A degenerate migration-consumption model in domains of arbitrary dimension - Winkler, Michael (Advanced Nonlinear Studies)
Abstract In a smoothly bounded convex domain Ω ⊂ R n ${\Omega}\subset {\mathbb{R}}^{n}$ with n ≥ 1, a no-flux initial-boundary value problem for u t = Δ u ϕ ( v ) , v t = Δ v − u v , $$\begin{cases}_{t}={\Delta}\left(u\phi \left(v\right)\right),\quad \hfill \\ {v}_{t}={\Delta}v-uv,\quad \hfill \end{cases}$$ is considered under the assumption that near the origin, the function ϕ suitably generalizes the prototype given by ϕ ( ξ ) = ξ α , ξ ∈ [ 0 , ξ 0 ] . $$\phi \left(\xi \right)={\xi }^{\alpha },\qquad \xi \in \left[0,{\xi }_{0}\right].$$ By means of separate approaches, it is shown that in both cases α ∈ (0, 1) and α ∈ [1, 2] some global weak solutions exist which, inter alia, satisfy C ( T ) ≔ ess sup t ∈ ( 0 , T ) ∫ Ω u ( ⋅ , t ) ln ⁡ u ( ⋅ , t ) < ∞ for all T > 0 , $$C\left(T\right){:=}\underset{t\in \left(0,T\right)}{\text{ess\,sup}}{\int }_{{\Omega}}u\left(\cdot ,t\right)\mathrm{ln}u\left(\cdot ,t\right){< }\infty \qquad \text{for\,all\,}T{ >}0,$$ with sup T >0 C ( T ) < ∞ if α ∈ [1, 2].

Externally forced blow-up and optimal spaces for source regularity in the two-dimensional Navier–Stokes system - Winkler, Michael (Mathematische Annalen)
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 .

Logarithmically refined Gagliardo–Nirenberg interpolation and application to blow-up exclusion in a singular chemotaxis–consumption system - Winkler, Michael (Annales de l'Institut Henri Poincaré C, Analyse non linéaire)
A family of interpolation inequalities is derived, which differ from estimates of classical Gagliardo–Nirenberg type through the appearance of certain logarithmic deviations from standard Lebesgue norms in zero-order expressions. Optimality of the obtained inequalities is shown. A subsequent application reveals that when posed under homogeneous Neumann boundary conditions in smoothly bounded planar domains and with suitably regular initial data, for any choice of α>0 the Keller–Segel-type migration–consumption system u_{t} = Δ (uv^{-α}) , v_{t} = Δ v-uv , admits a global classical solution.

A critical exponent in a quasilinear Keller–Segel system with arbitrarily fast decaying diffusivities accounting for volume-filling effects - Stinner, Christian; Winkler, Michael (Journal of Evolution Equations)
Abstract The quasilinear Keller–Segel system $$\begin{aligned} \left\{ \begin{array}{l} u_t=\nabla \cdot (D(u)\nabla u) - \nabla \cdot (S(u)\nabla v), \\ v_t=\Delta v-v+u, \end{array}\right. \end{aligned}$$ ut=∇·(D(u)∇u)-∇·(S(u)∇v),vt=Δv-v+u, endowed with homogeneous Neumann boundary conditions is considered in a bounded domain $$\Omega \subset {\mathbb {R}}^n$$ Ω⊂Rn , $$n \ge 3$$ n≥3 , with smooth boundary for sufficiently regular functions D and S satisfying $$D>0$$ D>0 on $$[0,\infty )$$ [0,∞) , $$S>0$$ S>0 on $$(0,\infty )$$ (0,∞) and $$S(0)=0$$ S(0)=0 . On the one hand, it is shown that if $$\frac{S}{D}$$ SD satisfies the subcritical growth condition $$\begin{aligned} \frac{S(s)}{D(s)} \le C s^\alpha \qquad \text{ for } \text{ all } s\ge 1 \qquad \text{ with } \text{ some } \alpha < \frac{2}{n} \end{aligned}$$ S(s)D(s)≤Csαforalls≥1withsomeα<2n and $$C>0$$ C>0 , then for any sufficiently regular initial data there exists a global weak energy solution such that $${ \mathrm{{ess}}} \sup _{t>0} \Vert u(t) \Vert _{L^p(\Omega )}<\infty $$ esssupt>0‖u(t)‖Lp(Ω)<∞ for some $$p > \frac{2n}{n+2}$$ p>2nn+2 . On the other hand, if $$\frac{S}{D}$$ SD satisfies the supercritical growth condition $$\begin{aligned} \frac{S(s)}{D(s)} \ge c s^\alpha \qquad \text{ for } \text{ all } s\ge 1 \qquad \text{ with } \text{ some } \alpha > \frac{2}{n} \end{aligned}$$ S(s)D(s)≥csαforalls≥1withsomeα>2n and $$c>0$$ c>0 , then the nonexistence of a global weak energy solution having the boundedness property stated above is shown for some initial data in the radial setting. This establishes some criticality of the value $$\alpha = \frac{2}{n}$$ α=2n for $$n \ge 3$$ n≥3 , without any additional assumption on the behavior of D ( s ) as $$s \rightarrow \infty $$ s→∞ , in particular without requiring any algebraic lower bound for D . When applied to the Keller–Segel system with volume-filling effect for probability distribution functions of the type $$Q(s) = \exp (-s^\beta )$$ Q(s)=exp(-sβ) , $$s \ge 0$$ s≥0 , for global solvability the exponent $$\beta = \frac{n-2}{n}$$ β=n-2n is seen to be critical.

