An interpolation inequality involving $L\log L$ 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.
Keywords: General Mathematics

Pairwise Disjoint Perfect Matchings in r-Edge-Connected r-Regular Graphs - Ma, Yulai; Mattiolo, Davide; Steffen, Eckhard; Wolf, Isaak Hieronymus (SIAM Journal on Discrete Mathematics)
Keywords: General Mathematics

Dequantizing the Quantum Singular Value Transformation: Hardness and Applications to Quantum Chemistry and the Quantum PCP Conjecture - Gharibian, Sevag; Le Gall, François (SIAM Journal on Computing)
Keywords: General Mathematics; General Computer Science

Error analysis for the numerical approximation of the harmonic map heat flow with nodal constraints - Bartels, Sören; Kovács, Balázs; Wang, Zhangxian (IMA Journal of Numerical Analysis)
Abstract An error estimate for a canonical discretization of the harmonic map heat flow into spheres is derived. The numerical scheme uses standard finite elements with a nodal treatment of linearized unit-length constraints. The analysis is based on elementary approximation results and only uses the discrete weak formulation.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Relaxation in a Keller-Segel-consumption system involving signal-dependent motilities - Li, Genglin; Winkler, Michael (Communications in Mathematical Sciences)
Keywords: Applied Mathematics; General Mathematics

Global generalized solvability in a strongly degenerate taxis-type parabolic system modeling migration–consumption interaction - Winkler, Michael (Zeitschrift für angewandte Mathematik und Physik)
Abstract The parabolic problem $$\begin{aligned} \left\{ \begin{array}{l} u_t=\Delta \big (u\phi (v)\big ), \\ v_t=\Delta v-uv, \end{array} \right. \end{aligned}$$ u t = Δ ( u ϕ ( v ) ) , v t = Δ v - u v , is considered in smoothly bounded subdomains of $${\mathbb {R}}^n$$ R n with arbitrary $$n\ge 1$$ n ≥ 1 . Under the assumptions that $$\phi \in C^0([0,\infty )) \cap C^3((0,\infty ))$$ ϕ ∈ C 0 ( [ 0 , ∞ ) ) ∩ C 3 ( ( 0 , ∞ ) ) is positive on $$(0,\infty )$$ ( 0 , ∞ ) and satisfies $$\begin{aligned} \liminf _{\xi \searrow 0} \frac{\phi (\xi )}{\xi ^\alpha }>0 \quad {\text{ and }} \quad \limsup _{\xi \searrow 0} \big \{ \xi ^\beta |\phi '(\xi )| \big \}<\infty \end{aligned}$$ lim inf ξ ↘ 0 ϕ ( ξ ) ξ α > 0 and lim sup ξ ↘ 0 { ξ β | ϕ ′ ( ξ ) | } < ∞ with some $$\alpha >0$$ α > 0 and $$\beta >0$$ β > 0 , for all reasonably regular initial data an associated no-flux type initial-boundary value problem is shown to admit a global solution in an appropriately generalized sense. This extends previously developed solution theories on problems of this form, which either concentrated on non-degenerate or weakly degenerate cases corresponding to the choices $$\alpha =0$$ α = 0 and $$\alpha \in (0,2)$$ α ∈ ( 0 , 2 ) , or were restricted to low-dimensional settings by requiring that $$n\le 2$$ n ≤ 2 .
Keywords: Applied Mathematics; General Physics and Astronomy; General Mathematics

