A regularity criterion in multiplier spaces to Navier-Stokes equations via the gradient of one velocity component

In this paper, we study regularity of weak solutions to the incompressible Navier-Stokes equations in $\mathbb{R}^{3}\times (0,T)$. The main goal is to establish the regularity criterion via the gradient of one velocity component in multiplier spaces.Comment: 9 pages. arXiv admin note: text overlap with arXiv:2005.1401

Exact two-dimensionalization of rapidly rotating large-Reynolds-number flows

We consider the flow of a Newtonian fluid in a three-dimensional domain, rotating about a vertical axis and driven by a vertically invariant horizontal body-force. This system admits vertically invariant solutions that satisfy the 2D Navier-Stokes equation. At high Reynolds number and without global rotation, such solutions are usually unstable to three-dimensional perturbations. By contrast, for strong enough global rotation, we prove rigorously that the 2D (and possibly turbulent) solutions are stable to vertically dependent perturbations: the flow becomes 2D in the long-time limit. These results shed some light on several fundamental questions of rotating turbulence: for arbitrary Reynolds number and small enough Rossby number, the system is attracted towards purely 2D flow solutions, which display no energy dissipation anomaly and no cyclone-anticyclone asymmetry. Finally, these results challenge the applicability of wave turbulence theory to describe stationary rotating turbulence in bounded domains.Comment: To be published in Journal of Fluid Mechanic

Cascades and transitions in turbulent flows

Turbulence is characterized by the non-linear cascades of energy and other inviscid invariants across a huge range of scales, from where they are injected to where they are dissipated. Recently, new experimental, numerical and theoretical works have revealed that many turbulent configurations deviate from the ideal 3D/2D isotropic cases characterized by the presence of a strictly direct/inverse energy cascade, respectively. We review recent works from a unified point of view and we present a classification of all known transfer mechanisms. Beside the classical cases of direct and inverse cascades, the different scenarios include: split cascades to small and large scales simultaneously, multiple/dual cascades of different quantities, bi-directional cascades where direct and inverse transfers of the same invariant coexist in the same scale-range and finally equilibrium states where no cascades are present, including the case when a condensate is formed. We classify all transitions as the control parameters are changed and we analyse when and why different configurations are observed. Our discussion is based on a set of paradigmatic applications: helical turbulence, rotating and/or stratified flows, MHD and passive/active scalars where the transfer properties are altered as one changes the embedding dimensions, the thickness of the domain or other relevant control parameters, as the Reynolds, Rossby, Froude, Peclet, or Alfven numbers. We discuss the presence of anomalous scaling laws in connection with the intermittent nature of the energy dissipation in configuration space. An overview is also provided concerning cascades in other applications such as bounded flows, quantum, relativistic and compressible turbulence, and active matter, together with implications for turbulent modelling. Finally, we present a series of open problems and challenges that future work needs to address.Comment: accepted for publication on Physics Reports 201

Supernova-driven Turbulence and Magnetic Field Amplification in Disk Galaxies

Supernovae are known to be the dominant energy source for driving turbulence in the interstellar medium. Yet, their effect on magnetic field amplification in spiral galaxies is still poorly understood. Analytical models based on the uncorrelated-ensemble approach predicted that any created field will be expelled from the disk before a significant amplification can occur. By means of direct simulations of supernova-driven turbulence, we demonstrate that this is not the case. Accounting for vertical stratification and galactic differential rotation, we find an exponential amplification of the mean field on timescales of 100Myr. The self-consistent numerical verification of such a "fast dynamo" is highly beneficial in explaining the observed strong magnetic fields in young galaxies. We, furthermore, highlight the importance of rotation in the generation of helicity by showing that a similar mechanism based on Cartesian shear does not lead to a sustained amplification of the mean magnetic field. This finding impressively confirms the classical picture of a dynamo based on cyclonic turbulence.Comment: 99 pages, 46 figures (in part strongly degraded), 8 tables, PhD thesis, University of Potsdam (2009). Resolve URN "urn:nbn:de:kobv:517-opus-29094" (e.g. via http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-29094) for a version with high-resolution figure

Regularisation and Long-Time Behaviour of Random Systems

Schenke A. Regularisation and Long-Time Behaviour of Random Systems. Bielefeld: Universität Bielefeld; 2020.In this work, we study several different aspects of systems modelled by partial differential equations (PDEs), both deterministic and stochastically perturbed. The thesis is structured as follows: Chapter I gives a summary of the contents of this work and illustrates the main results and ideas of the rest of the thesis. Chapter II is devoted to a new model for the flow of an electrically conducting fluid through a porous medium, the tamed magnetohydrodynamics (TMHD) equations. After a survey of regularisation schemes of fluid dynamical equations, we give a physical motivation for our system. We then proceed to prove existence and uniqueness of a strong solution to the TMHD equations, prove that smooth data lead to smooth solutions and finally show that if the onset of the effect of the taming term is deferred indefinitely, the solutions to the tamed equations converge to a weak solution of the MHD equations. In Chapter III we investigate a stochastically perturbed tamed MHD (STMHD) equation as a model for turbulent flows of electrically conducting fluids through porous media. We consider both the problem posed on the full space $\R^{3}$ as well as the problem with periodic boundary conditions. We prove existence of a unique strong solution to these equations as well as the Feller property for the associated semigroup. In the case of periodic boundary conditions, we also prove existence of an invariant measure for the semigroup. The last chapter deals with the long-time behaviour of solutions to SPDEs with locally monotone coefficients with additive L\'{e}vy noise. Under quite general assumptions, we prove existence of a random dynamical system as well as a random attractor. This serves as a unifying framework for a large class of examples, including stochastic Burgers-type equations, stochastic 2D Navier-Stokes equations, the stochastic 3D Leray-$\alpha$ model, stochastic power law fluids, the stochastic Ladyzhenskaya model, stochastic Cahn-Hilliard-type equations, stochastic Kuramoto-Sivashinsky-type equations, stochastic porous media equations and stochastic $p$-Laplace equations