choices give straight-up or nearly straight-up agreement with the Landau–Lifschitz
predictions for the number of degrees of freedom in 3D turbulent flow with m so large that (e.g. in the distinguished-class case for m
0 large enough) we would expect our solutions to be very good if not virtually indistinguishable approximants to standard NSE
solutions. We would expect lower choices of λ
m
(e.g. with a > 1) to still give good NSE approximation with lower powers on l
0/l
ε, showing the potential of the model to reduce the number of degrees of freedom needed in practical simulations. For the choice
, motivated by the Chapman–Enskog expansion in the case m = 0, the condition becomes , giving agreement with Landau–Lifschitz for smaller values of λ
m
then as above but still large enough to suggest good NSE approximation. Our final results establish the existence of a inertial
manifold for reasonably wide classes of the above models using the Foias/Sell/Temam theory. The first of these results obtains such
an of dimension N > m for the general class of operators A
φ if α > 5/2.
The special class of A
φ such that P
m
A
φ = 0 and Q
m
A
φ ≥ Q
m
A
α has a unique spectral-gap property which we can use whenever α ≥ 3/2 to show that we have an inertial manifold of dimension m if m is large enough. As a corollary, for most of the cases of the operators A
φ in the distinguished-class case that we expect will be typically used in practice we also obtain an , now of dimension m
0 for m
0 large enough, though under conditions requiring generally larger m
0 than the m in the special class. In both cases, for large enough m (respectively m
0), we have an inertial manifold for a system in which the inertial range essentially behaves according to standard NSE physics,
and in particular trajectories on are controlled by essentially NSE dynamics.
相似文献
9.
Robert Jensen Changyou Wang Yifeng Yu 《Archive for Rational Mechanics and Analysis》2008,190(2):347-370
For a bounded domain and , assume that is convex and coercive, and that has no interior points. Then we establish the uniqueness of viscosity solutions to the Dirichlet problem of Aronsson’s equation:
For H = H(p, x) depending on x, we illustrate the connection between the uniqueness and nonuniqueness of viscosity solutions to Aronsson’s equation and
that of the Hamilton–Jacobi equation .
Supported by NSF DMS 0601162.
Supported by NSF DMS 0601403. 相似文献
10.
A particle nonlinear two-scale turbulence model is proposed for simulating the anisotropic turbulent two-phase flow. The particle kinetic energy equation
for two-scale fluctuation, particle energy transfer rate equation for large-scale fluctuation, and particle turbulent kinetic
energy dissipation rate equation for small-scale fluctuation are derived and closed. This model is used to simulate gas–particle
flows in a sudden-expansion chamber. The simulation is compared with the experiment and with those obtained by using another
two kinds of tow-phase turbulence model, such as the single-scale two-phase turbulence model and the particle two-scale second-order moment (USM) two-phase turbulence model. It is shown that
the present model gives simulation in much better agreement with the experiment than the single-scale two-phase turbulence model does and is almost as good as the particle two-scale USM turbulence model.
The project supported by China Postdoctoral Science Foundation (2004036239). 相似文献
11.
12.
An analysis is presented for the unsteady laminar flow of an incompressible Newtonian fluid in an annulus between two concentric spheres rotating about a common axis of symmetry. A solution of the Navier-Stokes equations is obtained by employing an iterative technique. The solution is valid for small values of Reynolds numbers and acceleration parameters of the spheres. In applying the results of this analysis to a rotationally accelerating sphere, a virtual moment of intertia is introduced to account for the local inertia of the fluid.Nomenclature
R
i
radius of the inner sphere
-
R
o
radius of the outer sphere
-
radial coordinate
-
r
dimensionless radial coordinate,
-
meridional coordinate
-
azimuthal coordinate
-
time
-
t
dimensionless time,
- Re
i
instantaneous Reynolds number of the inner sphere,
i
R
k
2
/
- Re
o
instantaneous Reynolds number of the outer sphere,
o
R
o
2
/
-
radial velocity component
-
V
r
dimensionless radial velocity component,
-
meridional velocity component
- V
dimensionless meridional velocity component,
-
azimuthal velocity component
-
V
dimensionless azimuthal velocity component,
-
viscous torque
-
T
dimensionless viscous torque,
-
viscous torque at surface of inner sphere
-
T
i
dimensionless viscous torque at surface of inner sphere,
-
viscous torque at surface of outer sphere
-
T
o
dimensionless viscous torque at surface of outer sphere,
-
externally applied torque on inner sphere
-
T
p,i
dimensionless applied torque on inner sphere,
-
moment of inertia of inner sphere
-
Z
i
dimensionless moment of inertia of inner sphere,
-
virtual moment of inertia of inner sphere
-
Z
i,v
dimensionless virtual moment of inertia of inner sphere,
-
virtual moment of inertia of outer sphere
-
i
instantaneous angular velocity of the inner sphere
-
o
instantaneous angular velocity of the outer sphere
-
density of fluid
-
viscosity of fluid
-
kinematic viscosity of fluid,/
-
radius ratio,R
i/R
o
-
swirl function,
-
dimensionless swirl function,
-
stream function
-
dimensionless stream function,
-
i
acceleration parameter for the inner sphere,
-
o
acceleration parameter for the outer sphere,
-
shear stress
-
r
dimensionless shear stress,
相似文献
13.
A Jordan Curve Spanned by a Complete Minimal Surface 总被引:1,自引:0,他引:1
Francisco Martín Nikolai Nadirashvili 《Archive for Rational Mechanics and Analysis》2007,184(2):285-301
In this paper we construct complete (conformal) minimal immersions
which admit continuous extensions to the closed disk,
. Moreover,
is a homeomorphism and
is a (non-rectifiable) Jordan curve with Hausdorff dimension 1.