Radial blow-up in quasilinear Keller-Segel systems: approaching the full picture - Ding, Mengyao; Winkler, Michael (Nonlinearity)
Abstract The Neumann problem for the Keller-Segel system { u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( S ( u ) ∇ v ) , 0 = Δ v − μ + u , μ = − ∫ Ω u d x , is considered in n -dimensional balls Ω with n ⩾ 2 , with suitably regular and radially symmetric, radially nonincreasing initial data u 0 . The functions D and S are only assumed to belong to C 2 ( [ 0 , ∞ ) ) and to satisfy D > 0 and S ⩾ 0 on [ 0 , ∞ ) as well as S ( 0 ) = 0 ; in particular, diffusivities with arbitrarily fast decay are included. In this general context, it is shown that it is merely the asymptotic behavior as ξ → ∞ of the expression I ( ξ ) := S ( ξ ) ξ 2 n D ( ξ ) , ξ > 0 , which decides about the occurrence of blow-up: Namely, it is seen that • if lim ξ → ∞ I ( ξ ) = 0 , then any such solution is global and bounded, that • if lim sup ξ → ∞ I ( ξ ) < ∞ and ∫ Ω u 0 is suitably small, then the corresponding solution is global and bounded, and that • if lim inf ξ → ∞ I ( ξ ) > 0 , then at each appropriately large mass level m , there exist radial initial data u 0 such that ∫ Ω u 0 = m , and that the associated solution blows up either in finite or in infinite time. This especially reveals the presence of critical mass phenomena whenever lim ξ → ∞ I ( ξ ) ∈ ( 0 , ∞ ) exists.

Uniform $$L^p$$ Estimates for Solutions to the Inhomogeneous 2D Navier–Stokes Equations and Application to a Chemotaxis–Fluid System with Local Sensing - Fuest, Mario; Winkler, Michael (Journal of Mathematical Fluid Mechanics)
Abstract The chemotaxis-Navier–Stokes system $$\begin{aligned} \left\{ \begin{array}{rcl} n_t+u\cdot \nabla n & =& \Delta \big (n c^{-\alpha } \big ), \\ c_t+ u\cdot \nabla c & =& \Delta c -nc,\\ u_t + (u\cdot \nabla ) u & =& \Delta u+\nabla P + n\nabla \Phi , \qquad \nabla \cdot u=0, \end{array} \right. \end{aligned}$$ n t + u · ∇ n = Δ ( n c - α ) , c t + u · ∇ c = Δ c - n c , u t + ( u · ∇ ) u = Δ u + ∇ P + n ∇ Φ , ∇ · u = 0 , modelling the behavior of aerobic bacteria in a fluid drop, is considered in a smoothly bounded domain $$\Omega \subset \mathbb R^2$$ Ω ⊂ R 2 . For all $$\alpha > 0$$ α > 0 and all sufficiently regular $$\Phi $$ Φ , we construct global classical solutions and thereby extend recent results for the fluid-free analogue to the system coupled to a Navier–Stokes system. As a crucial new challenge, our analysis requires a priori estimates for u at a point in the proof when knowledge about n is essentially limited to the observation that the mass is conserved. To overcome this problem, we also prove new uniform-in-time $$L^p$$ L p estimates for solutions to the inhomogeneous Navier–Stokes equations merely depending on the space-time $$L^2$$ L 2 norm of the force term raised to an arbitrary small power.

