Differential inclusion

For $u:\Omega\subset\mathbb R^n\to\mathbb R^m$ , a first-order differential inclusion has the form

Du(x)\in K(x,u(x)),

where $Du$ is the derivative and $K(x,z)$ is a specified subset of the space of $m$ -by- $n$ matrices. The domain $\Omega$ is open. The required regularity of $u$ and whether inclusion holds everywhere or almost everywhere are part of the problem.

Linear equations plus a pointwise constraint

In fluid dynamics one often uses the same terminology for a linear differential system on enlarged variables together with a nonlinear pointwise constraint. For Euler, $\partial_t v+\operatorname{div}S+\nabla q=0$ , $\operatorname{div}v=0$ , and $S=v\otimes v-|v|^2I/n$ give such a formulation.

References

De Lellis and Székelyhidi, The Euler equations as a differential inclusion (2009).