Monotonicity formula

From DispersiveWiki


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

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

Thus if we have uniform bounds for , we automatically obtain type bounds for . 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.