# Monotonicity formula

A monotonicity formula for an equation is a formula of the form

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

where ${\displaystyle Q(t),R(t)}$ are integrals of the fields at time t, and ${\displaystyle R(t)}$ is either always non-negative, or always non-positive (so that ${\displaystyle Q(t)}$ is monotone in time). Thus for instance every conservation law is also a (rather trivial) example of a monotonicity formula.

From the fundamental theorem of calculus we see that

${\displaystyle \int _{0}^{T}|R(t)|\ dt=|\int _{0}^{T}R(t)\ dt|=|Q(T)-Q(0)|.}$

Thus if we have uniform bounds for ${\displaystyle Q}$, we automatically obtain ${\displaystyle L_{t}^{1}}$ type bounds for ${\displaystyle R}$. These type of spacetime integrablity bounds are particularly useful for obtaining scattering results.

Common classes of monotonicity formulae in nonlinear dispersive and wave equations include Morawetz inequalities and virial identities.