@article{63309,
abstract = {{AbstractThis manuscript is concerned with the problem of efficiently estimating chemotactic gradients, as forming a ubiquitous issue of key importance in virtually any proof of boundedness features in Keller–Segel type systems. A strategy is proposed which at its core relies on bounds for such quantities, conditional in the sense of involving certain Lebesgue norms of solution components that explicitly influence the signal evolution.Applications of this procedure firstly provide apparently novel boundedness results for two particular classes chemotaxis systems, and apart from that are shown to significantly condense proofs for basically well‐known statements on boundedness in two further Keller–Segel type problems.}},
author = {{Winkler, Michael}},
issn = {{0025-584X}},
journal = {{Mathematische Nachrichten}},
number = {{9}},
pages = {{1840--1862}},
publisher = {{Wiley}},
title = {{{A unifying approach toward boundedness in Keller–Segel type cross‐diffusion systems via conditional L∞$L^\infty$ estimates for taxis gradients}}},
doi = {{10.1002/mana.202000403}},
volume = {{295}},
year = {{2022}},
}
@article{63306,
author = {{Winkler, Michael}},
issn = {{0022-2518}},
journal = {{Indiana University Mathematics Journal}},
number = {{4}},
pages = {{1437--1465}},
publisher = {{Indiana University Mathematics Journal}},
title = {{{A critical blow-up exponent for flux limiation in a Keller-Segel system}}},
doi = {{10.1512/iumj.2022.71.9042}},
volume = {{71}},
year = {{2022}},
}
@article{63284,
abstract = {{ A no-flux initial-boundary value problem for the cross-diffusion system [Formula: see text] is considered in smoothly bounded domains [Formula: see text] with [Formula: see text]. It is shown that whenever [Formula: see text] is positive on [Formula: see text] and such that [Formula: see text] for some [Formula: see text], for all suitably regular positive initial data a global very weak solution, particularly preserving mass in its first component, can be constructed. This extends previous results which either concentrate on non-degenerate analogs, or are restricted to the special case [Formula: see text]. To appropriately cope with the considerably stronger cross-degeneracies thus allowed through [Formula: see text] when [Formula: see text] is large, in its core part the analysis relies on the use of the Moser–Trudinger inequality in controlling the respective diffusion rates [Formula: see text] from below. }},
author = {{Winkler, Michael}},
issn = {{1664-3607}},
journal = {{Bulletin of Mathematical Sciences}},
number = {{02}},
publisher = {{World Scientific Pub Co Pte Ltd}},
title = {{{Application of the Moser–Trudinger inequality in the construction of global solutions to a strongly degenerate migration model}}},
doi = {{10.1142/s1664360722500126}},
volume = {{13}},
year = {{2022}},
}
@article{63286,
abstract = {{ The chemotaxis system [Formula: see text] is considered in a ball [Formula: see text]. It is shown that if [Formula: see text] suitably generalizes the prototype given by [Formula: see text] with some [Formula: see text], and if diffusion is suitably weak in the sense that [Formula: see text] is such that there exist [Formula: see text] and [Formula: see text] fulfilling [Formula: see text] then for appropriate choices of sufficiently concentrated initial data, an associated no-flux initial-boundary value problem admits a global classical solution [Formula: see text] which blows up in infinite time and satisfies [Formula: see text] A major part of the proof is based on a comparison argument involving explicitly constructed subsolutions to a scalar parabolic problem satisfied by mass accumulation functions corresponding to solutions of ( ⋆ ). }},
