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:

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

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

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

the total momentum

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

and the total energy

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

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