Conservation Laws and Nonlocally Related Systems of Two-Dimensional Boundary Layer Models

2017 ◽  
Vol 72 (11) ◽  
pp. 1031-1051
Author(s):  
R. Naz ◽  
A.F. Cheviakov
Keyword(s):  
Boundary Layer ◽  
Conservation Laws ◽  
Direct Method ◽  
Local Conservation ◽  
Solution Sets ◽  
Nonlocal Symmetries ◽  
Pde Systems ◽  
Nonlocal Conservation Laws ◽  
Constitutive Parameters ◽  
Nonlocally Related Systems

AbstractLocal conservation laws, potential systems, and nonlocal conservation laws are systematically computed for three-equilibrium two-component boundary layer models that describe different physical situations: a plate flow, a flow parallel to the axis of a circular cylinder, and a radial jet striking a planar wall. First, local conservation laws of each model are computed using the direct method. For each of the three boundary layer models, two local conservation laws are found. The corresponding potential variables are introduced, and nonlocally related potential systems and subsystems are formed. Then nonlocal conservation laws are sought, arising as local conservation laws of nonlocally related systems. For each of the three physical models, similar nonlocal conservation laws arise. Further nonlocal variables that lead to further potential systems are considered. Trees of nonlocally related systems are constructed; their structure coincides for all three models. The three boundary layer models considered in this work provide rich and interesting examples of the construction of trees of nonlocally related systems. In particular, the trees involve spectral potential systems depending on a parameter; these spectral potential systems lead to nonlocal conservation laws. Moreover, potential variables that are not locally related on solution sets of some potential systems become local functions of each other on solution sets of other systems. The point symmetry analysis shows that the plate and radial jet flow models possess infinite-dimensional Lie algebras of point symmetries, whereas the Lie algebra of point symmetries for the cylinder flow model is three-dimensional. The computation of nonlocal symmetries reveals none that arise for the original model equations, which is common for partial differential equations (PDE) systems without constitutive parameters or functions, but does reveal nonlocal symmetries for some nonlocally related PDE systems.

Potential Systems and Nonlocal Conservation Laws of Prandtl Boundary Layer Equations on the Surface of a Sphere

2017 ◽  
Vol 72 (4) ◽  
pp. 351-357 ◽  
Author(s):  
R. Naz
Keyword(s):  
Boundary Layer ◽  
Conservation Laws ◽  
Linear Combination ◽  
Conservation Law ◽  
Local Conservation ◽  
Boundary Layer Equations ◽  
Potential System ◽  
Nonlocal Conservation Laws ◽  
Prandtl Boundary Layer ◽  
Nonlocally Related Systems

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.

Differential invariant method for seeking nonlocally related systems and nonlocal symmetries. I: General theory and examples

2021 ◽  
Vol 477 (2250) ◽  
pp. 20200908
Author(s):  
George W. Bluman ◽  
Rafael de la Rosa ◽  
María Santos Bruzón ◽  
María Luz Gandarias
Keyword(s):  
Differential Equation ◽  
Differential Invariant ◽  
Nonlocal Symmetries ◽  
Independent Variables ◽  
New Exact Solutions ◽  
Pde Systems ◽  
Nonlocal Conservation Laws ◽  
Two Independent Variables ◽  
Nonlocally Related Systems ◽  
Ode Systems

Nonlocally related systems, obtained through conservation law and symmetry-based methods, have proved to be useful for determining nonlocal symmetries, nonlocal conservation laws, non-invertible mappings and new exact solutions of a given partial differential equation (PDE) system. In this paper, it is shown that the symmetry-based method is a differential invariant-based method. It is shown that this allows one to naturally extend the symmetry-based method to ordinary differential equation (ODE) systems and to PDE systems with at least three independent variables. In particular, we present the situations for ODE systems, PDE systems with two independent variables and PDE systems with three or more independent variables, separately, and show that these three situations are directly connected. Examples are exhibited for each of the three situations.