author = {{Winkler, Michael}},
issn = {{0921-7134}},
journal = {{Asymptotic Analysis}},
number = {{1}},
pages = {{33--57}},
publisher = {{SAGE Publications}},
title = {{{Exponential grow-up rates in a quasilinear Keller–Segel system}}},
doi = {{10.3233/asy-221765}},
volume = {{131}},
year = {{2022}},
}
@article{63293,
abstract = {{<p style='text-indent:20px;'>The Cauchy problem in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^3 $\end{document}</tex-math></inline-formula> for the chemotaxis-Navier–Stokes system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{l} n_t + u\cdot\nabla n = \Delta n - \nabla \cdot (n\nabla c), \\ c_t + u\cdot\nabla c = \Delta c - nc, \\ u_t + (u\cdot\nabla) u = \Delta u + \nabla P + n\nabla\phi, \qquad \nabla \cdot u = 0, \ \end{array} \right. \end{eqnarray*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>is considered. Under suitable conditions on the initial data <inline-formula><tex-math id="M3">\begin{document}$ (n_0, c_0, u_0) $\end{document}</tex-math></inline-formula>, with regard to the crucial first component requiring that <inline-formula><tex-math id="M4">\begin{document}$ n_0\in L^1( \mathbb{R}^3) $\end{document}</tex-math></inline-formula> be nonnegative and such that <inline-formula><tex-math id="M5">\begin{document}$ (n_0+1)\ln (n_0+1) \in L^1( \mathbb{R}^3) $\end{document}</tex-math></inline-formula>, a globally defined weak solution with <inline-formula><tex-math id="M6">\begin{document}$ (n, c, u)|_{t = 0} = (n_0, c_0, u_0) $\end{document}</tex-math></inline-formula> is constructed. Apart from that, assuming that moreover <inline-formula><tex-math id="M7">\begin{document}$ \int_{ \mathbb{R}^3} n_0(x) \ln (1+|x|^2) dx $\end{document}</tex-math></inline-formula> is finite, it is shown that a weak solution exists which enjoys further regularity features and preserves mass in an appropriate sense.</p>}},
author = {{Kang, Kyungkeun and Lee, Jihoon and Winkler, Michael}},
issn = {{1078-0947}},
journal = {{Discrete and Continuous Dynamical Systems}},
number = {{11}},
publisher = {{American Institute of Mathematical Sciences (AIMS)}},
title = {{{Global weak solutions to a chemotaxis-Navier-Stokes system in $ \mathbb{R}^3 $}}},
doi = {{10.3934/dcds.2022091}},
volume = {{42}},
year = {{2022}},
}
@article{63290,
abstract = {{ This paper proposes a review focused on exotic chemotaxis and cross-diffusion models in complex environments. The term exotic is used to denote the dynamics of models interacting with a time-evolving external system and, specifically, models derived with the aim of describing the dynamics of living systems. The presentation first, considers the derivation of phenomenological models of chemotaxis and cross-diffusion models with particular attention on nonlinear characteristics. Then, a variety of exotic models is presented with some hints toward the derivation of new models, by accounting for a critical analysis looking ahead to perspectives. The second part of the paper is devoted to a survey of analytical problems concerning the application of models to the study of real world dynamics. Finally, the focus shifts to research perspectives within the framework of a multiscale vision, where different paths are examined to move from the dynamics at the microscopic scale to collective behaviors at the macroscopic scale. }},