An interpolation inequality involving LlogL$L\log L$ spaces and application to the characterization of blow‐up behavior in a two‐dimensional Keller–Segel–Navier–Stokes system - Wang, Yulan; Winkler, Michael (Journal of the London Mathematical Society)
Abstract In a smoothly bounded two‐dimensional domain and for a given nondecreasing positive unbounded , for each and the inequality is shown to hold for any positive fulfilling This is thereafter applied to nonglobal solutions of the Keller–Segel system coupled to the incompressible Navier–Stokes equations through transport and buoyancy, and it is seen that in any such blow‐up event the corresponding population density cannot remain uniformly integrable over near its explosion time.

Global smooth solutions in a chemotaxis system modeling immune response to a solid tumor - Tao, Youshan; Winkler, Michael (Proceedings of the American Mathematical Society)
This manuscript studies a no-flux initial-boundary value problem for a four-component chemotaxis system that has been proposed as a model for the response of cytotoxic T-lymphocytes to a solid tumor. In contrast to classical Keller-Segel type situations focusing on two-component interplay of chemotaxing populations with a signal directly secreted by themselves, the presently considered system accounts for a certain indirect mechanism of attractant evolution. Despite the presence of a zero-order exciting nonlinearity of quadratic type that forms a core mathematical feature of the model, the manuscript asserts the global existence of classical solutions for initial data of arbitrary size in three-dimensional domains.

L∞ blow-up in the Jordan–Moore–Gibson–Thompson equation - Nikolić, Vanja; Winkler, Michael (Nonlinear Analysis)

A singular growth phenomenon in a Keller–Segel–type parabolic system involving density‐suppressed motilities - Wang, Yulan; Winkler, Michael (Mathematische Nachrichten)
Abstract A no‐flux initial‐boundary value problem for is considered in a ball , where and . Under the assumption that , it is shown that for each , there exist and a positive with the property that whenever is nonnegative with , the global solutions to () emanating from the initial data have the property that

Complete infinite-time mass aggregation in a quasilinear Keller–Segel system - Winkler, Michael (Israel Journal of Mathematics)
Abstract Radially symmetric global unbounded solutions of the chemotaxis system $$\left\{ {\matrix{{{u_t} = \nabla · (D(u)\nabla u) - \nabla · (uS(u)\nabla v),} \hfill & {} \hfill \cr {0 = Δ v - μ + u,} \hfill & {μ = {1 \over {|Ω |}}\int_Ω {u,} } \hfill \cr } } \right.$$ { u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( u S ( u ) ∇ v ) , 0 = Δ v − μ + u , μ = 1 | Ω | ∫ Ω u , are considered in a ball Ω = B R (0) ⊂ ℝ n , where n ≥ 3 and R > 0. Under the assumption that D and S suitably generalize the prototypes given by D ( ξ ) = ( ξ + ι ) m−1 and S ( ξ ) = ( ξ + 1) −λ−1 for all ξ > 0 and some m ∈ ℝ, λ >0 and ι ≥ 0 fulfilling $$m + λ < 1 - {2 \over n}$$ m + λ < 1 − 2 n , a considerably large set of initial data u 0 is found to enforce a complete mass aggregation in infinite time in the sense that for any such u 0 , an associated Neumann type initial-boundary value problem admits a global classical solution ( u, v ) satisfying $${1 \over C} · {(t + 1)^{{1 \over λ }}} \le ||u( · ,t)|{|_{{L^∞ }(Ω )}} \le C · {(t + 1)^{{1 \over λ }}}\,\,\,{\rm{for}}\,\,{\rm{all}}\,\,t > 0$$ 1 C ⋅ ( t + 1 ) 1 λ ≤ | | u ( ⋅ , t ) | | L ∞ ( Ω ) ≤ C ⋅ ( t + 1 ) 1 λ f o r a l l t > 0 as well as $$||u( · \,,t)|{|_{{L^1}(Ω \backslash {B_{{r_0}}}(0))}} → 0\,\,\,{\rm{as}}\,\,t → ∞ \,\,\,{\rm{for}}\,\,{\rm{all}}\,\,{r_0} \in (0,R)$$ | | u ( ⋅ , t ) | | L 1 ( Ω \ B r 0 ( 0 ) ) → 0 as t → ∞ for all r 0 ∈ ( 0 , R ) with some C > 0.

