Monotonicity formula

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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 $L^1_t$ type bounds for $R$ . These type of spacetime integrablity bounds are particularly useful for obtaining scattering results.