Singular Limits — Low Stratification

Abstract We study the thermodynamic equilibrium spectra of the Charney- Hasegawa-Mima (CHM) equation in its weakly nonlinear limit. In this limit, the equation has three adiabatic invariants, in contrast to the two invariants of the 2D Euler or Gross-Pitaevskii equations, which are examples for comparison. We explore how the third invariant considerably enriches the variety of equilibrium spectra that the CHM system can access. In particular we characterise the singular limits of these spectra in which condensates occur, i.e. a single Fourier mode (or pair of modes) accumulate(s) a macroscopic fraction of the total invariants. We show that these equilibrium condensates provide a simple explanation for the characteristic structures observed in CHM systems of finite size: highly anisotropic zonal flows, large-scale isotropic vortices, and vortices at small scale. We show how these condensates are associated with combinations of negative thermodynamic potentials (e.g. temperature).

Download Full-text

Singular perturbation of a nonlinear problem with multiple solutions

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500013858 ◽

1985 ◽

Vol 100 (3-4) ◽

pp. 327-341

Author(s):

Anne-Marie Lefevere

Keyword(s):

Free Boundary ◽

Boundary Value Problem ◽

Multiple Solutions ◽

Nonlinear Boundary ◽

Boundary Value ◽

Critical Value ◽

Free Boundary Value Problem ◽

Singular Limits ◽

Natural Extensions ◽

Barrier Parameter

SynopsisA nonlinear boundary value problem (P) having positive parameters L and a is considered. We associate with it a family of perturbed problems () affected by the presence of a barrier parameter γ related to L and a. There is a critical value L*(a) of the parameter L such that for L >L*(a), (P) has no regular solution. Then some natural extensions of (P), solutions of a free boundary value problem, arise as singular limits of ().

Download Full-text

Construction of singular limits for four-dimensional elliptic problems with exponentially dominated nonlinearity

Bulletin des Sciences Mathématiques ◽

10.1016/j.bulsci.2006.11.002 ◽

2007 ◽

Vol 131 (7) ◽

pp. 670-685 ◽

Cited By ~ 3

Author(s):

Sami Baraket ◽

Imen Bazarbacha ◽

Nihed Trabelsi

Keyword(s):

Elliptic Problems ◽

Singular Limits

Download Full-text

Singular limits in compressible fluid dynamics

Archive for Rational Mechanics and Analysis ◽

10.1007/bf00387711 ◽

1994 ◽

Vol 128 (4) ◽

pp. 313-327 ◽

Cited By ~ 23

Author(s):

H. Beirão da Veiga

Keyword(s):

Fluid Dynamics ◽

Compressible Fluid ◽

Singular Limits ◽

Compressible Fluid Dynamics

Download Full-text

On Viscosity and Equivalent Notions of Solutions for Anisotropic Geometric Equations

Abstract and Applied Analysis ◽

10.1155/2020/7545983 ◽

2020 ◽

Vol 2020 ◽

pp. 1-14

Author(s):

Cecilia De Zan ◽

Pierpaolo Soravia

Keyword(s):

Mean Curvature Flow ◽

Group Structure ◽

Reaction Diffusion ◽

Reaction Diffusion Equations ◽

Carnot Groups ◽

Singular Limits ◽

Euclidian Space ◽

Characteristic Points ◽

Definition Of ◽

Homogeneous Norm

We prove that viscosity solutions of geometric equations in step two Carnot groups can be equivalently reformulated by restricting the set of test functions at the singular points. These are characteristic points for the level sets of the solutions and are usually difficult to deal with. A similar property is known in the Euclidian space, and in Carnot groups, it is based on appropriate properties of a suitable homogeneous norm. We also use this idea to extend to Carnot groups the definition of generalised flow, and it works similarly to the Euclidian setting. These results simplify the handling of the singularities of the equation, for instance, to study the asymptotic behaviour of singular limits of reaction diffusion equations. We provide examples of using the simplified definition, showing, for instance, that boundaries of strictly convex subsets in the Carnot group structure become extinct in finite time when subject to the horizontal mean curvature flow even if characteristic points are present.

Download Full-text