L∞ bounds in a two-dimensional doubly degenerate nutrient taxis system with general cross-diffusive flux - Winkler, Michael (Journal of Differential Equations)

Integrated, bright broadband, two-colour parametric down-conversion source - Pollmann, René; Roeder, Franz; Quiring, Victor; Ricken, Raimund; Eigner, Christof; Brecht, Benjamin; Silberhorn, Christine (Optics Express)
Broadband quantum light is a vital resource for quantum metrology and spectroscopy applications such as quantum optical coherence tomography or entangled two photon absorption. For entangled two photon absorption in particular, very high photon flux combined with high time-frequency entanglement is crucial for observing a signal. So far these conditions could be met by using high power lasers driving degenerate, type 0 bulk-crystal spontaneous parametric down conversion (SPDC) sources. This naturally limits the available wavelength ranges and precludes deterministic splitting of the generated output photons. In this work we demonstrate an integrated two-colour SPDC source utilising a group-velocity matched lithium niobate waveguide, reaching both exceptional brightness 1.52⋅10 6 pairssmWGHz and large bandwidth (7.8 THz FWHM) while pumped with a few mW of continuous wave (CW) laser light. By converting a narrow band pump to broadband pulses the created photon pairs show correlation times of Δ τ ≈ 120 fs while maintaining the narrow bandwidth Δ ω p ≪ 1 MHz of the CW pump light, yielding strong time-frequency entanglement. Furthermore our process can be adapted to a wide range of central wavelengths.

Ultra-broadband non-degenerate guided-wave bi-photon source in the near and mid-infrared - Roeder, Franz; Gnanavel, Abira; Pollmann, René; Brecht, Olga; Stefszky, Michael; Padberg, Laura; Eigner, Christof; Silberhorn, Christine; Brecht, Benjamin (New Journal of Physics)
The latest applications in ultrafast quantum metrology require bright, broadband bi-photon sources with one of the photons in the mid-infrared and the other in the visible to near infrared. However, existing sources based on bulk crystals are limited in brightness due to the short interaction length and only allow for limited dispersion engineering. Here, we present an integrated PDC source based on a Ti:LiNbO3 waveguide that generates broadband bi-photons with central wavelengths at 860 nm and 2800 nm. Their spectral bandwidth exceeds 25 THz and is achieved by simultaneous matching of the group velocities (GVs) and cancellation of GV dispersion for the signal and idler field. We provide an intuitive understanding of the process by studying our source’s behavior at different temperatures and pump wavelengths, which agrees well with simulations.

In-situ computed tomography analysis of the failure mechanisms of thermomechanically manufactured joints with auxiliary joining element - Borgert, Thomas; Köhler, D; Wiens, Eugen; Kupfer, R; Troschitz, J; Homberg, Werner; Gude, M (Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications)
Lightweight design by using low-density and load-adapted materials can reduce the weight of vehicles and the emissions generated during operation. However, the usage of different materials requires innovative joining technologies with increased versatility. In this investigation, the focus is on describing and characterising the failure behaviour of connections manufactured by an innovative thermomechanical joining process with adaptable auxiliary joining elements in single-lap tensile-shear tests. In order to analyse the failure development in detail, the specimens are investigated using in-situ computed tomography (in-situ CT). Here, the tensile-shear test is interrupted at points of interest and CT scans are conducted under load. In addition, the interrupted in-situ testing procedure is validated by comparing the loading behaviour with conventional continuous tensile-shear tests. The results of the in-situ investigations of joints with varying material combinations clearly describe the cause of failure, allowing conclusions towards an improved joint design.

Inverses Verfahren zur Identifikation piezoelektrischer Materialparameter unterstützt durch neuronale Netze - Koch, Kevin; Claes, Leander; Jurgelucks, Benjamin; Meihost, Lars; Henning, Bernd (Fortschritte der Akustik - DAGA 2024)

Estimation of temperature-dependent piezoelectric material parameters using ring-shaped specimens - Friesen, Olga; Claes, Leander; Scheidemann, Claus; Feldmann, Nadine; Hemsel, Tobias; Henning, Bernd (2023 International Congress on Ultrasonics, Beijing, China)

Randomised material parameter impedance dataset of piezoelectric rings - Koch, Kevin; Friesen, Olga; Claes, Leander

Randomised material parameter piezoelectric impedance dataset with structured electrodes - Koch, Kevin; Claes, Leander

Machine learning in inverse measurement problems: An application to piezoelectric material characterisation - Claes, Leander; Koch, Kevin; Friesen, Olga; Meihost, Lars