Conservation law

Definition. @olver86 section 4.3.
Consider a system of DEs $Δ_{ν} (x, u^{(n)}) = 0$ . A conservation law is an expression

Div P = 0

which vanishes for all solutions $u = f (x)$ of the system. Here $P = (P_{1} (x, u^{(n)}), \dots, P_{p} (x, u^{(n)}))$ is a $p$ -tuple of smooth functions and $Div$ is the total divergence.
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Particular cases

System of ODEs

In the particular case of a system of ODEs, where we have only 1 independent variable and $u$ represents functions

u : x \mapsto (u^{1}, \dots, u^{q})

a conservation law is of the form $D_{x} P = 0$ for every solution of the system, that is, $P (x, u^{(n)})$ must be constant along solutions. So in this case a conservation law is nothing but a first integral of the system.

In a dynamical context

Suppose that our system of DE arise in a dynamical context, and that one of the independent variables is the time $t$ and the other $x_{1}, \dots, x_{p}$ are spatial variables. The conservation law is given by

D_{t} T + {Div}_{s p a c e} X = 0

The function $T$ is called the conserved density and $X = (X_{1}, \dots, X_{p})$ are called the associated flux. This is completely related to the note four-current . Indeed, an example of conservation law is the continuity equation.

From local to global

Integrating the conserved density over a spatial domain $Ω$ (we write the density as $ρ$ and the flux as $j$ , as in the continuity equation),

\frac{d}{d t} \int_{Ω} ρ d x = - \int_{\partial Ω} j \cdot n d S,

gives a global quantity that is constant whenever the flux through the boundary vanishes (or decays at infinity). In $1 + 1$ dimensions this reads $d Q / d t = - \int_{Ω} j_{x} d x = - j |_{\partial Ω}$ .

This is the connection with evolution equation: the conserved density is not constant in time — the flow of the equation redistributes it through the flux — but its integral is conserved under the appropriate boundary conditions. In other words, $Q$ is a conserved functional on the space of solutions: the analogue for PDEs of a first integral.

EDS formulation

If $ω$ is the differential form corresponding to the current, a conservation law is geometrically represented by

d ω \in I

(i.e. $d ω$ vanishes modulo the equation ideal, see exterior differential system). Compare this with the condition for a first integral:

$d F \in I$ $\Rightarrow$ $F$ is constant on the solution manifold $S$ ;
$d ω \in I$ $\Rightarrow$ $ω$ satisfies a local balance law on $S$ : the density changes locally only because the current flows through the boundary.

The density need not be constant at all. For ODEs the two conditions collapse into one, since there is only one independent variable; for PDEs they separate. See first integral vs conservation law.

Conservation laws are generated by Noether symmetries of a variational problem; see Noether's theorem.

Characteristic of a conservation law

Page 266 @olver86.