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1.
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
2.
Craig Cowan Pierpaolo Esposito Nassif Ghoussoub Amir Moradifam 《Archive for Rational Mechanics and Analysis》2010,198(3):763-787
We study the regularity of the extremal solution of the semilinear biharmonic equation ${{\Delta^2} u=\frac{\lambda}{(1-u)^2}}We study the regularity of the extremal solution of the semilinear biharmonic equation
D2 u=\fracl(1-u)2{{\Delta^2} u=\frac{\lambda}{(1-u)^2}}, which models a simple micro-electromechanical system (MEMS) device on a ball
B ì \mathbbRN{B\subset{\mathbb{R}}^N}, under Dirichlet boundary conditions u=?n u=0{u=\partial_\nu u=0} on ?B{\partial B}. We complete here the results of Lin and Yang [14] regarding the identification of a “pull-in voltage” λ* > 0 such that a stable classical solution u
λ with 0 < u
λ < 1 exists for l ? (0,l*){\lambda\in (0,\lambda^*)}, while there is none of any kind when λ > λ*. Our main result asserts that the extremal solution ul*{u_{\lambda^*}} is regular (supB ul* < 1 ){({\rm sup}_B u_{\lambda^*} <1 )} provided
N \leqq 8{N \leqq 8} while ul*{u_{\lambda^*}} is singular (supB ul* = 1){({\rm sup}_B u_{\lambda^*} =1)} for
N \geqq 9{N \geqq 9}, in which case
1-C0|x|4/3 \leqq ul* (x) \leqq 1-|x|4/3{1-C_0|x|^{4/3} \leqq u_{\lambda^*} (x) \leqq 1-|x|^{4/3}} on the unit ball, where
C0:=(\fracl*[`(l)])\frac13{C_0:=\left(\frac{\lambda^*}{\overline{\lambda}}\right)^\frac{1}{3}} and
[`(l)]: = \frac89(N-\frac23)(N- \frac83){\bar{\lambda}:= \frac{8}{9}\left(N-\frac{2}{3}\right)\left(N- \frac{8}{3}\right)}. 相似文献
3.
Matthias Geissert Matthias Hess Matthias Hieber Céline Schwarz Kyriakos Stavrakidis 《Journal of Mathematical Fluid Mechanics》2010,12(1):47-60
Introducing a new localization method involving Bogovskiĭ's operator we give a short and new proof for maximal Lp – Lq-estimates for the solution of the Stokes equation. Moreover, it is shown that, up to constants, the Stokes operator is an
R{\mathcal{R}}-sectorial operator in Lps(W)L^{p}_{\sigma}(\Omega), 1 < p < ¥1 < p < \infty, of R{\mathcal{R}}-angle 0, for bounded or exterior domains of Ω. 相似文献
4.
This paper deals with the rational function approximation of the irrational transfer function
G(s) = \fracX(s)E(s) = \frac1[(t0s)2m + 2z(t0s)m + 1]G(s) = \frac{X(s)}{E(s)} = \frac{1}{[(\tau _{0}s)^{2m} + 2\zeta (\tau _{0}s)^{m} + 1]} of the fundamental linear fractional order differential equation
(t0)2m\fracd2mx(t)dt2m + 2z(t0)m\fracdmx(t)dtm + x(t) = e(t)(\tau_{0})^{2m}\frac{d^{2m}x(t)}{dt^{2m}} + 2\zeta(\tau_{0})^{m}\frac{d^{m}x(t)}{dt^{m}} + x(t) = e(t), for 0<m<1 and 0<ζ<1. An approximation method by a rational function, in a given frequency band, is presented and the impulse and
the step responses of this fractional order system are derived. Illustrative examples are also presented to show the exactitude
and the usefulness of the approximation method. 相似文献
5.
Alessio Figalli 《Archive for Rational Mechanics and Analysis》2010,195(2):533-560
Given two densities f and g, we consider the problem of transporting a fraction ${m \in [0,\min\{\|f\|_{L^1},\|g\|_{L^1}\}]}Given two densities f and g, we consider the problem of transporting a fraction m ? [0,min{||f||L1,||g||L1}]{m \in [0,\min\{\|f\|_{L^1},\|g\|_{L^1}\}]} of the mass of f onto g minimizing a transportation cost. If the cost per unit of mass is given by |x − y|2, we will see that uniqueness of solutions holds for m ? [||fùg||L1,min{||f||L1,||g||L1}]{m \in [\|f\wedge g\|_{L^1},\min\{\|f\|_{L^1},\|g\|_{L^1}\}]} . This extends the result of Caffarelli and McCann in Ann Math (in print), where the authors consider two densities with disjoint supports. The free boundaries of the active
regions are shown to be (n − 1)-rectifiable (provided the supports of f and g have Lipschitz boundaries), and under some weak regularity assumptions on the geometry of the supports they are also locally
semiconvex. Moreover, assuming f and g supported on two bounded strictly convex sets
W,L ì \mathbb Rn{{\Omega,\Lambda \subset \mathbb {R}^n}} , and bounded away from zero and infinity on their respective supports, C0,aloc{C^{0,\alpha}_{\rm loc}} regularity of the optimal transport map and local C
1 regularity of the free boundaries away from W?L{{\Omega\cap \Lambda}} are shown. Finally, the optimal transport map extends to a global homeomorphism between the active regions. 相似文献
6.
