Lagrangian Approach to Evolution Equations: Symmetries and Conservation Laws

We show that one can apply a Lagrangian approach to certain evolution equations by considering them together with their associated equations. Consequently, one can employ Noether's theorem and derive conservation laws from symmetries of coupled systems of evolution equations. We discuss in detail the linear and non-linear heat equations as well as the Burgers equation and obtain new non-local conservation laws for the non-linear heat and the Burgers equations by extending their symmetries to the associated equations. We also provide Lagrangians for non-linear Schrödinger and Korteweg—de Vries type systems.     

Potential Symmetries and Associated Conservation Laws with Application to Wave Equations

It has been shown that one can generate a class of nontrivial conservation laws for second-order partial differential equations using some recent results dealing with the action of any Lie–Bäcklund symmetry generator of the equivalentfirst-order system on the respective conservation law. These conservedvectors are nonlocal as they are constructed from associatednonlocal symmetries of the partial differential equation. The method canbe successfully extended to association with genuine nonlocal(potential) symmetries. However, it usually involves solving moredifficult systems of partial differential equations which may not alwaysbe easy to uncouple.     

On Dual Conservation Laws in Linear Elasticity: Stress Function Formalism

Dual conservation laws of linear planar elasticity theory have been systematically studied based on stress function formalism. By employing generalized symmetry transformation or the Lie—Bäcklund transformation, a class of new dual conservation laws in planar elasticity have been discovered based on the Noether theorem and its Bessel—Hagen generalization. The physical implications of these dual conservation laws are discussed briefly.     

Lie Point Symmetries,Conservation and Balance Laws in Linear Gradient Elastodynamics

The aim of this work is the derivation of Lie point symmetries, conservation and balance laws in linear gradient elastodynamics of grade-2 (up to second gradients of the displacement vector and the first gradient of the velocity). The conservation and balance laws of translational, rotational, scaling variational symmetries and addition of solutions are derived using Noether’s theorem. It turns out that the scaling symmetry is not a strict variational symmetry in gradient elasticity.      

New Conservation Laws of Energy and C-D Inequalities in Continua Without Microstructure

ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｔｉｎｕｕｍｍｅｃｈａｎｉｃｓｉｓｎｏｔｏｎｌｙａｎｏｌｄａｎｄｂｕｔａｌｓｏａｙｏｕｎｇｓｃｉｅｎｔｉｆｉｃｄｉｓｃｉｐｌｉｎｅ.Ｉｔｃｏｎｓｉｓｔｓｏｆｓｏｍｅｆｕｎｄａｍｅｎｔａｌｌａｗｓ,ｗｈｉｃｈａｒｅｖａｌｉｄｆｏｒａｌｌｂｏｄｉｅｓｉｒｒｅｓｐｅｃｔｉｖｅｏｆｔｈｅｉｒｓｈａｐｅｓ,ｃｏｎｓｔｉｔｕｔｉｏｎｓａｎｄｃｏｎｓｔｉｔｕｔｉｖｅｒｅｌａｔｉｏｎｓ,ｗｈｉｃｈｍｕｓｔｒｅｆｌｅｃｔｔｈｅｎａｔｕｒｅｏｆｔｈｅｍａｔｅｒｉａｌａｎｄｔｈｅｃｏｎｓｔｉ…     

The L1-Stability of Boundary Layers for Scalar Viscous Conservation Laws

Let v=v(x) be a non-trivial bounded steady solution of a viscous scalar conservation law u _t+f(u) _x =u _xx on a half-line R⁺, with a Dirichlet boundary condition. The semi-group of this IBVP is known to be contractive for the distance d(u, u)u–u₁ induced by L ¹(R⁺). We prove here that v is asymptotically stable with respect to d: if u ₀–vL ¹, then u(t)–v₁0 as t+. When v is a constant, we show that this property holds if and only if f(v)0. These results complement our study of the Cauchy problem [2].     