author = {{Bellomo, N. and Outada, N. and Soler, J. and Tao, Y. and Winkler, Michael}},
issn = {{0218-2025}},
journal = {{Mathematical Models and Methods in Applied Sciences}},
number = {{04}},
pages = {{713--792}},
publisher = {{World Scientific Pub Co Pte Ltd}},
title = {{{Chemotaxis and cross-diffusion models in complex environments: Models and analytic problems toward a multiscale vision}}},
doi = {{10.1142/s0218202522500166}},
volume = {{32}},
year = {{2022}},
}
@article{63295,
abstract = {{AbstractWe introduce a generalized concept of solutions for reaction–diffusion systems and prove their global existence. The only restriction on the reaction function beyond regularity, quasipositivity and mass control is special in that it merely controls the growth of cross-absorptive terms. The result covers nonlinear diffusion and does not rely on an entropy estimate.}},
author = {{Lankeit, Johannes and Winkler, Michael}},
issn = {{1424-3199}},
journal = {{Journal of Evolution Equations}},
number = {{1}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Global existence in reaction–diffusion systems with mass control under relaxed assumptions merely referring to cross-absorptive effects}}},
doi = {{10.1007/s00028-022-00768-9}},
volume = {{22}},
year = {{2022}},
}
@article{63299,
author = {{Tao, Youshan and Winkler, Michael}},
issn = {{0036-1410}},
journal = {{SIAM Journal on Mathematical Analysis}},
number = {{4}},
pages = {{4806--4864}},
publisher = {{Society for Industrial & Applied Mathematics (SIAM)}},
title = {{{Existence Theory and Qualitative Analysis for a Fully Cross-Diffusive Predator-Prey System}}},
doi = {{10.1137/21m1449841}},
volume = {{54}},
year = {{2022}},
}
@article{63298,
author = {{Stevens, Angela and Winkler, Michael}},
issn = {{0360-5302}},
journal = {{Communications in Partial Differential Equations}},
number = {{12}},
pages = {{2341--2362}},
publisher = {{Informa UK Limited}},
title = {{{Taxis-driven persistent localization in a degenerate Keller-Segel system}}},
doi = {{10.1080/03605302.2022.2122836}},
volume = {{47}},
year = {{2022}},
}
@article{63266,
abstract = {{AbstractIn a ball $$\Omega =B_R(0)\subset \mathbb {R}^n$$
Ω
=
B
R
(
0
)
⊂
R
n
, $$n\ge 2$$
n
≥
2
, the chemotaxis system $$\begin{aligned} \left\{ \begin{array}{l}u_t = \nabla \cdot \big ( D(u) \nabla u \big ) - \nabla \cdot \big ( uS(u)\nabla v\big ), \\ 0 = \Delta v - \mu + u, \qquad \mu =\frac{1}{|\Omega |} \int _\Omega u, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$
u
t
=
∇
·
(
D
(
u
)
∇
u
)
-
∇
·
(
u
S
(
u
)
∇
v
)
,
0
=
Δ
v
-
μ
+
u
,
μ
=
1
|
Ω
|
∫
Ω
u
,
(
⋆
)
is considered under no-flux boundary conditions, with a focus on nonlinearities $$S\in C^2([0,\infty ))$$
S
∈
C
2
(
[
0
,
∞
)
)
which exhibit super-algebraically fast decay in the sense that with some $$K_S>0, \beta \in [0,1)$$
K
S
>
0
,
β
∈
[
0
,
1
)
and $$\xi _0>0$$
ξ
0
>
0
, $$\begin{aligned} S(\xi )>0 \quad \text{ and } \quad S'(\xi ) \le -K_S\xi ^{-\beta } S(\xi ) \qquad \text{ for } \text{ all } \xi \ge \xi _0. \end{aligned}$$
S
(
ξ
)
>
0
and
S
′
(
ξ
)
≤
-
K
S
ξ
-
β
S
(
ξ
)
for
all
ξ
≥
ξ
0
.
