Regularity of Lie groups - Hanusch, Maximilian (Communications in Analysis and Geometry)
Keywords: regularity of Lie groups

Analysis 1 und 2 Skript/Buch - Hanusch, Maximilian

Stabilization of arbitrary structures in a doubly degenerate reaction-diffusion system modeling bacterial motion on a nutrient-poor agar. - Winkler, Michael (Calculus of Variations and Partial Differential Equations)

Chemotaxis-Stokes interaction with very weak diffusion enhancement: Blow-up exclusion via detection of absorption-induced entropy structures involving multiplicative couplings. - Winkler, Michael (Advanced Nonlinear Studies)

Small-signal solutions of a two-dimensional doubly degenerate taxis system modeling bacterial motion in a nutrient-poor environments. - Winkler, Michael (Nonlinear Analysis: Real World Applications)

A unifying approach toward boundedness in Keller-Segel type cross-diffusion systems via conditional $L^∞$ estimates for taxis gradients. - Winkler, Michael (Mathematische Nachrichten)

Oscillatory decay in a degenerate parabolic equation. - Winkler, Michael (Partial Differential Equations and Applications)

A critical blow-up exponent for flux limitation in a Keller-Segel system - Winkler, Michael (Indiana University Mathematics Journal)

Unlimited growth in logarithmic Keller-Segel systems - Winkler, Michael (Journal of Differential Equations)

Radial solutions to a chemotaxis-consumption model involving prescribed signal concentrations on the boundary - Lankeit, Johannes; Winkler, Michael (Nonlinearity)

Reaction-driven relaxation in threee-dimensional Keller-Segel-Navier-Stokes interaction. - Winkler, Michael (Communications in Mathematical Physics)

A family of mass-critical Keller-Segel systems. - Winkler, Michael (Proceedings of the London Mathematical Society)

Global weak solutions and absorbing sets in a chemotaxis-Navier–Stokes system with prescribed signal concentration on the boundary - Black, Tobias; Winkler, Michael (Mathematical Models and Methods in Applied Sciences)
An initial-boundary value problem for a coupled chemotaxis-Navier–Stokes model with porous medium type diffusion is considered. Previous related literature has provided profound knowledge in cases when the system is augmented with no-flux/no-flux/no-slip boundary conditions for the density of cells, the chemical concentration and the fluid velocity field, respectively; in particular, available qualitative results strongly indicate that only trivial solution behavior can be expected on large time scales. In line with refined modeling approaches to oxygen evolution near fluid-air interfaces, this study now focuses on situations involving a fixed chemoattractant concentration on the boundary. Despite an apparent loss of mathematically favorable energy structures thereby induced, by means of an alternative variational approach a basic theory of global existence is developed in a natural framework of weak solvability. Beyond this, some additional qualitative information on the large time behavior of these solutions is derived by identifying a certain global relaxation property. Specifically, a second result asserts, within a suitable topological setting, the existence of a bounded set which eventually absorbs each individual of the obtained trajectories, and the diameter of which is bounded only by the physically relevant quantities of total population size and prescribed boundary concentration of the chemical signal.
Keywords: Applied Mathematics; Modeling and Simulation

Global weak solutions to a chemotaxis-Navier-Stokes system in $R^3$. - Kang, Kyungkeun; Lee, Jihoon; Winkler, Michael (Discrete and Continuous Dynamical Systems)

Elementary symmetric polynomials and martingales for Heckman-Opdam processes - Rösler, Margit; Voit, Michael (Contemporary Mathematics)
We consider the generators $L_k$ of Heckman-Opdam diffusion processes in the compact and non-compact case in $N$ dimensions for root systems of type $A$ and $B$, with a multiplicity function of the form $k=κk_0$ with some fixed value $k_0$ and a varying constant $κ\in\,[0,\infty[$. Using elementary symmetric functions, we present polynomials which are simultaneous eigenfunctions of the $L_k$ for all $κ\in\,]0,\infty[$. This leads to martingales associated with the Heckman-Opdam diffusions $ (X_{t,1},\ldots,X_{t,N})_{t\ge0}$. As our results extend to the freezing case $κ=\infty$ with a deterministic limit after some renormalization, we find formulas for the expectations $\mathbb E(\prod_{j=1}^N(y-X_{t,j})),$ $y\in\mathbb C$.

