Conservation law

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A conservation law is a system of homogeneous first-order quasilinear hyperbolic equation that (in one space dimension) can be put into the form:[1]

 \mathbf y_t + \mathbf A(\mathbf y) \mathbf y_x = 0

where the dependent varible y is called the density of a conserved quantity, and A(y) is called the current jacobian, and the subscript notation for partial derivatives has been employed. The more general inhomogenous case:

 \mathbf y_t + \mathbf A(\mathbf y) \mathbf y_x = \mathbf s

is not a conservation equation but the general kind of balance equation describing a dissipative system. The dependent variable y is called a nonconserved quantity, and the inhomogeneous term s(y,x,t) is the a (quantity)-source, or dissipation. For example balance equations of this kind are the momentum and energy Navier-Stokes equations, or the entropy balance for a general isolated system.

In mathematical physics, conservation equations express a particular measurable property of an isolated physical system does not change as the system evolves. One particularly important result concerning conservation equations is Noether's theorem, which states that there is a one-to-one correspondence between each one of them and a differentiable symmetry in the system. For example, the conservation of energy follows from the time-invariance of physical systems, and the fact that physical systems behave the same regardless of how they are oriented in space gives rise to the conservation of angular momentum.

Differential forms[edit]

In the one-dimensional space a conservation equation is a first-order quasilinear hyperbolic equation that can be put into the advection form:

 y_t + a(y) y_x = 0

where the dependent variable y(x,t) is called the density of the conserved (scalar) quantity (c.q.(d.) = conserved quantity (density)) , and a(y) is called the current coefficient, usually corresponding to the partial derivative in the conserved quantity of a current density (c.d.) of the conserved quantity j(y):[1]

 a(y) = j_y (y)

In this case since the chain rule applies:

 j_x= j_y (y) y_x = a(y) y_x

the conservation equation can be put into the current density form:

 y_t + j_x (y)= 0

In a space with more than one dimension the former definition can be extended to an equation that can be put into the form:

 y_t + \mathbf a(y) \cdot \nabla y = 0

where the conserved quantity is y(r,t), \cdot denotes the scalar product, is the nabla operator, here indicating a gradient, and a(y) is a vector of current coefficients, analogously corresponding to the divergence of a vector c.d. associated to the c.q. j(y):

 y_t + \nabla \cdot \mathbf  j(y) = 0

This is the case for the continuity equation:

 \rho_t + \nabla \cdot (\rho \mathbf u) = 0

Here the conserved quantity is the mass, with density ρ(r,t) and current density ρu, identical to the momentum density, while u(r,t) is the flow velocity.

In the general case a conservation equation can be also a system of this kind of equations (a vector equation) in the form:[1]

 \mathbf y_t + \mathbf A(\mathbf y) \cdot \nabla \mathbf y = \mathbf 0

where y is called the conserved (vector) quantity, ∇ y is its gradient, 0 is the zero vector, and A(y) is called the Jacobian of the current density. In fact as in the former scalar case, also in the vector case A(y) usually corresponding to the Jacobian of a current density matrix J(y):

 \mathbf A( \mathbf y) = \mathbf J_{\mathbf y} (\mathbf y)

and the conservation equation can be put into the form:

 \mathbf y_t + \nabla \cdot \mathbf J (\mathbf y)= \mathbf 0

For example this the case for Euler equations (fluid dynamics). In the simple incompressible case they are:


\begin{align}
\nabla\cdot \bold u=0\\[1.2ex]
{\partial \bold u \over\partial t}+ \bold u \cdot
\nabla \bold u + \nabla s =\bold{0},
\end{align}

where:

It can be shown that the conserved (vector) quantity and the c.d. matrix for these equations are respectively:


{\bold y}=\begin{pmatrix}1 \\  \bold u \end{pmatrix}; \qquad
{\bold J}=\begin{pmatrix}\bold u\\ \bold u \otimes \bold u + s \bold I\end{pmatrix};\qquad

where \otimes denotes the tensor product.

Integral and weak forms[edit]

Conservation equations can be also expressed in integral form: the advantage of the latter is substantially that it requires less smoothness of the solution, which paves the way to weak form, extending the class of admissible solutions to include discontinuous solutions.[2] By integrating in any space-time domain the current density form in 1-D space:

 y_t + j_x (y)= 0

and by using Green's theorem, the integral form is:

 \int_{- \infty}^{\infty} y dx + \int_{0}^{\infty} j (y) dt = 0

In a similar fashion, for the scalar multidimensional space, the integral form is:

 \oint [y d^N r + j (y) dt] = 0

where the line integration is performed along the boundary of the domain, in an anticlock-wise manner.[2]

Moreover, by defining a test function φ(r,t) continuously differentiable both in time and space with compact support, the weak form can be obtained pivoting on the initial condition. In 1-D space it is:

 \int_{0}^{\infty} \int_{- \infty}^{\infty} \phi_t y + \phi_x j(y) dx dt = - \int_{-\infty}^{\infty} \phi(x,0) y(x,0) dt

Note that in the weak form all the partial derivatives of the density and current density have been passed on to the test function, which with the former hypothesis is sufficiently smooth to admit these derivatives.[2]

Exact laws[edit]

A partial listing of physical conservation equations due to symmetry that are said to be exact laws, or more precisely have never been [proven to be] violated:

Conservation Law Respective Noether symmetry invariance Number of dimensions
Conservation of mass-energy Time invariance Lorentz invariance symmetry 1 translation about time axis
Conservation of linear momentum Galilean invariance 3 translation about x,y,z position
Conservation of angular momentum Rotation invariance 3 rotation about x,y,z axes
CPT symmetry (combining charge, parity and time conjugation) Lorentz invariance 1+1+1 (charge inversion q→-q) + (position inversion r→-r) + (time inversion t→-t)
Conservation of electric charge Gauge invariance 1⊗4 scalar field (1D) in 4D spacetime (x,y,z + time evolution)
Conservation of color charge SU(3) Gauge invariance 3 r,g,b
Conservation of weak isospin SU(2)L Gauge invariance 1 weak charge
Conservation of probability Probability invariance 1⊗4 total probability always=1 in whole x,y,z space, during time evolution

Approximate laws[edit]

There are also approximate conservation laws. These are approximately true in particular situations, such as low speeds, short time scales, or certain interactions.

See also[edit]

Examples and applications[edit]

Notes[edit]

  1. ^ a b c see Toro, p.43
  2. ^ a b c see Toro, p.62-63

References[edit]

  • Philipson, Schuster, Modeling by Nonlinear Differential Equations: Dissipative and Conservative Processes, World Scientific Publishing Company 2009.
  • Victor J. Stenger, 2000. Timeless Reality: Symmetry, Simplicity, and Multiple Universes. Buffalo NY: Prometheus Books. Chpt. 12 is a gentle introduction to symmetry, invariance, and conservation laws.
  • Toro, E.F. (1999). "Chapter 2. Notions on Hyperbolic PDEs". Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer-Verlag. ISBN 3-540-65966-8. 
  • E. Godlewski and P.A. Raviart, Hyperbolic systems of conservation laws, Ellipses, 1991.

External links[edit]