It is, inter alia, shown that if furthermore $$D\in C^2((0,\infty ))$$
D
∈
C
2
(
(
0
,
∞
)
)
is positive and suitably small in relation to S by satisfying $$\begin{aligned} \frac{\xi S(\xi )}{D(\xi )} \ge K_{SD}\xi ^\lambda \qquad \text{ for } \text{ all } \xi \ge \xi _0 \end{aligned}$$
ξ
S
(
ξ
)
D
(
ξ
)
≥
K
SD
ξ
λ
for
all
ξ
≥
ξ
0
with some $$K_{SD}>0$$
K
SD
>
0
and $$\lambda >\frac{2}{n}$$
λ
>
2
n
, then throughout a considerably large set of initial data, ($$\star $$
⋆
) admits global classical solutions (u, v) fulfilling $$\begin{aligned} \frac{z(t)}{C} \le \Vert u(\cdot ,t)\Vert _{L^\infty (\Omega )} \le Cz(t) \qquad \text{ for } \text{ all } t>0, \end{aligned}$$
z
(
t
)
C
≤
‖
u
(
·
,
t
)
‖
L
∞
(
Ω
)
≤
C
z
(
t
)
for
all
t
>
0
,
with some $$C=C^{(u,v)}\ge 1$$
C
=
C
(
u
,
v
)
≥
1
, where z denotes the solution of $$\begin{aligned} \left\{ \begin{array}{l}z'(t) = z^2(t) \cdot S\big ( z(t)\big ), \qquad t>0, \\ z(0)=\xi _0, \end{array} \right. \end{aligned}$$
z
′
(
t
)
=
z
2
(
t
)
·
S
(
z
(
t
)
)
,
t
>
0
,
z
(
0
)
=
ξ
0
,
which is seen to exist globally, and to satisfy $$z(t)\rightarrow +\infty $$
z
(
t
)
→
+
∞
as $$t\rightarrow \infty $$
t
→
∞
. As particular examples, exponentially and doubly exponentially decaying S are found to imply corresponding infinite-time blow-up properties in ($$\star $$
⋆
) at logarithmic and doubly logarithmic rates, respectively.}},
author = {{Winkler, Michael}},
issn = {{1040-7294}},
journal = {{Journal of Dynamics and Differential Equations}},
number = {{2}},
pages = {{1677--1702}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Slow Grow-up in a Quasilinear Keller–Segel System}}},
doi = {{10.1007/s10884-022-10167-w}},
volume = {{36}},
year = {{2022}},
}
@article{63272,
author = {{Tao, Youshan and Winkler, Michael}},
issn = {{0022-0396}},
journal = {{Journal of Differential Equations}},
pages = {{390--418}},
publisher = {{Elsevier BV}},
title = {{{Global solutions to a Keller-Segel-consumption system involving singularly signal-dependent motilities in domains of arbitrary dimension}}},
doi = {{10.1016/j.jde.2022.10.022}},
volume = {{343}},
year = {{2022}},
}
@article{63268,
author = {{Desvillettes, Laurent and Laurençot, Philippe and Trescases, Ariane and Winkler, Michael}},
issn = {{0362-546X}},
journal = {{Nonlinear Analysis}},
publisher = {{Elsevier BV}},
title = {{{Weak solutions to triangular cross diffusion systems modeling chemotaxis with local sensing}}},
doi = {{10.1016/j.na.2022.113153}},
volume = {{226}},
year = {{2022}},
}
@article{63278,
abstract = {{Abstract
The Neumann problem for (0.1)$$ \begin{align}& V_t = \Delta V-aV+f(x,t) \end{align}$$is considered in bounded domains $\Omega \subset {\mathbb {R}}^n$ with smooth boundary, where $n\ge 1$ and $a\in {\mathbb {R}}$. By means of a variational approach, a statement on boundedness of the quantities $$ \begin{eqnarray*} \sup_{t\in (0,T)} \int_\Omega \big|\nabla V(\cdot,t)\big|^p L^{\frac{n+p}{n+2}} \Big( \big|\nabla V(\cdot,t)\big| \Big) \end{eqnarray*}$$in dependence on the expressions (0.2)$$ \begin{align}& \sup_{t\in (0,T-\tau)} \int_t^{t+\tau} \int_\Omega |f|^{\frac{(n+2)p}{n+p}} L\big( |f|\big) \end{align}$$is derived for $p\ge 2$, $\tau>0$, and $T\ge 2\tau $, provided that $L\in C^0([0,\infty ))$ is positive, strictly increasing, unbounded, and slowly growing