求解多维双曲守恒律方程组的四阶半离散格式

提出了求解多维双曲守恒律方程组的四阶半离散格式。该方法以中心加权基本无振荡(CWENO)重构为基础,同时考虑到在R iemann扇内波传播的局部速度,从而回避了计算过程中的网格交错,建立了数值耗散较小的介于迎风格式和中心格式之间的半离散格式。本文的四阶半离散格式是Kurganov等人的三阶半离散格式的高阶推广。大量的数值算例充分说明了本文方法的高分辨率和稳定性。     

Lie–Bäcklund and Noether Symmetries with Applications

New identities relating the Euler–Lagrange, Lie–Bäcklund and Noether operators are obtained. Some important results are shown to be consequences of these fundamental identities. Furthermore, we generalise an interesting example presented by Noether in her celebrated paper and prove that any Noether symmetry is equivalent to a strict Noether symmetry, i.e. a Noether symmetry with zero divergence. We then use the symmetry based results deduced from the new identities to construct Lagrangians for partial differential equations. In particular, we show how the knowledge of a symmetry and its corresponding conservation law of a given partial differential equation can be utilised to construct a Lagrangian for the equation. Several examples are given.     

New Conservation Laws of Energy and C-D Inequalities in Continua with Microstructure

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｏｕｒｐｒｅｃｅｅｄｉｎｇｐａｐｅｒ[1 ]ｔｈｅｆｕｎｄａｍｅｎｔａｌｌａｗｓ,ｂａｌａｎｃｅｅｑｕａｔｉｏｎｓａｎｄＣ＿Ｄｉｎｅｑｕａｌｉｔｉｅｓｉｎｃｏｎｔｉｎｕａｗｉｔｈｏｕｔｍｉｃｒｏｓｔｒｕｃｔｕｒｅａｒｅｓｙｓｔｅｍａｔｉｃａｌｌｙｒｅｓｔｕｄｉｅｄ ,ａｎｄｔｈｅｎｅｗｃｏｎｓｅｒｖａｔｉｏｎｌａｗｓａｎｄｔｈｅｒｅｌａｔｅｄＣ＿Ｄｉｎｅｑｕａｌｉｔｉｅｓａｒｅｅｓｔａｂｌｉｓｈｅｄ .Ｉｎｔｈｉｓｐａｐｅｒｓｏｍｅｒｅｍａｒｋｓｏｎｔｈｅｒｅｓｕｌｔｓｉｎｃ…     

On group properties and conservation laws for second-order quasi-linear differential equations

A sufficient condition for the absence of tangent transformations admitted by second-order quasi-linear differential equations and a sufficient condition for linear autonomy of operators of the Lie group of transformations admitted by second-order weakly nonlinear differential equations are found. A theorem on the structure of the first-order conservation laws for second-order weakly nonlinear differential equations is proved. A classification of second-order linear differential equations with two independent variables in terms of first-order conservation laws is proposed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 3, pp. 64–70, May–June, 2009.     

Flux and source term discretization in two‐dimensional shallow water models with porosity on unstructured grids

Two‐dimensional shallow water models with porosity appear as an interesting path for the large‐scale modelling of floodplains with urbanized areas. The porosity accounts for the reduction in storage and in the exchange sections due to the presence of buildings and other structures in the floodplain. The introduction of a porosity into the two‐dimensional shallow water equations leads to modified expressions for the fluxes and source terms. An extra source term appears in the momentum equation. This paper presents a discretization of the modified fluxes using a modified HLL Riemann solver on unstructured grids. The source term arising from the gradients in the topography and in the porosity is treated in an upwind fashion so as to enhance the stability of the solution. The Riemann solver is tested against new analytical solutions with variable porosity. A new formulation is proposed for the macroscopic head loss in urban areas. An application example is presented, where the large scale model with porosity is compared to a refined flow model containing obstacles that represent a schematic urban area. The quality of the results illustrates the potential usefulness of porosity‐based shallow water models for large scale floodplain simulations. Copyright © 2005 John Wiley & Sons, Ltd.