We consider, for
ρ
∈
0
,
1
and
ε
>
0
small, the nonautonomous weakly damped wave equation with a singularly oscillating external force
∂
t
2
u
−
Δ
u
+
γ
∂
t
u
=
−
f
(
u
)
+
g
0
(
t
)
+
ε
−
ρ
g
1
(
...t
/
ε
)
,
together with the
averaged equation
∂
t
2
u
−
Δ
u
+
γ
∂
t
u
=
−
f
(
u
)
+
g
0
(
t
)
.
Under suitable assumptions on the nonlinearity and the external force, we prove the uniform (with respect to
ε) boundedness of the attractors
A
ε
in the weak energy space. If
ρ
<
1
, we establish the convergence of the attractor
A
ε
of the first equation to the attractor
A
0
of the second one, as
ε
→
0
+
. On the other hand, if
ρ
=
1
, this convergence may fail. When
A
0
is exponential, then the convergence rate of
A
ε
to
A
0
is controlled by
M
ε
η
, for some
M
⩾
0
and some
η
=
η
(
ρ
)
∈
(
0
,
1
)
.
Pour tout
ρ
∈
0
,
1
et pour
ε
>
0
suffisamment petit, on considère l'équation des ondes non autonome faiblement amortie avec une force extérieure singulière et oscillatoire
∂
t
2
u
−
Δ
u
+
γ
∂
t
u
=
−
f
(
u
)
+
g
0
(
t
)
+
ε
−
ρ
g
1
(
t
/
ε
)
,
et le problème
moyenné
∂
t
2
u
−
Δ
u
+
γ
∂
t
u
=
−
f
(
u
)
+
g
0
(
t
)
.
Avec des hypothèses adéquates sur la nonlinéarité et sur la force, on obtient une borne uniforme (par rapport à
ε) pour les attracteurs
A
ε
dans l'espace faible d'énergie. Si
ρ
<
1
, on démontre la convergence de l'attracteur
A
ε
de la première équation vers l'attracteur
A
0
de la deuxième équation lorsque
ε
→
0
+
. D'autre part, si
ρ
=
1
, cette convergence peut ne pas avoir lieu. Quand
A
0
est exponentiel, la vitesse de convergence de
A
ε
vers
A
0
est bornée par
M
ε
η
, pour certains
M
⩾
0
et
η
=
η
(
ρ
)
∈
(
0
,
1
)
.
We study the global attractor of the non-autonomous 2D Navier-Stokes system with time-dependent external force g(x,t). We assume that g(x,t) is a translation compact function and the corresponding ...Grashof number is small. Then the global attractor has a simple structure: it is the closure of all the values of the unique bounded complete trajectory of the Navier-Stokes system. In particular, if g(x,t) is a quasiperiodic function with respect to t, then the attractor is a continuous image of a torus. Moreover the global attractor attracts all the solutions of the NS system with exponential rate, that is, the attractor is exponential. We also consider the 2D Navier-Stokes system with rapidly oscillating external force g(x,t,t/ε), which has the average as ε arrow right 0+. We assume that the function g(x,t,z) has a bounded primitive with respect to z and the averaged NS system has a small Grashof number that provides a simple structure of the averaged global attractor. Then we prove that the distance from the global attractor of the original NS system to the attractor of the averaged NS system is less than a small power of ε. PUBLICATION ABSTRACT
Near total reflection regime has been widely used in x-ray science, specifically in grazing incidence small angle x-ray scattering and in hard x-ray photoelectron spectroscopy (XPS). In this work, we ...introduce some practical aspects of using near total reflection (NTR) in ambient pressure XPS and apply this technique to study chemical concentration gradients in a substrate/photoresist system. Experimental data are accompanied by x-ray optical and photoemission simulations to quantitatively probe the photoresist and the interface with the depth accuracy of ~1 nm. Together, our calculations and experiments confirm that NTR XPS is a suitable method to extract information from buried interfaces with highest depth-resolution, which can help address open research questions regarding our understanding of concentration profiles, electrical gradients, and charge transfer phenomena at such interfaces. Furthermore, the presented methodology is especially attractive for solid/liquid interface studies, since it provides all the strengths of a Bragg-reflection standing-wave spectroscopy without the need of an artificial multilayer mirror serving as a standing wave generator, thus dramatically simplifying the sample synthesis.
The lamellar cuprate superconductors exhibit the highest ambient-pressure superconducting transition temperatures ( T c ), and after more than three decades of extraordinary research activity, ...continue to pose formidable scientific challenges. A major experimental obstacle has been to distinguish universal phenomena from materials- or technique-dependent ones. Angle-resolved photoemission spectroscopy (ARPES) measures momentum-dependent single-particle electronic excitations and has been invaluable in the endeavor to determine the anisotropic momentum-space properties of the cuprates.Hg Ba2 Cu O4+δ(Hg1201) is a singleCu O2-layer cuprate with a particularly high optimal Tc and a simple crystal structure, yet there exists little information from ARPES about the electronic properties of this model system. Here we present an ARPES study of doping-, temperature-, and momentum-dependent systematics of near-nodal dispersion anomalies in Hg1201. The data reveal a hierarchy of three distinct energy scales: a subgap low-energy kink, an intermediate-energy kink near 55 meV, and a peak-dip-hump structure. The first two features are attributed to the coupling of electrons to Ba-derived optical phonons and in-plane bond-stretching phonons, respectively. The nodal peak-dip-hump structure appears to have a common doping dependence in several single-layer cuprates and is interpreted as a manifestation of pseudogap physics at the node. These results establish several universal phenomena, both in terms of connecting multiple experimental techniques for a single material and in terms of connecting comparable spectral features in multiple structurally similar cuprates
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