# Conservation law

## 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:

${\displaystyle \partial _{t}Q(t)=0.}$

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

${\displaystyle M(t):=\int _{\mathbb {R} ^{d}}|u(t,x)|^{2}\ dx}$

the total momentum

${\displaystyle p(t):=\int _{\mathbb {R} ^{d}}\Im ({\overline {u(t,x)}}\nabla u(t,x))\ dx}$

and the total energy

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

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 ${\displaystyle \rho (t,x)}$ of the fields at or near ${\displaystyle (t,x)}$ which obeys the continuity equation

${\displaystyle \partial _{t}\rho (t,x)+\partial _{i}j_{i}(t,x)=0}$

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