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Funder: John Simon Guggenheim Memorial Foundation; doi: https://doi.org/10.13039/100005851, Funder: Einstein Visiting Fellow Program, The Rayleigh–Bénard system with stress-free boundary conditions is shown to have a global attractor in each affine space where velocity has fixed spatial average. The physical problem is shown to be equivalent to one with periodic boundary conditions and certain symmetries. This enables a Gronwall estimate on enstrophy. That estimate is then used to bound the L 2 norm of the temperature gradient on the global attractor, which, in turn, is used to find a bounding region for the attractor in the enstrophy–palinstrophy plane. All final bounds are algebraic in the viscosity and thermal diffusivity, a significant improvement over previously established estimates. The sharpness of the bounds are tested with numerical simulations.

In an attempt to understand the soliton resolution conjecture, we consider the Sine-Gordon equation on a spherically symmetric wormhole spacetime. We show that within each topological sector (indexed by a positive integer degree $n$) there exists a unique linearly stable soliton, which we call the $n$-kink. We give numerical evidence that the $n$-kink is a global attractor in the evolution of any smooth, finite energy solutions of degree $n$. When the radius of the wormhole throat $a$ is large enough, the convergence to the $n$-kink is shown to be governed by internal modes that slowly decay due to the resonant transfer of energy to radiation. We compute the exact asymptotics of this relaxation process for the $1$-kink using the Soffer-Weinstein weakly nonlinear perturbation theory., Comment: 19 pages, 10 figures, final version

Funder: University of Cambridge Centre for Doctoral Training, We consider the nearest-neighbour tight binding model of the honeycomb lattice in magnetic fields and discover surprizing new analytical results that fully explain fractal spectra and experimentally observed asymmetries in the density of states of molecular graphene. We describe a fractal Cantor spectrum for irrational magnetic flux through a honeycomb, and establish the existence of zero energy Dirac cones for each rational flux with fully explicit estimates on the cone angle. Our results give a substantially more refined description of subtleties in the de Haas–van Alphen and quantum Hall effects, and provide the first quantitative bounds on transport coefficients for the tight-binding model under disorder.

In this work we introduce the r-Hunter-Saxton equation, a generalisation of the Hunter-Saxton equation arising as extremals of an action principle posed in L_r. We characterise solutions to the Cauchy problem, quantifying the blow-up time and studying various symmetry reductions. We construct piecewise linear functions and show that they are weak solutions to the r-Hunter-Saxton equation., Revised after referee comments

Funder: John Simon Guggenheim Memorial Foundation; doi: https://doi.org/10.13039/100005851, In this work we study the global solvability of the primitive equations for the atmosphere coupled to moisture dynamics with phase changes for warm clouds, where water is present in the form of water vapor and in the liquid state as cloud water and rain water. This moisture model contains closures for the phase changes condensation and evaporation, as well as the processes of auto conversion of cloud water into rainwater and the collection of cloud water by the falling rain droplets. It has been used by Klein and Majda in [17] and corresponds to a basic form of the bulk microphysics closure in the spirit of Kessler [16] and Grabowski and Smolarkiewicz [12]. The moisture balances are strongly coupled to the thermodynamic equation via the latent heat associated to the phase changes. In [14] we assumed the velocity field to be given and proved rigorously the global existence and uniqueness of uniformly bounded solutions of the moisture balances coupled to the thermodynamic equation. In this paper we present the solvability of a full moist atmospheric flow model, where the moisture model is coupled to the primitive equations of atmospherical dynamics governing the velocity field. For the derivation of a priori estimates for the velocity field we thereby use the ideas of Cao and Titi [6], who succeeded in proving the global solvability of the primitive equations.

This study assessed the presence, amount, and sensitivity of personally identifiable information and personal health information found in hospital recycling bins in Toronto, Ontario, Canada.

A mathematical or computational model in evolutionary biology should necessary combine several comparatively fast processes, which actually drive natural selection and evolution, with a very slow process of evolution. As a result, several very different time scales are simultaneously present in the model; this makes its analytical study an extremely difficult task. However, the significant difference of the time scales implies the existence of a possibility of the model order reduction through a process of time separation. In this paper we conduct the procedure of model order reduction for a reasonably simple model of RNA virus evolution reducing the original system of three integro-partial derivative equations to a single equation. Computations confirm that there is a good fit between the results for the original and reduced models.