Sharp decay estimates in a bioconvection model with quadratic degradation in bounded domains

The paper studies large time behaviour of solutions to the Keller–Segel system with quadratic degradation in a liquid environment, as given byunder Neumann boundary conditions in a bounded domain Ω ⊂ ℝn, where n ≥ 1 is arbitrary. It is shown that whenever U : Ω × (0,∞) → ℝn is a bounded and sufficiently regular solenoidal vector field any non-trivial global bounded solution of (⋆) approaches the trivial equilibrium at a rate that, with respect to the norm in either of the spaces L1(Ω) and L∞(Ω), can be controlled from above and below by appropriate multiples of 1/(t + 1). This underlines that, even up to this quantitative level of accuracy, the large time behaviour in (⋆) is essentially independent not only of the particular fluid flow, but also of any effect originating from chemotactic cross-diffusion. The latter is in contrast to the corresponding Cauchy problem, for which known results show that in the n = 2 case the presence of chemotaxis can significantly enhance biomixing by reducing the respective spatial L1 norms of solutions.

Global regularity for a model Navier–Stokes equations on ℝ3

We study a nonlinear parabolic system for a time dependent solenoidal vector field on ℝ

STABILITY WITH RESPECT TO DOMAIN OF THE LOW MACH NUMBER LIMIT OF COMPRESSIBLE VISCOUS FLUIDS

We study the asymptotic limit of solutions to the barotropic Navier–Stokes system, when the Mach number is proportional to a small parameter ε → 0 and the fluid is confined to an exterior spatial domain Ωε that may vary with ε. As ε → 0, it is shown that the fluid density becomes constant while the velocity converges to a solenoidal vector field satisfying the incompressible Navier–Stokes equations on a limit domain. The velocities approach the limit strongly (a.a.) on any compact set, uniformly with respect to a certain class of domains. The proof is based on spectral analysis of the associated wave propagator (Neumann Laplacian) governing the motion of acoustic waves.

Helicity and the Călugăreanu invariant

The helicity of a localized solenoidal vector field (i.e. the integrated scalar product of the field and its vector potential) is known to be a conserved quantity under ‘frozen field’ distortion of the ambient medium. In this paper we present a number of results concerning the helicity of linked and knotted flux tubes, particularly as regards the topological interpretation of helicity in terms of the Gauss linking number and its limiting form (the Călugăreanu invariant). The helicity of a single knotted flux tube is shown to be intimately related to the Călugăreanu invariant and a new and direct derivation of this topological invariant from the invariance of helicity is given. Helicity is decomposed into writhe and twist contributions, the writhe contribution involving the Gauss integral (for definition, see equation (4.8)), which admits interpretation in terms of the sum of signed crossings of the knot, averaged over all projections. Part of the twist contribution is shown to be associated with the torsion of the knot and part with what may be described as ‘intrinsic twist’ of the field lines in the flux tube around the knot (see equations (5.13) and (5.15)). The generic behaviour associated with the deformation of the knot through a configuration with points of inflexion (points at which the curvature vanishes) is analysed and the role of the twist parameter is discussed. The derivation of the Călugăreanu invariant from first principles of fluid mechanics provides a good demonstration of the relevance of fluid dynamical techniques to topological problems.

Magnetfeldbeschreibung mit verallgemeinerten poloidalen und toroidalen Skalaren

Representation of Magnetic Fields by Generalized poloidal and Toroidal Scalars Every solenoidal vector field can be represented by unique poloidal and toroidal scalars. This description is especially appropriate to the geometry of a sphere. A generalization which can be applied to a more or less complicated geometry could be elaborated by means of transforming integrability conditions of space into integrability conditions of surfaces. This formalism enables us to give simple proofs of other important representations of vector fields by two scalars (magnetic coordinates, complex-lamellar fields).