It turns out that the set of Jordan curves
constructed by the above procedure is dense in the space of Jordan curves of
with the Hausdorff metric. 相似文献
14.
A system is described which allows the recreation of the three-dimensional motion and deformation of a single hydrogen bubble time-line in time and space. By digitally interfacing dualview video sequences of a bubble time-line with a computer-aided display system, the Lagrangian motion of the bubble-line can be displayed in any viewing perspective desired. The u and v velocity history of the bubble-line can be rapidly established and displayed for any spanwise location on the recreated pattern. The application of the system to the study of turbulent boundary layer structure in the near-wall region is demonstrated.List of Symbols
Reynolds number based on momentum thickness u
/v
- t+
nondimensional time
-
u
shear velocity
-
u
local streamwise velocity, x-direction
-
u
+
nondimensional streamwise velocity
-
v
local normal velocity, -direction
-
x
+
nondimensional coordinate in streamwise direction
- +
nondimensional coordinate normal to wall
-
+
wire
wire nondimensional location of hydrogen bubble-wire normal to wall
-
z
+
nondimensional spanwise coordinate
-
momentum thickness
-
v
kinematic viscosity
-
W
wall shear stress 相似文献
15.
16.
Changyou Wang 《Archive for Rational Mechanics and Analysis》2008,188(2):351-369
For any compact n-dimensional Riemannian manifold (M, g) without boundary, a compact Riemannian manifold without boundary, and 0 < T ≦ +∞, we prove that for n ≧ 4, if u : M × (0, T] → N is a weak solution to the heat flow of harmonic maps such that , then u ∈C
∞(M × (0, T], N). As a consequence, we show that for n ≧3, if 0 < T < +∞ is the maximal time interval for the unique smooth solution u ∈C
∞(M × [0, T), N) of (1.1), then blows up as t ↑ T. 相似文献
17.
The response of a turbulent boundary layer to a short roughness strip is investigated using laser Doppler velocimetry (LDV) and laser induced fluorescence (LIF). Skin friction coefficients are inferred from accurate near-wall measurements. There is an undershoot in
, where
is the undisturbed smooth wall skin friction coefficient, immediately after the strip. Downstream of the strip,
overshoots before relaxing back to unity in an oscillatory manner. The roughness strip has a major effect on the turbulent stresses
; these quantities increase, relative to the undisturbed smooth wall, in the region between the two internal layers originating at the upstream and downstream edges of the strip. The increase in the ratio
suggests a decrease in near-wall anisotropy. From the flow visualizations, it is inferred that streamwise vortical structures are weakened immediately downstream of the strip. Consistently, streamwise length scales are also reduced; direct support for this is provided by measured two-point velocity correlations. 相似文献
18.
Philippe G. Ciarlet Liliana Gratie Cristinel Mardare 《Archive for Rational Mechanics and Analysis》2008,188(3):457-473
The fundamental theorem of surface theory classically asserts that, if a field of positive-definite symmetric matrices (a
αβ
) of order two and a field of symmetric matrices (b
αβ
) of order two together satisfy the Gauss and Codazzi-Mainardi equations in a simply connected open subset ω of , then there exists an immersion such that these fields are the first and second fundamental forms of the surface , and this surface is unique up to proper isometries in . The main purpose of this paper is to identify new compatibility conditions, expressed again in terms of the functions a
αβ
and b
αβ
, that likewise lead to a similar existence and uniqueness theorem. These conditions take the form of the matrix equation
where A
1 and A
2 are antisymmetric matrix fields of order three that are functions of the fields (a
αβ
) and (b
αβ
), the field (a
αβ
) appearing in particular through the square root U of the matrix field The main novelty in the proof of existence then lies in an explicit use of the rotation field R that appears in the polar factorization of the restriction to the unknown surface of the gradient of the canonical three-dimensional extension of the unknown immersion . In this sense, the present approach is more “geometrical” than the classical one. As in the recent extension of the fundamental
theorem of surface theory set out by S. Mardare [20–22], the unknown immersion is found in the present approach to exist in function spaces “with little regularity”, such as , p > 2. This work also constitutes a first step towards the mathematical justification of models for nonlinearly elastic shells
where rotation fields are introduced as bona fide unknowns. 相似文献
19.
Forced convective heat transfer coefficients and friction factors for flow of water in microchannels with a rectangular cross
section were measured. An integrated microsystem consisting of five microchannels on one side and a localized heater and seven
polysilicon temperature sensors along the selected channels on the other side was fabricated using a double-polished-prime
silicon wafer. For the microchannels tested, the friction factor constant
obtained are values between 53.7 and 60.4, which are close to the theoretical value from a correlation for macroscopic dimension,
56.9 for D
h
= 100 μm. The heat transfer coefficients obtained by measuring the wall temperature along the micro channels were linearly
dependent on the wall temperature, in turn, the heat transfer mechanism is strongly dependent on the fluid properties such
as viscosity. The measured Nusselt number in the laminar flow regime tested could be correlated by which is quite different from the constant value obtained in macrochannels. 相似文献
20.
We prove the existence of multiscale Young measures associated with almost periodic homogenization. We give applications of
this tool in the homogenization of nonlinear partial differential equations with an almost periodic structure, such as scalar
conservation laws, nonlinear transport equations, Hamilton–Jacobi equations and fully nonlinear elliptic equations. Motivated
by the application in nonlinear transport equations, we also prove basic results on flows generated by Lipschitz almost periodic
vector fields, which are of interest in their own. In our analysis, an important role is played by the so-called Bohr compactification
of ; this is a natural parameter space for the Young measures. Our homogenization results provide also the asymptotic behavior
for the whole set of -translates of the solutions, which is in the spirit of recent studies on the homogenization of stationary ergodic processes. 相似文献