in the sense that $\limsup _{s\to \infty } \frac {L(s^{\lambda _0})}{L(s)} <\infty $ for some $\lambda _0>1$. In the particular case when $p=n\ge 2$, an additional condition on growth of $L$, particularly satisfied by $L(\xi ):=\ln ^\alpha (\xi +b)$ whenever $b>0$ and $\alpha>\frac {(n+2)(n-1)}{2n}$, is identified as sufficient to ensure that as a consequence of the above, bounds for theintegrals in (0.2) even imply estimates for the spatio-temporal modulus of continuity of solutions to (0.1). A subsequent application to the Keller–Segel system $$ \begin{eqnarray*} \left\{ \begin{array}{l} u_t = \nabla \cdot \big( D(v)\nabla u\big) - \nabla \cdot \big( uS(v)\nabla v\big) + ru - \mu u^2, \\[1mm] v_t = \Delta v-v+u, \end{array} \right. \end{eqnarray*}$$shows that when $n=2$, $r\in {\mathbb {R}}$, $0<D\in C^2([0,\infty ))$, and $S\in C^2([0,\infty )) \cap W^{1,\infty }((0,\infty ))$ and thus especially in the presence of arbitrarily strong diffusion degeneracies implied by rapid decay of $D$, any choice of $\mu>0$ excludes blowup in the sense that for all suitably regular nonnegative initial data, an associated initial-boundary value problem admits a global bounded classical solution.}},
author = {{Winkler, Michael}},
issn = {{1073-7928}},
journal = {{International Mathematics Research Notices}},
number = {{19}},
pages = {{16336--16393}},
publisher = {{Oxford University Press (OUP)}},
title = {{{A Result on Parabolic Gradient Regularity in Orlicz Spaces and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion System}}},
doi = {{10.1093/imrn/rnac286}},
volume = {{2023}},
year = {{2022}},
}
@article{63279,
abstract = {{
In a smoothly bounded convex domain
\Omega \subset \mathbb{R}^3
, we consider the chemotaxis-Navier–Stokes model
\begin{cases} n_t + u\cdot\nabla n = \Delta n - \nabla \cdot (n\nabla c), & x\in \Omega, \, t>0, \\ c_t + u\cdot\nabla c = \Delta c -nc, & x\in \Omega, \, t>0, \\ u_t + (u\cdot\nabla) u = \Delta u + \nabla P + n\nabla\Phi, \quad \nabla\cdot u=0, & x\in \Omega, \, t>0, \end{cases} \quad (\star)
proposed by Goldstein et al. to describe pattern formation in populations of aerobic bacteria interacting with their liquid environment via transport and buoyancy. Known results have asserted that under appropriate regularity assumptions on
\Phi
and the initial data, a corresponding no-flux/no-flux/Dirichlet initial-boundary value problem is globally solvable in a framework of so-called weak energy solutions, and that any such solution eventually becomes smooth and classical.
Going beyond this, the present work focuses on the possible extent of unboundedness phenomena also on short timescales, and hence investigates in more detail the set of times in
(0,\infty)
at which solutions may develop singularities. The main results in this direction reveal the existence of a global weak energy solution which coincides with a smooth function throughout
\overline{\Omega}\times E
, where
E
denotes a countable union of open intervals which is such that
|(0,\infty)\setminus E|=0
. In particular, this indicates that a similar feature of the unperturbed Navie–Stokes equations, known as Leray’s structure theorem, persists even in the presence of the coupling to the attractive and hence potentially destabilizing cross-diffusive mechanism in the full system (
\star
).
