Conservation laws via the partial Lagrangian and group invariant solutions for radial and two-dimensional free jets

The conservation laws for Prandtl’s boundary layer equations for an incompressible fluid governing the flow in radial and two-dimensional jets are investigated. For both radial and two-dimensional jets the partial Lagrangian method is used to derive conservation laws for the system of two differential equations for the velocity components. The Lie point symmetries are calculated for both cases and a symmetry is associated with the conserved vector that is used to establish the conserved quantity for the jet. This associated symmetry is then used to derive the group invariant solution for the system governing the flow in the free jet.     

Conservation laws for laminar axisymmetric jet flows with weak swirl

The conservation laws for laminar axisymmetric jet flows with weak swirl are studied here. The multiplier approach is used to derive the conservation laws for the system of three boundary layer equations for the velocity components governing flow in laminar axisymmetric jet flows with weak swirl. Conservation laws for the system of two partial differential equations for the stream function are also derived.     

Conservation laws and symmetries of semilinear radial wave equations

Classifications of symmetries and conservation laws are presented for a variety of physically and analytically interesting wave equations with power nonlinearities in n spatial dimensions: a radial hyperbolic equation, a radial Schrödinger equation and its derivative variant, and two proposed radial generalizations of modified Korteweg-de Vries equations, as well as Hamiltonian variants. The mains results classify all admitted local point symmetries and all admitted local conserved densities depending on up to first order spatial derivatives, including any that exist only for special powers or dimensions. All such cases for which these wave equations admit, in particular, dilational energies or conformal energies and inversion symmetries are determined. In addition, potential systems arising from the classified conservation laws are used to determine nonlocal symmetries and nonlocal conserved quantities admitted by these equations. As illustrative applications, a discussion is given of energy norms, conserved Hs norms, critical powers for blow-up solutions, and one-dimensional optimal symmetry groups for invariant solutions.     

Formal conserved quantities for isothermic surfaces

Isothermic surfaces in \(S^n\) are characterised by the existence of a pencil \(\nabla ^t\) of flat connections. Such a surface is special of type \(d\) if there is a family \(p(t)\) of \(\nabla ^t\) -parallel sections whose dependence on the spectral parameter \(t\) is polynomial of degree \(d\) . We prove that any isothermic surface admits a family of \(\nabla ^t\) -parallel sections which is a formal Laurent series in \(t\) . As an application, we give conformally invariant conditions for an isothermic surface in \(S^3\) to be special.     

Conservation laws for linear viscoelasticity

In the brief note entitled On Conservation Laws for Dissipative Systems [4], a new method for constructing conservation laws was proposed. This method was termed the Neutral Action (NA) method in [5]. For any system governed by a set of differential equations, the NA method offers a systematic approach for determination of conservation laws applicable to the system. It is the purpose of the present paper to establish conservation laws for one- and two-dimensional viscoelasticy (Voigt model) via the NA method. The conservation laws derived should prove useful in studies of fracture and defects in a viscoelastic material.     

Non-classical laws with conserved quantities from the arbitrary functions included in general solution to elasticity and application

The general solution to static and/or dynamic linear elasticity is a transformation between the displacements and new arbitrary functions, whose conservativeness depends on some independent partial differential equations (PDEs) satisfied by the new arbitrary functions. Zhang's general solutions are mathematically appropriate since the displacements are expressed in terms of two new arbitrary functions, and the sum of the highest order derivative added together from the independent PDEs satisfied by the two new arbitrary functions is the same as that of Navier–Cauchy equations. Therefore, the following points should be emphasized: (i) the independent PDEs come from the Laplace and D'Alembert operators acting on the two new arbitrary functions in static and dynamic general solutions, respectively, and it is found that the two new arbitrary functions are related to the rotations, first strain invariant and distortion; (ii) especially, conservation laws constructed from the equations satisfied by the spatial integrals of functions hold true, although some arbitrary functions of the spatial integrals have been canceled. Based on these facts, since Noether's identity not only can be applied to a Lagrangian but also can be used to construct a functional for widespread PDEs, the functionals relating to the rotations, first strain invariant and distortion are constructed with arbitrary integer order spatial derivative or integral, and the conservation laws follow. This kind of non-classical conservation laws does not come from the Lagrangian density of an elastic body and belongs to the deep-level natures of symmetries of elastic field derived by standard techniques. Availability is shown by two examples, from which the field intensity of a vertical load applied to the surface of an elastic half-space and the path-independent integrals in a coordinate system moving with Galilean transformation are presented for comparison.     

