Definition
Convex integration
A method using increasingly fine oscillations to realize a nonlinear differential constraint from relaxed data.
Convex integration is a family of construction methods for differential inclusions and related equations. One starts with a relaxed constraint, often involving a suitable convex hull, and introduces finer oscillations to approach the original nonlinear constraint. The proof must control errors, convergence, and passage to the limit in the chosen solution class; a formal oscillatory ansatz alone is not a solution.
For a weak formulation, the limit must satisfy every required test identity.
Fluid equations
Fluid constructions often reduce a stress defect through an iteration. The convergence topology must justify the quadratic momentum term as well as any claimed energy property. Convex integration can yield nonuniqueness, but the conclusion depends on the precise equation and class. Buckmaster and Vicol constructed nonunique finite energy weak solutions of three-dimensional periodic Navier–Stokes. Finite energy weak solutions in that result should not be identified with Leray–Hopf solutions satisfying the energy inequality.