}},
author = {{Winkler, Michael}},
issn = {{1435-9855}},
journal = {{Journal of the European Mathematical Society}},
number = {{4}},
pages = {{1423--1456}},
publisher = {{European Mathematical Society - EMS - Publishing House GmbH}},
title = {{{Does Leray’s structure theorem withstand buoyancy-driven chemotaxis-fluid interaction?}}},
doi = {{10.4171/jems/1226}},
volume = {{25}},
year = {{2022}},
}
@article{63274,
abstract = {{In a ball $\Omega \subset \mathbb {R}^{n}$ with $n\ge 2$, the chemotaxis system
\[ \left\{ \begin{array}{@{}l} u_t = \nabla \cdot \big( D(u)\nabla u\big) + \nabla\cdot \big(\dfrac{u}{v} \nabla v\big), \\ 0=\Delta v - uv \end{array} \right. \]is considered along with no-flux boundary conditions for $u$ and with prescribed constant positive Dirichlet boundary data for $v$. It is shown that if $D\in C^{3}([0,\infty ))$ is such that $0< D(\xi ) \le {K_D} (\xi +1)^{-\alpha }$ for all $\xi >0$ with some ${K_D}>0$ and $\alpha >0$, then for all initial data from a considerably large set of radial functions on $\Omega$, the corresponding initial-boundary value problem admits a solution blowing up in finite time.}},
author = {{Wang, Yulan and Winkler, Michael}},
issn = {{0308-2105}},
journal = {{Proceedings of the Royal Society of Edinburgh: Section A Mathematics}},
number = {{4}},
pages = {{1150--1166}},
publisher = {{Cambridge University Press (CUP)}},
title = {{{Finite-time blow-up in a repulsive chemotaxis-consumption system}}},
doi = {{10.1017/prm.2022.39}},
volume = {{153}},
year = {{2022}},
}
@article{63282,
abstract = {{ The chemotaxis system [Formula: see text] is considered in a ball [Formula: see text], [Formula: see text], where the positive function [Formula: see text] reflects suitably weak diffusion by satisfying [Formula: see text] for some [Formula: see text]. It is shown that whenever [Formula: see text] is positive and satisfies [Formula: see text] as [Formula: see text], one can find a suitably regular nonlinearity [Formula: see text] with the property that at each sufficiently large mass level [Formula: see text] there exists a globally defined radially symmetric classical solution to a Neumann-type boundary value problem for (⋆) which satisfies [Formula: see text] }},
author = {{Winkler, Michael}},
issn = {{0219-1997}},
journal = {{Communications in Contemporary Mathematics}},
number = {{10}},
publisher = {{World Scientific Pub Co Pte Ltd}},
title = {{{Arbitrarily fast grow-up rates in quasilinear Keller–Segel systems}}},
doi = {{10.1142/s0219199722500626}},
volume = {{25}},
year = {{2022}},
}
@inproceedings{40212,
author = {{Haucke-Korber, Barnabas and Schenke, Maximilian and Wallscheid, Oliver}},
booktitle = {{IKMT 2022; 13. GMM/ETG-Symposium}},
pages = {{1--6}},
title = {{{Reinforcement Learning-Based Deep Q Direct Torque Control with Adaptable Switching Frequency Towards Six-Step Operation of Permanent Magnet Synchronous Motors}}},
year = {{2022}},
}
@inbook{63428,
author = {{Kirschtein, Claudia}},
booktitle = {{Wie beeinflussen Gefühle und Sprache den (Online-) Lernprozess? Tagungsband zum 21. E-Learning Tag der FH JOANNEUM am 21.09.2022}},
editor = {{Pauschenwein, Jutta and Hernády, Birgit and Michelitsch, Linda}},
pages = {{73--84}},
publisher = {{FH JOANNEUM Gesellschaft}},
title = {{{Mediendidaktische Konzeption mit Emotion}}},
year = {{2022}},
}
@inproceedings{6553,
author = {{Claes, Leander and Feldmann, Nadine and Schulze, Veronika and Jurgelucks, Benjamin and Walther, Andrea and Henning, Bernd}},
booktitle = {{Fortschritte der Akustik - DAGA 2022}},
location = {{Stuttgart}},
pages = {{1326--1329}},
title = {{{Identification of piezoelectric material parameters using optimised multi-electrode specimens}}},
year = {{2022}},
}
@misc{6558,
author = {{Friesen, Olga and Claes, Leander and Feldmann, Nadine and Henning, Bernd}},
title = {{{Estimation of piezoelectric material parameters of ring-shaped specimens}}},
year = {{2022}},
}