In this paper we study the following coupled Schr?dinger system, which can be seen as a critically coupled perturbed Brezis–Nirenberg problem: {ll-Du +l1 u = m1 u3+buv2, x ? W,-Dv +l2 v = m2 v3+bvu2, x ? W,u\geqq 0, v\geqq 0 in W, u=v=0 on ?W.\left\{\begin{array}{ll}-\Delta u +\lambda_1 u = \mu_1 u^3+\beta uv^2, \quad x\in \Omega,\\-\Delta v +\lambda_2 v =\mu_2 v^3+\beta vu^2, \quad x\in \Omega,\\u\geqq 0, v\geqq 0\, {\rm in}\, \Omega,\quad u=v=0 \quad {\rm on}\, \partial\Omega.\end{array}\right. 相似文献
7.
S. H. Saker 《Nonlinear Oscillations》2011,13(3):407-428
Our aim is to establish some sufficient conditions for the oscillation of the second-order quasilinear neutral functional
dynamic equation
( p(t)( [ y(t) + r(t)y( t(t) ) ]D )g )D + f( t,y( d(t) ) = 0, t ? [ t0,¥ )\mathbbT, {\left( {p(t){{\left( {{{\left[ {y(t) + r(t)y\left( {\tau (t)} \right)} \right]}^\Delta }} \right)}^\gamma }} \right)^\Delta } + f\left( {t,y\left( {\delta (t)} \right)} \right. = 0,\quad t \in {\left[ {{t_0},\infty } \right)_\mathbb{T}}, 相似文献
8.
We prove a regularity result for the anisotropic linear elasticity equation ${P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f}
9.
S. I. Maksymenko 《Nonlinear Oscillations》2010,13(2):196-227
Let
D2 ì \mathbbR2 {D^2} \subset {\mathbb{R}^2} be a closed unit 2-disk centered at the origin
O ? \mathbbR2 O \in {\mathbb{R}^2} and let F be a smooth vector field such that O is a unique singular point of F and all other orbits of F are simple closed curves wrapping once around O. Thus, topologically O is a “center” singularity. Let q:D2\{ O } ? ( 0, + ¥ ) \theta :D2\backslash \left\{ O \right\} \to \left( {0, + \infty } \right) be the function associating with each z ≠ O its period with respect to F. In general, such a function cannot be even continuously defined at O. Let also D+ (F) {\mathcal{D}^{+} }(F) be the group of diffeomorphisms of D
2 that preserve orientation and leave invariant each orbit of F. It is proved that θ smoothly extends to all of D
2 if and only if the 1-jet of F at O is a “rotation,” i.e.,
j1F(O) = - y\frac??x + x\frac??y {j^1}F(O) = - y\frac{\partial }{{\partial x}} + x\frac{\partial }{{\partial y}} . Then D+ (F) {\mathcal{D}^{+} }(F) is homotopy equivalent to a circle. 相似文献
10.
G. H. Keetels W. Kramer H. J. H. Clercx G. J. F. van Heijst 《Theoretical and Computational Fluid Dynamics》2011,25(5):293-300
Recently, numerical studies revealed two different scaling regimes of the peak enstrophy Z and palinstrophy P during the collision of a dipole with a no-slip wall [Clercx and van Heijst, Phys. Rev. E 65, 066305, 2002]: Z μ Re0.8{Z\propto{\rm Re}^{0.8}} and P μ Re2.25{P\propto {\rm Re}^{2.25}} for 5 × 102 ≤ Re ≤ 2 × 104 and Z μ Re0.5{Z\propto{\rm Re}^{0.5}} and P μ Re1.5{P\propto{\rm Re}^{1.5}} for Re ≥ 2 × 104 (with Re based on the velocity and size of the dipole). A critical Reynolds number Re
c
(here, Rec ? 2×104{{\rm Re}_c\approx 2\times 10^4}) is identified below which the interaction time of the dipole with the boundary layer depends on the kinematic viscosity
ν. The oscillating plate as a boundary-layer problem can then be used to mimick the vortex-wall interaction and the following
scaling relations are obtained: Z μ Re3/4, P μ Re9/4{Z\propto{\rm Re}^{3/4}, P\propto {\rm Re}^{9/4}} , and dP/dt μ Re11/4{\propto {\rm Re}^{11/4}} in agreement with the numerically obtained scaling laws. For Re ≥ Re
c
the interaction time of the dipole with the boundary layer becomes independent of the kinematic viscosity and, applying flat-plate
boundary-layer theory, this yields: Z μ Re1/2{Z\propto{\rm Re}^{1/2}} and P μ Re3/2{P\propto {\rm Re}^{3/2}}. 相似文献
11.