Classical solutions to Cauchy problems for parabolic–elliptic systems of Keller-Segel type - Winkler, Michael (Open Mathematics)
Abstract The Cauchy problem in R n {{\mathbb{R}}}^{n} , n ≥ 2 n\ge 2 , for u t = Δ u − ∇ ⋅ ( u S ⋅ ∇ v ) , 0 = Δ v + u , ( ⋆ ) \begin{array}{r}\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{l}{u}_{t}=\Delta u-\nabla \cdot \left(uS\cdot \nabla v),\\ 0=\Delta v+u,\end{array}\right.\hspace{2.0em}\hspace{2.0em}\hspace{2.0em}\left(\star )\end{array} is considered for general matrices S ∈ R n × n S\in {{\mathbb{R}}}^{n\times n} . A theory of local-in-time classical existence and extensibility is developed in a framework that differs from those considered in large parts of the literature by involving bounded classical solutions. Specifically, it is shown that for all non-negative initial data belonging to BUC ( R n ) ∩ L p ( R n ) {\rm{BUC}}\left({{\mathbb{R}}}^{n})\cap {L}^{p}\left({{\mathbb{R}}}^{n}) with some p ∈ [ 1 , n ) p\in \left[1,n) , there exist T max ∈ ( 0 , ∞ ] {T}_{\max }\in \left(0,\infty ] and a uniquely determined u ∈ C 0 ( [ 0 , T max ) ; BUC ( R n ) ) ∩ C 0 ( [ 0 , T max ) ; L p ( R n ) ) ∩ C ∞ ( R n × ( 0 , T max ) ) u\in {C}^{0}\left(\left[0,{T}_{\max });\hspace{0.33em}{\rm{BUC}}\left({{\mathbb{R}}}^{n}))\cap {C}^{0}\left(\left[0,{T}_{\max });\hspace{0.33em}{L}^{p}\left({{\mathbb{R}}}^{n}))\cap {C}^{\infty }\left({{\mathbb{R}}}^{n}\times \left(0,{T}_{\max })) such that with v ≔ Γ ⋆ u v:= \Gamma \star u , and with Γ \Gamma denoting the Newtonian kernel on R n {{\mathbb{R}}}^{n} , the pair ( u , v ) \left(u,v) forms a classical solution of ( ⋆ \star ) in R n × ( 0 , T max ) {{\mathbb{R}}}^{n}\times \left(0,{T}_{\max }) , which has the property that if T max < ∞ , then both limsup t ↗ T max ‖ u ( ⋅ , t ) ‖ L ∞ ( R n ) = ∞ and limsup t ↗ T max ‖ ∇ v ( ⋅ , t ) ‖ L ∞ ( R n ) = ∞ . \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}{T}_{\max }\lt \infty ,\hspace{1.0em}\hspace{0.1em}\text{then both}\hspace{0.1em}\hspace{0.33em}\mathop{\mathrm{limsup}}\limits_{t\nearrow {T}_{\max }}\Vert u\left(\cdot ,t){\Vert }_{{L}^{\infty }\left({{\mathbb{R}}}^{n})}=\infty \hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}\mathop{\mathrm{limsup}}\limits_{t\nearrow {T}_{\max }}\Vert \nabla v\left(\cdot ,t){\Vert }_{{L}^{\infty }\left({{\mathbb{R}}}^{n})}=\infty . An exemplary application of this provides a result on global classical solvability in cases when ∣ S + 1 ∣ | S+{\bf{1}}| is sufficiently small, where 1 = diag ( 1 , … , 1 ) {\bf{1}}={\rm{diag}}\hspace{0.33em}\left(1,\ldots ,1) .
Keywords: General Mathematics

Keller–Segel–Stokes interaction involving signal‐dependent motilities - Tian, Yu; Winkler, Michael (Mathematical Methods in the Applied Sciences)
The chemotaxis‐Stokes system is considered along with homogeneous boundary conditions of no‐flux type for and , and of Dirichlet type for , in a smoothly bounded domain . Under the assumption that , that is bounded on each of the intervals with arbitrary , and that with some and , we have It is shown that for any suitably regular initial data, an associated initial‐boundary value problem admits a global very weak solution.
Keywords: General Engineering; General Mathematics

New examples of harmonic maps to the hyperbolic plane via Bäcklund transformation - Polychrou, G.; Papageorgiou, Efthymia; Fotiadis, A.; Daskaloyannis, C. (Revista Matemática Complutense)
Abstract We study harmonic maps from a subset of the complex plane to a subset of the hyperbolic plane. In Fotiadis and Daskaloyannis (Nonlinear Anal 214, 112546, 2022), harmonic maps are related to the sinh-Gordon equation and a Bäcklund transformation is introduced, which connects solutions of the sinh-Gordon and sine-Gordon equation. We develop this machinery in order to construct new harmonic maps to the hyperbolic plane.
Keywords: General Mathematics

Implementing profession orientation as a design principle for overcoming Klein’s second discontinuity – preservice teacher’s perspectives on interface activities in the context of a geometry course - Hoffmann, Max; Biehler, Rolf (ZDM – Mathematics Education)
Intending to counteract Klein’s second discontinuity in teacher education, we explored and applied the innovation of “interface ePortfolio” in the context of a geometry course for preservice teachers (PSTs). The tool offers the possibility of implementing the design principle of profession orientation. In the article, we theoretically clarify what we understand by this principle and locate our innovative concept against this theoretical background. We empirically investigate the extent to which counteraction against the second discontinuity is successful by analyzing reflection texts created in the interface ePortfolio, focusing on PSTs’ perspectives. Our qualitative content analysis shows that most of them perceive the innovation as helpful in the intended sense and indicates that the course concept, in general, and the interface ePortfolio, in particular, have helped establish relevant links between the course content and their later work as teachers.
Keywords: General Mathematics; Education