Monotonicity formula

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

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

where $Q(t),R(t)$ are integrals of the fields at time t, and $R(t)$ is either always non-negative, or always non-positive (so that $Q(t)$ 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

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

Thus if we have uniform bounds for $Q$, we automatically obtain $L_{t}^{1}$ type bounds for $R$. 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.