Local conservation law and dark radiation in cosmological braneworld

Abstract:The potential systems and nonlocal conservation laws of Prandtl boundary layer equations on the surface of a sphere have been investigated. The multiplier approach yields two local conservation laws for the Prandtl boundary layer equations on the surface of a sphere. Two potential variables ψ and ϕ are introduced corresponding to first and second conservation law. Moreover, another potential variable p is introduced by considering the linear combination of both conservation laws. Two level one potential systems involving a single nonlocal variable ψ or ϕ are constructed. One level two potential system involving both nonlocal variables ψ and ϕ is established. The nonlocal variable p is utilised to derive a spectral potential system. The nonlocal conservation laws of Prandtl boundary layer equations on the surface of a sphere are derived by computing the local conservation laws of its potential systems. The nonlocal conservation laws are utilised to derive the further nonlocally related systems.

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Direct construction method for conservation laws of partial differential equations Part I: Examples of conservation law classifications

European Journal of Applied Mathematics ◽

10.1017/s095679250100465x ◽

2002 ◽

Vol 13 (5) ◽

pp. 545-566 ◽

Cited By ~ 245

Author(s):

STEPHEN C. ANCO ◽

GEORGE BLUMAN

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Conservation Laws ◽

Conservation Law ◽

Wave Equations ◽

Explicit Construction ◽

Nonlinear Wave Equations ◽

Local Conservation ◽

Partial Differential ◽

Direct Construction

An effective algorithmic method is presented for finding the local conservation laws for partial differential equations with any number of independent and dependent variables. The method does not require the use or existence of a variational principle and reduces the calculation of conservation laws to solving a system of linear determining equations similar to that for finding symmetries. An explicit construction formula is derived which yields a conservation law for each solution of the determining system. In the first of two papers (Part I), examples of nonlinear wave equations are used to exhibit the method. Classification results for conservation laws of these equations are obtained. In a second paper (Part II), a general treatment of the method is given.

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Generalized Ertel’s theorem and infinite hierarchies of conserved quantities for three-dimensional time-dependent Euler and Navier–Stokes equations

Journal of Fluid Mechanics ◽

10.1017/jfm.2014.611 ◽

2014 ◽

Vol 760 ◽

pp. 368-386 ◽

Cited By ~ 20

Author(s):

Alexei F. Cheviakov ◽

Martin Oberlack

Keyword(s):

Conservation Laws ◽

Conservation Law ◽

Evolution Equations ◽

Stokes Equations ◽

Three Dimensional ◽

Infinite Family ◽

Time Dependent ◽

Local Conservation ◽

Direct Construction ◽

Primitive Variables

AbstractLocal conservation laws are systematically constructed for three-dimensional time-dependent viscous and inviscid incompressible fluid flows, in primitive variables and vorticity formulation, using the direct construction method. Complete sets of local conservation laws in primitive variables are derived for the case of conservation law multipliers depending on derivatives up to the second order. In the vorticity formulation, there exists an infinite family of vorticity-dependent conservation laws involving an arbitrary differentiable function of space and time, holding for both viscous and inviscid cases. The infinite conservation law family is used to generate further independent hierarchies of conservation laws that essentially involve vorticity and arbitrary flow parameters, which are determined by known evolution equations such as those for momentum, energy or helicity, though not necessarily in the form of a conservation law. The new conservation laws are not restricted to any reduced flow geometry such as planar or axisymmetric limits. Examples are considered.

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