Incompressible Euler equations

The incompressible Euler equations are

\partial_tu+(u\cdot\nabla)u+\nabla p=f, \qquad\nabla\cdot u=0.

They model constant-density inviscid motion: the momentum equation contains no viscous stress. Initial data, the spatial domain, boundary conditions, and the solution class are additional parts of an Euler problem.

Relation to viscosity

Formally setting $\nu=0$ in the Navier–Stokes equations gives this system. Justifying convergence of viscous solutions as $\nu\downarrow0$ is a separate limiting problem, particularly near boundaries.

Terminology

“Euler equations” can also refer to compressible fluid systems. The divergence-free, constant-density version is the one specified here.

References

Lenya Ryzhik, Lecture Notes for Math 256B (2024), Sections 1–3.

Relaxed formulations

Euler–Reynolds equations track a tensor defect. An Euler subsolution expresses a pointwise energy relaxation used in convex integration.