Conservation laws for the relativistic P-system

Abstract

This note is devoted to the existence of rigorous asymptotic expansions for some boundary layer problems. We follow ideas of geometric optics and show that, generically, the study of such expansions is linked to the kernel and range of suitable projectors. We apply this remark to some classical geophysical systems, and recover in particular the results of (Grenier, E., Masmoudi, N. (1997). Ekman layers of rotating fluids, the case of well prepared initial data. Comm. Partial Differential Equations 22(5–6):953–975) with some improvements.     

Conservation laws for two-phase filtration models

The paper is devoted to investigation of group properties of a one-dimensional model of two-phase filtration in porous medium. Along with the general model, some of its particular cases widely used in oil-field development are discussed. The Buckley–Leverett model is considered in detail as a particular case of the one-dimensional filtration model. This model is constructed under the assumption that filtration is one-dimensional and horizontally directed, the porous medium is homogeneous and incompressible, the filtering fluids are also incompressible. The model of “chromatic fluid” filtration is also investigated. New conservation laws and particular solutions are constructed using symmetries and nonlinear self-adjointness of the system of equations.     

非线性对流扩散方程的守恒律

利用直接方法研究了非线性对流扩散方程的守恒律,得到了关于非线性对流扩散方程的守恒律乘子性质的一个定理.利用这个定理,可以简化守恒律乘子的确定方程.随后通过对确定方程中的变量函数进行分析,发现在四种情况下乘子的确定方程是可解的.最后解出这些守恒律乘子,利用积分公式法分别得到了四种情况下对应于各个守恒律乘子的守恒律.     

Conservation laws for nonlinear telegraph equations

A complete conservation law classification is given for nonlinear telegraph (NLT) systems with respect to multipliers that are functions of independent and dependent variables. It turns out that a very large class of NLT systems admits four nontrivial local conservation laws. The results of this work are summarized in tables which display all multipliers, fluxes and densities for the corresponding conservation laws. A physical example is considered for possible applications.     

Strange conserved quantities for the wave equation for elastic membranes

In this letter, we examine conservation laws for the vibrating membrane which arise from Noether's Theorem. We show that for linear membranes Noether's Theorem gives conservation laws which appear to contradict the basic assumptions on the model. We then show that the appropriate interpretation of these laws must follow from the nonlinear theory.     

Projection methods for stochastic differential equations with conserved quantities

In this paper, we consider the numerical methods preserving single or multiple conserved quantities, and these methods are able to reach high order of strong convergence simultaneously based on some kinds of projection methods. The mean-square convergence orders of these methods under certain conditions are given, which can reach order 1.5 or even 2 according to the supporting methods embedded in the projection step. Finally, three numerical experiments are taken into account to show the superiority of the projection methods.     

Conservation laws for a reaction-diffusion type system

For a system of two equations of the reaction-diffusion type in the space of three independent variables, the necessary and sufficient conditions are found for the existence of nontrivial first-order conservation laws. Some examples are given of the construction of conservation laws both for the abstract reaction-diffusion type systems and the available models: the FitzHugh-Nagumo model, the predator-prey model, the Brusselator model, and the chemical kinetic model.     

Conservation laws for a consistent plate theory

Within the framework of a consistent second-order plate theory, three material conservation laws are established, which are useful to describe the energy-release rates and the material forces connected with the change of configuration of inhomogeneities or defects within the plate. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Conservation laws for conformally invariant variational problems

We succeed in writing 2-dimensional conformally invariant non-linear elliptic PDE (harmonic map equation, prescribed mean curvature equations,..., etc.) in divergence form. These divergence-free quantities generalize to target manifolds without symmetries the well known conservation laws for weakly harmonic maps into homogeneous spaces. From this form we can recover, without the use of moving frame, all the classical regularity results known for 2-dimensional conformally invariant non-linear elliptic PDE (see [Hel]). It enables us also to establish new results. In particular we solve a conjecture by E. Heinz asserting that the solutions to the prescribed bounded mean curvature equation in arbitrary manifolds are continuous and we solve a conjecture by S. Hildebrandt [Hil1] claiming that critical points of continuously differentiable elliptic conformally invariant Lagrangian in two dimensions are continuous.     

Conservation laws for a modified lubrication equation

In this work we study the conservation laws of a modified lubrication equation, which describes the dynamics of the interfacial motion in phase transition. We show that the equation is nonlinear self-adjoint and has an exact Lagrangian with an auxiliary function. As a result, by a general theorem on conservation laws proved by Nail Ibragimov recently and Noether’s theorem, some new conservation laws for the equation are obtained. Our results show that the non-locally defined conservation laws generated by Noether’s theorem are equivalent to the local ones given by Ibragimov’s theorem.     

Conservation laws and their hyperbolic regularizations

The problems of the kinetics for hyperbolic regularizations of conservation laws are studied.