scholarly journals ON INFINITE SERIES OF NONLOCAL CONSERVATION LAWS FOR PARTIAL DIFFERENTIAL EQUATIONS

2018 ◽  
Vol 21 (3) ◽  
pp. 150-159
Author(s):  
N. G. Khor’kova
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Infinite Series ◽  
Conservation Law ◽  
Integrability Condition ◽  
Normal System ◽  
Partial Differential ◽  
Nonlocal Symmetries ◽  
Nonlocal Conservation Laws

Популярное в математике понятие интегрируемости дифференциальных уравнений (и столь же разнообразно трактуемое) тесно связано с существованием симметрий и законов сохранения. Все известные интегрируемые дифференциальные уравнения обладают бесконечными сериями симметрий и (или) законов сохранения. Однако также имеется целый ряд уравнений, важных для приложений, но имеющих крайне скудный запас симметрий или законов сохранения. Попытки расширить понятия симметрии и закона сохранения предпринимались разными авторами, и на эту тему имеется обширная литература. В данной статье представлен следующий результат. Если ℓ-нормальная система дифференциальных уравнений в частных производных имеет когомологически нетривиальный закон сохранения, то этот закон сохранения порождает бесконечную серию нелокальных законов сохранения. Этот факт обобщает аналогичный результат статьи автора для дифференциальных уравнений (не систем). Результат получен в рамках геометрической теории дифференциальных уравнений в частных производных. Согласно геометрическому подходу, многообразие, снабженное конечномерным распределением, удовлетворяющим условиям интегрируемости Фробениуса, называется диффеотопом (diffiety), если локально оно имеет вид бесконечно продолженного уравнения Ɛ∞. Диффеотопы являются объектами категории дифференциальных уравнений, введенной А.М. Виноградовым. Под симметриями уравнения понимают преобразования (конечные или инфинитизимальные) бесконечного продолжения уравнения, которые сохраняют распределение Картана, а под законами сохранения – (n-1)-e классы когомологий горизонтального комплекса де Рама уравнения, где n – число независимых переменных уравнения. Накрытием называется эпиморфизм  τ:Ɛ⟶ Ɛ∞ в категории дифференциальных уравнений, порождающий изоморфизм распределений. Симметрии и законы сохранения диффеотопа ࣟƐ называются нелокальными симметриями и законами сохранения уравнения ࣟƐ  Выбор подходящего накрытия позволяет получать новые (нелокальные) симметрии и законы сохранения исследуемого уравнения. В работе приведена конструкция одного накрытия и доказано существование бесконечных серий нелокальных законов сохранения у широкого класса систем дифференциальных уравнений в частных производных.системы дифференциальных уравнений в частных производных; накрытия дифференциальных уравнений; нелокальные симметрии и законы сохранения  The notion of integrability of differential equations is closely connected with the existence of symmetries and conservation laws. All known integrable differential equations have infinite series of symmetries and (or) conservation laws. However, there is also a number of equations that are important for applications, but with an extremely scarce stock of symmetries or conservation laws. Attempts to extend the concepts of symmetry and conservation law were made by different authors. This article presents the following result. If a ℓ-normal system of partial differential equations has a cohomologically nontrivial conservation law, then this conservation law generates an infinite series of non-local conservation laws. This fact generalizes the analogous result of the author for differential equations (not systems). The result is obtained within the framework of geometrical theory of partial differential equations (PDE). A manifold supplied with an infinite-dimensional distribution satisfying the Frobenius complete integrability condition is called a diffiety, if it is locally in the form of  Ɛ∞. Diffieties are objects of the category of differential equations introduced by A.M. Vinogradov. Symmetries of PDE are transformations (finite or infinitesimal) of the infinite prolongation  Ɛ∞ preserving the Cartan distribution, while conservation laws are (n-1)-cohomology classes of the horizontal de Rham cohomology. If a covering τ:Ɛ⟶ Ɛ∞ is given, then symmetries and conservation laws of the diffiety Ɛ are called nonlocal symmetries and conservation laws of the equation Ɛ .In appropriate coverings one can get new (nonlocal) symmetries and conservation laws for an equation under consideration. In this paper we investigate one covering and prove the existence of infinite series of nonlocal conservation laws.   

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