Conservation law

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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:

\partial_t Q(t) = 0.

The conserved quantity Q(t) is typically an integral over space. For instance, in NLS, examples of conserved quantities include the total mass

M(t) := \int_{\R^d} |u(t,x)|^2\ dx

the total momentum

p(t) := \int_{\R^d} \Im( \overline{u(t,x)} \nabla u(t,x) )\ dx

and the total energy

E(t) := \int_{\R^d} \frac{1}{2} |\nabla u(t,x)|^2 + \frac{1}{p+1} |u(t,x)|^{p+1}\ dx.

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 ρ(t,x) of the fields at or near (t,x) which obeys the continuity equation

\partial_t \rho(t,x) + \partial_i j_i(t,x) = 0

for some other local functions ji(t,x) of the fields near (t,x). Note from Stokes' theorem that this implies (in flat space at least) that the integral Q(t) := \int_{\R^d} \rho(t,x)\ dx is a global conserved quantity. By modifying ρ using spatial or frequency cutoffs one can also create almost conserved quantities, virial identities and monotonicity formulae.

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