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== Global conservation laws ==

A global or integral '''conservation law''' for an evolution equation is any quantity ''Q(t)'' depending on the value of all the fields at time ''t''
which is (formally) constant in time:
<center><math>\partial_t Q(t) = 0.</math></center>
The '''conserved quantity''' Q(t) is typically an integral over space.  For instance, in [[NLS]], examples of conserved quantities include the total mass
<center><math>M(t) := \int_{\R^d} |u(t,x)|^2\ dx</math></center>
the total momentum
<center><math>p(t) := \int_{\R^d} \Im( \overline{u(t,x)} \nabla u(t,x) )\ dx</math></center>
and the total energy
<center><math>E(t) := \int_{\R^d} \frac{1}{2} |\nabla u(t,x)|^2 + \frac{1}{p+1} |u(t,x)|^{p+1}\ dx.</math></center>

[[Noether's theorem]] relates conserved quantities to symmetries of the underlying equation, in the case that the equation is [[Hamiltonian]] or Lagrangian.

== Local conservation laws ==

A local or pointwise '''conservation law''' for any equation is any local function <math>\rho(t,x)</math> of the fields at or near <math>(t,x)</math> which obeys the continuity equation
<center><math>\partial_t \rho(t,x) + \partial_i j_i(t,x) = 0</math></center>
for some other local functions <math>j_i(t,x)</math> of the fields near <math>(t,x)</math>.  Note from Stokes' theorem that this implies (in flat space at least) that the integral <math>Q(t) := \int_{\R^d} \rho(t,x)\ dx</math> is a global conserved quantity.  By modifying <math>\rho</math> using spatial or frequency cutoffs one can also create [[almost conserved]] quantities, [[virial identities]] and [[monotonicity formula]]e.

[[Category:Concept]]