Chemotaxis and cross-diffusion models in complex environments: Models and analytic problems toward a multiscale vision. - Bellomo, Nicolas; Outada, Nisrine; Soler, Juan; Tao, Youshan; Winkler, Michael (Mathematical Models & Methods in Applied Sciences)

Global existence in reaction-diffusion systems with mass control under relaxed assumptions merely referring to cross-absorptiva effects. - Lankeit, Johannes; Winkler, Michael (Journal of Evolution Equations)

A double critical mass phenomenon in a no-flux-Dirichlet Keller-Segel system. - Fuhrmann, Jan; Lankeit, Johannes; Winkler, Michael (Journal de Mathématiques Pures et Appliquées)

A smallness condition ensuring boundedness in a two-dimensional chemotaxis-Navier-Stokes system involving Dirichlet boundary conditions for the signal. - Wang, Yulan; Winkler, Michael; Xiang, Zhaoyin (Acta Mathematica Sinica (English Series))

Global mass-preserving solutions to a chemotaxis-fluid model involving Dirichlet boundary conditions for the signal. - Wang, Yulan; Winkler, Michael; Xiang, Zhaoyin (Analysis and Applications)

Potential kernels for radial Dunkl Laplacians - Graczyk, P.; Luks, Tomasz; Sawyer, P. (Canadian Journal of Mathematics)

Asymptotic stability of spatial homogeneity in a haptotaxis model for oncolytic virotherapy. - Tao, Youshan; Winkler, Michael (Proceedings of the Royal Society of Edinburgh Section A: Mathematics)

Existence theory and qualitative analysis for a fully cross-diffusive predator-prey system - Tao, Youshan; Winkler, Michael (SIAM Journal on Mathematical Analysis)

Nonnegative solutions to a doubly degenerate nutrient taxis system - Li, Genglin; Winkler, Michael (Communications on Pure and Applied Analysis)

Prescribed signal concentration on the boundary: eventual smoothness in a chemotaxis-Navier–Stokes system with logistic proliferation - Black, Tobias; Wu, Chunyan (Calculus of Variations and Partial Differential Equations)
Keywords: Applied Mathematics; Analysis

Semiclassical formulae For Wigner distributions - Barkhofen, Sonja; Schütte, Philipp; Weich, Tobias (Journal of Physics A: Mathematical and Theoretical)
In this paper we give an overview over some aspects of the modern mathematical theory of Ruelle resonances for chaotic, i.e. uniformly hyperbolic, dynamical systems and their implications in physics. First we recall recent developments in the mathematical theory of resonances, in particular how invariant Ruelle distributions arise as residues of weighted zeta functions. Then we derive a correspondence between weighted and semiclassical zeta functions in the setting of negatively curved surfaces. Combining this with results of Hilgert, Guillarmou and Weich yields a high frequency interpretation of invariant Ruelle distributions as quantum mechanical matrix coefficients in constant negative curvature. We finish by presenting numerical calculations of phase space distributions in the more physical setting of 3-disk scattering systems.

