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Johannes Lankeit

Publications and source records attributed to Johannes Lankeit.

61 records · Page 4Linked to original sources
arXiv2014

Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source

We prove existence of global weak solutions to the chemotaxis system $ u_t=Δu - \nabla\cdot (u\nabla v) +κu -μu^2 $ $ v_t=Δv-v+u $ under homogeneous Neumann boundary conditions in a smooth bounded convex domain $Ω\subset R^n$, for arbitrarily small values of $μ>0$. Additionally, we show that in the three-dimensional setting, after some time, these solutions become classical solutions, provided that $κ$ is not too large. In this case, we also consider their large-time behaviour: We prove decay if $κ\leq 0$ and the existence of an absorbing set if $κ>0$ is sufficiently small. Keywords: chemotaxis, logistic source, existence, weak solutions, eventual smoothness

math.AP↗
arXiv2014

Chemotaxis can prevent thresholds on population density

We define and (for $q>n$) prove uniqueness and an extensibility property of $W^{1,q}$-solutions to $u_t =-\nabla\cdot(u\nabla v)+κu-μu^2$ $ 0 =Δv-v+u$ $\partial_νv|_{\partialΩ} = \partial_νu|_{\partialΩ}=0,$ $ u(0,\cdot)=u_0 $ in balls in $\mathbb{R}^n$, which we then use to obtain a criterion guaranteeing some kind of structure formation in a corresponding chemotaxis system - thereby extending recent results of Winkler to the higher dimensional (radially symmetric) case. Keywords: chemotaxis, logistic source, blow-up, hyperbolic-elliptic system

math.AP↗
arXiv2013

On Grioli's minimum property and its relation to Cauchy's polar decomposition

The unitary polar factor of a matrix F is the unitary matrix Q realizing the minimum of the norm of F-Q over all unitary matrices Q. Tracing back the development on the optimality of the polar factor to its presumable roots, in this paper we present a commented translation of a note by G. Grioli from 1940 (Boll.Un.Math.Ital.2,452-455(1940)) showing the minimization property for the Frobenius norm in dimension 3; in his words: "We show that for the homogeneous displacement, tangent to any finite displacement, there exists a minimum property completely analogous to a well known property valid for infinitesimal displacements." Keywords: polar decomposition, optimality of the polar factor, Euclidean distance, geodesic distance, Euclidean movement

math.NA↗
arXiv2013

The minimization of matrix logarithms - on a fundamental property of the unitary polar factor

We show that the unitary factor U in the polar decomposition of a nonsingular matrix Z = U H is the minimizer for both ||Log(Q^* Z)|| and ||sym (Log(Q^*Z))|| over unitary Q, for any given invertible complex n-times-n matrix Z, for any unitarily invariant norm and any n. We prove that U is the unique matrix with this property. As important tools we use a generalized Bernstein trace inequality and the theory of majorization.

math.CA↗
arXiv2013

Sum of squared logarithms - An inequality relating positive definite matrices and their matrix logarithm

Let y1, y2, y3, a1, a2, a3 > 0 be such that y1 y2 y3 = a1 a2 a3 and y1 + y2 + y3 >= a1 + a2 + a3, y1 y2 + y2 y3 + y1 y3 >= a1 a2 + a2 a3 + a1 a3. Then the following inequality holds (log y1)^2 + (log y2)^2 + (log y3)^2 >= (log a1)^2 + (log a2)^2 + (log a3)^2. This can also be stated in terms of real positive definite 3x3-matrices P1, P2: If their determinants are equal det P1 = det P2, then tr P1 >= tr P2 and tr Cof P1 >= tr Cof P2 implies norm(log P1) >= norm(log P2), where log is the principal matrix logarithm and norm(P) denotes the Frobenius matrix norm. Applications in matrix analysis and nonlinear elasticity are indicated.

math.CA↗