An infinite scale of incompressible twisting solutions to the nonlinear elliptic system L [u; A, B] = ∇P and the discriminant ∆(h, g)

Morrison, George and Taheri, Ali (2018) An infinite scale of incompressible twisting solutions to the nonlinear elliptic system L [u; A, B] = ∇P and the discriminant ∆(h, g). Nonlinear Analysis Theory Methods & Applications, 173. pp. 209-219. ISSN 0362-546X

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Abstract

In this paper we consider the second order nonlinear elliptic system

div[A(|x|, |u| 2 , |∇u| 2 )∇u] + B(|x|, |u| 2 , |∇u| 2 )u = [cof ∇u]∇P,

where the unknown vector field u satisfies the incompressibility constraint det ∇u = 1 a.e. along with suitable boundary conditions and P = P(x) is an a priori unknown hydrostatic pressure field. Here, A = A(r, s, ξ) and B = B(r, s, ξ) are sufficiently regular scalar functions satisfying natural structural properties. Most notably in the case of a finite symmetric annulus we prove the existence of a countably infinite scale of topologically distinct twisting solutions to the system in all even dimensions. In sharp contrast in odd dimensions the only twisting solution is the map u ≡ x. We study a related class of systems by introducing the novel notion of a discriminant. Using this a complete and explicit characterisation of all twisting solutions for n ≥ 2 is given and a curious dichotomy in the behaviour of the system and its solutions captured and analyse

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Research Centres and Groups: Analysis and Partial Differential Equations Research Group
Depositing User: Ali Taheri
Date Deposited: 04 May 2018 10:12
Last Modified: 13 Jul 2018 07:41
URI: http://srodev.sussex.ac.uk/id/eprint/75608

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