Vieri Benci Marco Ghimenti Anna Maria Micheletti 《Archive for Rational Mechanics and Analysis》2012,205(2):467-492
We study the behavior of the soliton solutions of the equation i\frac?y?t = - \frac12m Dy+ \frac12We¢(y) + V(x)y,i\frac{\partial\psi}{{\partial}t} = - \frac{1}{2m} \Delta\psi + \frac{1}{2}W_{\varepsilon}^{\prime}(\psi) + V(x){\psi}, 相似文献
12.
Katrin Schumacher 《Journal of Mathematical Fluid Mechanics》2009,11(4):552-571
We investigate the solvability of the instationary Navier–Stokes equations with fully inhomogeneous data in a bounded domain
W ì \mathbbRn \Omega \subset {{\mathbb{R}}^{n}} . The class of solutions is contained in Lr(0, T; Hb, qw (W))L^{r}(0, T; H^{\beta, q}_{w} (\Omega)), where Hb, qw (W){H^{\beta, q}_{w}} (\Omega) is a Bessel-Potential space with a Muckenhoupt weight w. In this context we derive solvability for small data, where this smallness can be realized by the restriction to a short
time interval. Depending on the order of this Bessel-Potential space we are dealing with strong solutions, weak solutions,
or with very weak solutions. 相似文献
13.
Mahesh Nerurkar 《Journal of Dynamics and Differential Equations》2011,23(3):451-473
Consider the class of C
r
-smooth
SL(2, \mathbb R){SL(2, \mathbb R)} valued cocycles, based on the rotation flow on the two torus with irrational rotation number α. We show that in this class,
(i) cocycles with positive Lyapunov exponents are dense and (ii) cocycles that are either uniformly hyperbolic or proximal
are generic, if α satisfies the following Liouville type condition:
|a-\fracpnqn| £ C exp (-qr+1+kn)\left|\alpha-\frac{p_n}{q_n}\right| \leq C {\rm exp} (-q^{r+1+\kappa}_{n}), where C > 0 and 0 < k < 1{0 < \kappa <1 } are some constants and
\fracPnqn{\frac{P_n}{q_n}} is some sequence of irreducible fractions. 相似文献
14.
Roberto Alicandro Marco Cicalese Antoine Gloria 《Archive for Rational Mechanics and Analysis》2011,200(3):881-943
This article is devoted to the study of the asymptotic behavior of a class of energies defined on stochastic lattices. Under
polynomial growth assumptions, we prove that the energy functionals Fe{F_\varepsilon} stored in the deformation of an e{{\varepsilon}}-scaling of a stochastic lattice Γ-converge to a continuous energy functional when e{{\varepsilon}} goes to zero. In particular, the limiting energy functional is of integral type, and deterministic if the lattice is ergodic.
We also generalize, to systems and nonlinear settings, well-known results on stochastic homogenization of discrete elliptic
equations. As an application of the main result, we prove the convergence of a discrete model for rubber towards the nonlinear
theory of continuum mechanics. We finally address some mechanical properties of the limiting models, such as frame-invariance,
isotropy and natural states. 相似文献
15.
We prove the existence of a global semigroup for conservative solutions of the nonlinear variational wave equation u
tt
− c(u)(c(u)u
x
)
x
= 0. We allow for initial data u|
t = 0 and u
t
|
t=0 that contain measures. We assume that
0 < k-1 \leqq c(u) \leqq k{0 < \kappa^{-1} \leqq c(u) \leqq \kappa}. Solutions of this equation may experience concentration of the energy density (ut2+c(u)2ux2)dx{(u_t^2+c(u)^2u_x^2){\rm d}x} into sets of measure zero. The solution is constructed by introducing new variables related to the characteristics, whereby
singularities in the energy density become manageable. Furthermore, we prove that the energy may focus only on a set of times
of zero measure or at points where c′(u) vanishes. A new numerical method for constructing conservative solutions is provided and illustrated with examples. 相似文献
16.
Fix a strictly increasing right continuous with left limits function ${W: \mathbb{R} \to \mathbb{R}}
17.
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
18.
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
19.
M. M. Cavalcanti V. N. Domingos Cavalcanti R. Fukuoka J. A. Soriano 《Archive for Rational Mechanics and Analysis》2010,197(3):925-964
Let (M, g) be a n-dimensional ( ${n\geqq 2}
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