Airy equation

The (homogeneous) Airy equation is given by

$u_{t}+u_{xxx}=0.$

It is the linear component of many equations of KdV type, including of course the Korteweg-de Vries equation itself. For applications to such nonlinear perturbations of the homogeneous Airy equation, it is often important to study the more general inhomogeneous Airy equation

$u_{t}+u_{xxx}=F$

for various forcing terms F. Of course, the inhomogeneous and homogeneous equations are related by Duhamel's formula.

A large number of linear, bilinear, trilinear, and multilinear estimates for this equation are known; see the page on Airy estimates for more details.