Poisson transforms for trees of bounded degree - Bux, Kai-Uwe; Hilgert, Joachim; Weich, Tobias (Journal of Spectral Theory)
Keywords: Geometry and Topology; Mathematical Physics; Statistical and Nonlinear Physics

Poisson transforms for trees of bounded degree - Hilgert, Joachim; Weich, Tobias; Bux, K.-U. (J. of Spectral Theory)

Ethan D. Bolker und Maura B. Mast: Common Sense Mathematics, Second Edition. AMS/MAA Press 2021 - Hilgert, Joachim (Mathematische Semesterberichte)

Asymptotic expansion of generalized Witten integrals for Hamiltonian circle actions - Delarue, Benjamin; Ramacher, Pablo (Journal of Symplectic Geometry)

Decompositions of locally compact contraction groups, series and extensions - Glöckner, Helge; Willis, George A. (Journal of Algebra)
A locally compact contraction group is a pair (G,α), where G is a locally compact group and α:G→G an automorphism such that αn(x)→e pointwise as n→∞. We show that every surjective, continuous, equivariant homomorphism between locally compact contraction groups admits an equivariant continuous global section. As a consequence, extensions of locally compact contraction groups with abelian kernel can be described by continuous equivariant cohomology. For each prime number p, we use 2-cocycles to construct uncountably many pairwise non-isomorphic totally disconnected, locally compact contraction groups (G,α) which are central extensions0→Fp((t))→G→Fp((t))→0 of the additive group of the field of formal Laurent series over Fp=Z/pZ by itself. By contrast, there are only countably many locally compact contraction groups (up to isomorphism) which are torsion groups and abelian, as follows from a classification of the abelian locally compact contraction groups.
Keywords: Contraction group; Torsion group; Extension; Cocycle; Section; Equivariant cohomology; Abelian group; Nilpotent group; Isomorphism types

Locally pro-p contraction groups are nilpotent - Glöckner, Helge; Willis, George A. (J. Reine Angew. Math.)
Keywords: 22D05; 22A05; 20E18

Direct limits of regular Lie groups - Glöckner, Helge (Mathematische Nachrichten)

Contraction groups and the big cell for endomorphisms of Lie groups over local fields - Glöckner, Helge (arXiv:2101.02981)
Let $G$ be a Lie group over a totally disconnected local field and $α$ be an analytic endomorphism of $G$. The contraction group of $α$ ist the set of all $x\in G$ such that $α^n(x)→ e$ as $n→∞$. Call sequence $(x_{-n})_{n≥ 0}$ in $G$ an $α$-regressive trajectory for $x\in G$ if $α(x_{-n})=x_{-n+1}$ for all $n≥ 1$ and $x_0=x$. The anti-contraction group of $α$ is the set of all $x\in G$ admitting an $α$-regressive trajectory $(x_{-n})_{n≥ 0}$ such that $x_{-n}→ e$ as $n→∞$. The Levi subgroup is the set of all $x\in G$ whose $α$-orbit is relatively compact, and such that $x$ admits an $α$-regressive trajectory $(x_{-n})_{n≥ 0}$ such that $\{x_{-n}\colon n≥ 0\}$ is relatively compact. The big cell associated to $α$ is the set $Ω$ of all all products $xyz$ with $x$ in the contraction group, $y$ in the Levi subgroup and $z$ in the anti-contraction group. Let $π$ be the mapping from the cartesian product of the contraction group, Levi subgroup and anti-contraction group to $Ω$ which maps $(x,y,z)$ to $xyz$. We show: $Ω$ is open in $G$ and $π$ is étale for suitable immersed Lie subgroup structures on the three subgroups just mentioned. Moreover, we study group-theoretic properties of contraction groups and anti-contraction groups.

Resonances and weighted zeta functions for obstacle scattering via smooth models - Schütte, Philipp; Weich, Tobias; Delarue, Benjamin
We consider a geodesic billiard system consisting of a complete Riemannian manifold and an obstacle submanifold with boundary at which the trajectories of the geodesic flow experience specular reflections. We show that if the geodesic billiard system is hyperbolic on its trapped set and the latter is compact and non-grazing the techniques for open hyperbolic systems developed by Dyatlov and Guillarmou can be applied to a smooth model for the discontinuous flow defined by the non-grazing billiard trajectories. This allows us to obtain a meromorphic resolvent for the generator of the billiard flow. As an application we prove a meromorphic continuation of weighted zeta functions together with explicit residue formulae. In particular, our results apply to scattering by convex obstacles in the Euclidean plane.

Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces - Küster, Benjamin; Weich, Tobias (International Mathematics Research Notices)
Abstract For a compact Riemannian locally symmetric space $\mathcal M$ of rank 1 and an associated vector bundle $\mathbf V_{τ }$ over the unit cosphere bundle $S^{* }\mathcal M$, we give a precise description of those classical (Pollicott–Ruelle) resonant states on $\mathbf V_{τ }$ that vanish under covariant derivatives in the Anosov-unstable directions of the chaotic geodesic flow on $S^{* }\mathcal M$. In particular, we show that they are isomorphically mapped by natural pushforwards into generalized common eigenspaces of the algebra of invariant differential operators $D(G,σ )$ on compatible associated vector bundles $\mathbf W_{σ }$ over $\mathcal M$. As a consequence of this description, we obtain an exact band structure of the Pollicott–Ruelle spectrum. Further, under some mild assumptions on the representations $τ$ and $σ$ defining the bundles $\mathbf V_{τ }$ and $\mathbf W_{σ }$, we obtain a very explicit description of the generalized common eigenspaces. This allows us to relate classical Pollicott–Ruelle resonances to quantum eigenvalues of a Laplacian in a suitable Hilbert space of sections of $\mathbf W_{σ }$. Our methods of proof are based on representation theory and Lie theory.
Keywords: General Mathematics

Wave Front Sets of Nilpotent Lie Group Representations - Weich, Tobias; Budde, Julia (arXiv:2103.02968v1)

Symmetries of analytic curves - Hanusch, Maximilian (Differential Geometry and its Applications)
Keywords: Geometry and Topology; Analysis

Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with gradient-dependent flux limitation. - Winkler, Michael (Nonlinear Analysis: Real World Applications)

Suppressing blow-up by gradient-dependent flux limitation in a planar Keller-Segel-Navier-Stokes system. - Winkler, Michael (Zeitschrift für Angewandte Mathematik und Physik)

Can rotational fluxes impede the tendency toward spatial homogeneity in nutrient taxis (-Stokes) systems? - Winkler, Michael (International Mathematics Research Notices)

Boundedness in a three-dimensional Keller-Segel-Stokes system with subcritical sensitivity. - Winkler, Michael (Applied Mathematics Letters)

Does spatial homogeneity ultimately prevail in nutrient taxis systems? A paradigm for structure support by rapid diffusion decay in an autonomous parabolic flow. - Winkler, Michael (Transactions of the American Mathematical Society)

Positive intertwiners for Bessel functions of type B - Rösler, Margit; Voit, Michael (Proceedings of the American Mathematical Society)
Keywords: Applied Mathematics; General Mathematics

A fully cross-diffusive two-component evolution system: Existence and qualitative analysis via entropy-consistent thin-film-type approximation. - Tao, Youshan; Winkler, Michael (Journal of Functional Analysis)

The dampening role of large repulsive convection in a chemotaxis system modeling tumor angiogenesis. - Tao, Youshan; Winkler, Michael (Nonlinear Analysis, Theory, Methods and Applications)

Immediate regularization of measure-type population densities in a two-dimensional chemotaxis system with signal consumption. - Wang, Yulan; Winkler, Michael; Xiang, Zhaoyin (Science China Mathematics)

Small-density solutiions in Keller-Segel systems involving rapidly decaying diffusivities. - Ding, Mengyao; Winkler, Michael (Nonlinear Differential Equatiions and Applicatiions)

Global smooth solutions in a two-dimensional cross-diffusion system modeling propagation of urban crime. - Tao, Youshan; Winkler, Michael (Communications in Mathematical Sciences)

Taxis-driven formation of singular hotspots in a May-Nowak type model for virus infection. - Tao, Youshan; Winkler, Michael (SIAM Journal on Mathematical Analysis)