@article{63342,
abstract = {{AbstractIn a bounded planar domain $\varOmega $
Ω
with smooth boundary, the initial-boundary value problem of homogeneous Neumann type for the Keller-Segel-fluid system
$$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad }l} n_{t} + \nabla \cdot (nu) = \Delta n - \nabla \cdot (n\nabla c), & x\in \varOmega , \ t>0, \\ 0 = \Delta c -c+n, & x\in \varOmega , \ t>0, \end{array}\displaystyle \right . \end{aligned}$$
{
n
t
+
∇
⋅
(
n
u
)
=
Δ
n
−
∇
⋅
(
n
∇
c
)
,
x
∈
Ω
,
t
>
0
,
0
=
Δ
c
−
c
+
n
,
x
∈
Ω
,
t
>
0
,
is considered, where $u$
u
is a given sufficiently smooth velocity field on $\overline {\varOmega }\times [0,\infty )$
Ω
‾
×
[
0
,
∞
)
that is tangential on $\partial \varOmega $
∂
Ω
but not necessarily solenoidal.It is firstly shown that for any choice of $n_{0}\in C^{0}(\overline {\varOmega })$
n
0
∈
C
0
(
Ω
‾
)
with $\int _{\varOmega}n_{0}<4\pi $
∫
Ω
n
0
<
4
π
, this problem admits a global classical solution with $n(\cdot ,0)=n_{0}$
n
(
⋅
,
0
)
=
n
0
, and that this solution is even bounded whenever $u$
u
is bounded and $\int _{\varOmega}n_{0}<2\pi $
∫
Ω
n
0
<
2
π
. Secondly, it is seen that for each $m>4\pi $
m
>
4
π
one can find a classical solution with $\int _{\varOmega}n(\cdot ,0)=m$
∫
Ω
n
(
⋅
,
0
)
=
m
which blows up in finite time, provided that $\varOmega $
Ω
satisfies a technical assumption requiring $\partial \varOmega $
∂
Ω
to contain a line segment.In particular, this indicates that the value $4\pi $
4
π
of the critical mass for the corresponding fluid-free Keller-Segel system is left unchanged by any fluid interaction of the considered type, thus marking a considerable contrast to a recent result revealing some fluid-induced increase of critical blow-up masses in a related Cauchy problem in the entire plane.}},
author = {{Winkler, Michael}},
issn = {{0167-8019}},
journal = {{Acta Applicandae Mathematicae}},
number = {{1}},
pages = {{577--591}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Can Fluid Interaction Influence the Critical Mass for Taxis-Driven Blow-up in Bounded Planar Domains?}}},
doi = {{10.1007/s10440-020-00312-2}},
volume = {{169}},
year = {{2020}},
}
@article{8753,
abstract = {{In a wide range of applications it is desirable to optimally control a dynamical system with respect to concurrent, potentially competing goals. This gives rise to a multiobjective optimal control problem where, instead of computing a single optimal solution, the set of optimal compromises, the so-called Pareto set, has to be approximated. When the problem under consideration is described by a partial differential equation (PDE), as is the case for fluid flow, the computational cost rapidly increases and makes its direct treatment infeasible. Reduced order modeling is a very popular method to reduce the computational cost, in particular in a multi query context such as uncertainty quantification, parameter estimation or optimization. In this article, we show how to combine reduced order modeling and multiobjective optimal control techniques in order to efficiently solve multiobjective optimal control problems constrained by PDEs. We consider a global, derivative free optimization method as well as a local, gradient-based approach for which the optimality system is derived in two different ways. The methods are compared with regard to the solution quality as well as the computational effort and they are illustrated using the example of the flow around a cylinder and a backward-facing-step channel flow.}},
author = {{Peitz, Sebastian and Ober-Blöbaum, Sina and Dellnitz, Michael}},
issn = {{0167-8019}},
journal = {{Acta Applicandae Mathematicae}},
number = {{1}},
pages = {{171–199}},
title = {{{Multiobjective Optimal Control Methods for the Navier-Stokes Equations Using Reduced Order Modeling}}},
doi = {{10.1007/s10440-018-0209-7}},
volume = {{161}},
year = {{2018}},
}
@article{63368,
author = {{Winkler, Michael}},
issn = {{0167-8019}},
journal = {{Acta Applicandae Mathematicae}},
number = {{1}},
pages = {{1--17}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Boundedness in a Chemotaxis-May-Nowak Model for Virus Dynamics with Mildly Saturated Chemotactic Sensitivity}}},
doi = {{10.1007/s10440-018-0211-0}},
volume = {{163}},
year = {{2018}},
}
@article{39948,
author = {{Rösler, Margit and Voit, Michael}},
issn = {{0167-8019}},
journal = {{Acta Applicandae Mathematicae}},
keywords = {{Applied Mathematics}},
number = {{1-2}},
pages = {{179--195}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{SU(d)-Biinvariant Random Walks on SL(d,C) and their Euclidean Counterparts}}},
doi = {{10.1007/s10440-006-9035-4}},
volume = {{90}},
year = {{2006}},
}