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Start Over Topic 010102 general mathematics Topic general mathematics
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1. Algebraic bounds on the Rayleigh–Bénard attractor

Author

Michael S. Jolly, Edriss S. Titi, Yu Cao, Jared P. Whitehead, Jolly, Michael S [0000-0002-7158-0933], Titi, Edriss S [0000-0002-5004-1746], Apollo - University of Cambridge Repository, Jolly, MS [0000-0002-7158-0933], and Titi, ES [0000-0002-5004-1746]

Subjects
Paper, General Mathematics, General Physics and Astronomy, global attractor, Enstrophy, 01 natural sciences, 76F35, Attractor, Periodic boundary conditions, Boundary value problem, 0101 mathematics, Algebraic number, Rayleigh–Bénard convection, math.AP, Mathematical Physics, Mathematics, Rayleigh-Benard convection, Plane (geometry), Applied Mathematics, 010102 general mathematics, Mathematical analysis, Statistical and Nonlinear Physics, 76E06, Nonlinear Sciences::Chaotic Dynamics, 010101 applied mathematics, 34D06, Homogeneous space, Affine space, synchronization, 35Q35
Abstract

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.

Published
2021

2. The r-Hunter-Saxton equation, smooth and singular solutions and their approximation

Author

Colin J. Cotter, Tristan Pryer, Jacob Deasy, Cotter, Colin J [0000-0001-7962-8324], Apollo - University of Cambridge Repository, and Engineering & Physical Science Research Council (EPSRC)

Subjects
Paper, singular solutions, GEODESIC-FLOW, Work (thermodynamics), General Mathematics, Mathematics, Applied, HYPERBOLIC VARIATIONAL EQUATION, Mathematics::Analysis of PDEs, General Physics and Astronomy, FOS: Physical sciences, 01 natural sciences, Piecewise linear function, 37K06, Mathematics - Analysis of PDEs, 0102 Applied Mathematics, 37K05, FOS: Mathematics, Hunter–Saxton equation, Applied mathematics, Initial value problem, Lie symmetries, 0101 mathematics, nlin.SI, math.AP, Mathematical Physics, Mathematics, Science & Technology, Nonlinear Sciences - Exactly Solvable and Integrable Systems, Physics, Applied Mathematics, 010102 general mathematics, 4901 Applied Mathematics, 4904 Pure Mathematics, Statistical and Nonlinear Physics, Action (physics), Symmetry (physics), Physics, Mathematical, 010101 applied mathematics, 35Q53, Nonlinear Sciences::Exactly Solvable and Integrable Systems, nonlinear PDEs, Physical Sciences, 49 Mathematical Sciences, 37K58, Exactly Solvable and Integrable Systems (nlin.SI), Analysis of PDEs (math.AP)
Abstract

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

Published
2019