@article{59343,
abstract = {{Abstract
On a finite regular graph, (co)resonant states are eigendistributions of the transfer operator associated to the shift on one-sided infinite non-backtracking paths. We introduce two pairings of resonant and coresonant states, the vertex pairing which involves only the dependence on the initial/terminal vertex of the path, and the geodesic pairing which is given by integrating over all geodesics the evaluation of the coresonant state on the first half of the geodesic times the resonant state on the second half. The main result is that these two pairings coincide up to a constant which depends on the resonance, i.e. the corresponding eigenvalue of the transfer operator.}},
author = {{Arends, Christian and Frahm, Jan and Hilgert, Joachim}},
issn = {{0025-5831}},
journal = {{Mathematische Annalen}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{A pairing formula for resonant states on finite regular graphs}}},
doi = {{10.1007/s00208-025-03140-7}},
year = {{2025}},
}
@article{63248,
abstract = {{Abstract
The Navier–Stokes system
$$\begin{aligned} \left\{ \begin{array}{l} u_t + (u\cdot \nabla ) u =\Delta u+\nabla P + f(x,t), \\ \nabla \cdot u=0, \end{array} \right. \end{aligned}$$
u
t
+
(
u
·
∇
)
u
=
Δ
u
+
∇
P
+
f
(
x
,
t
)
,
∇
·
u
=
0
,
is considered along with homogeneous Dirichlet boundary conditions in a smoothly bounded planar domain
$$\Omega $$
Ω
. It is firstly, inter alia, observed that if
$$T>0$$
T
>
0
and
$$\begin{aligned} \int _0^T \bigg \{ \int _\Omega |f(x,t)| \cdot \ln ^\frac{1}{2} \big (|f(x,t)|+1\big ) dx \bigg \}^2 dt <\infty , \end{aligned}$$
∫
0
T
{
∫
Ω
|
f
(
x
,
t
)
|
·
ln
1
2
(
|
f
(
x
,
t
)
|
+
1
)
d
x
}
2
d
t
<
∞
,
then for all divergence-free
$$u_0\in L^2(\Omega ;{\mathbb {R}}^2)$$
u
0
∈
L
2
(
Ω
;
R
2
)
, a corresponding initial-boundary value problem admits a weak solution u with
$$u|_{t=0}=u_0$$
u
|
t
=
0
=
u
0
. For any positive and nondecreasing
$$L\in C^0([0,\infty ))$$
L
∈
C
0
(
[
0
,
∞
)
)
such that
$$\begin{aligned} \frac{L(\xi )}{\ln ^\frac{1}{2} \xi } \rightarrow 0 \qquad \text{ as } \xi \rightarrow \infty , \end{aligned}$$
L
(
ξ
)
ln
1
2
ξ
→
0
as
ξ
→
∞
,
this is complemented by a statement on nonexistence of such a solution in the presence of smooth initial data and a suitably constructed
$$f:\Omega \times (0,T)\rightarrow {\mathbb {R}}^2$$
f
:
Ω
×
(
0
,
T
)
→
R
2
fulfilling
$$\begin{aligned} \int _0^T \bigg \{ \int _\Omega |f(x,t)| \cdot L\big (|f(x,t)|\big ) dx \bigg \}^2 dt < \infty . \end{aligned}$$
∫
0
T
{
∫
Ω
|
f
(
x
,
t
)
|
·
L
(
|
f
(
x
,
t
)
|
)
d
x
}
2
d
t
<
∞
.
This resolves a fine structure in the borderline case
$$p=1$$
p
=
1
and
$$q=2$$
q
=
2
appearing in results on existence of weak solutions for sources in
$$L^q((0,T);L^p(\Omega ;{\mathbb {R}}^2))$$
L
q
(
(
0
,
T
)
;
L
p
(
Ω
;
R
2
)
)
when
$$p\in (1,\infty ]$$
p
∈
(
1
,
∞
]
and
$$q\in [1,\infty ]$$
q
∈
[
1
,
∞
]
satisfy
$$\frac{1}{p}+\frac{1}{q}\le \frac{3}{2}$$
1
p
+
1
q
≤
3
2
, and on nonexistence if here
$$p\in [1,\infty )$$
p
∈
[
1
,
∞
)
and
$$q\in [1,\infty )$$
q
∈
[
1
,
∞
)
are such that
$$\frac{1}{p}+\frac{1}{q}>\frac{3}{2}$$
1
p
+
1
q
>
3
2
.}},
author = {{Winkler, Michael}},
issn = {{0025-5831}},
journal = {{Mathematische Annalen}},
number = {{2}},
pages = {{3023--3054}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Externally forced blow-up and optimal spaces for source regularity in the two-dimensional Navier–Stokes system}}},
doi = {{10.1007/s00208-024-02987-6}},
volume = {{391}},
year = {{2024}},
}
@article{64272,
abstract = {{AbstractIn the present paper we further the study of the compression cone of a real spherical homogeneous space $$Z=G/H$$
Z
=
G
/
H
. In particular we provide a geometric construction of the little Weyl group of Z introduced recently by Knop and Krötz. Our technique is based on a fine analysis of limits of conjugates of the subalgebra $$\mathrm{Lie}(H)$$
Lie
(
H
)
along one-parameter subgroups in the Grassmannian of subspaces of $$\mathrm{Lie}(G)$$
Lie
(
G
)
. The little Weyl group is obtained as a finite reflection group generated by the reflections in the walls of the compression cone.}},
author = {{Kuit, Job J. and Sayag, Eitan}},
issn = {{0025-5831}},
journal = {{Mathematische Annalen}},
number = {{1-2}},
pages = {{433--498}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{On the little Weyl group of a real spherical space}}},
doi = {{10.1007/s00208-022-02473-x}},
volume = {{387}},
year = {{2022}},
}
@article{63361,
author = {{Winkler, Michael}},
issn = {{0025-5831}},
journal = {{Mathematische Annalen}},
number = {{3-4}},
pages = {{1237--1282}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{How unstable is spatial homogeneity in Keller-Segel systems? A new critical mass phenomenon in two- and higher-dimensional parabolic-elliptic cases}}},
doi = {{10.1007/s00208-018-1722-8}},
volume = {{373}},
year = {{2018}},
}
@article{31267,
author = {{Guillarmou, Colin and Hilgert, Joachim and Weich, Tobias}},
issn = {{0025-5831}},
journal = {{Mathematische Annalen}},
keywords = {{General Mathematics}},
number = {{3-4}},
pages = {{1231--1275}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Classical and quantum resonances for hyperbolic surfaces}}},
doi = {{10.1007/s00208-017-1576-5}},
volume = {{370}},
year = {{2017}},
}
@article{53189,
author = {{Januszewski, Fabian}},
issn = {{0025-5831}},
journal = {{Mathematische Annalen}},
keywords = {{General Mathematics}},
number = {{3-4}},
pages = {{1805--1881}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Rational structures on automorphic representations}}},
doi = {{10.1007/s00208-017-1567-6}},
volume = {{370}},
year = {{2017}},
}