Flexibility of two-dimensional Euler flows with integrable vorticity

We propose a new convex integration scheme in fluid mechanics, and we provide an application to the two-dimensional Euler equations. We prove the flexibility and nonuniqueness of L∞L2 weak solutions with vorticity in L∞Lp for some p>1, surpassing for the first time the critical scaling of the standard convex integration technique. To achieve this, we introduce several new ideas, including: a new family of building blocks built from the Lamb–Chaplygin dipole; a new method to cancel the error based on time averages and nonperiodic, spatially anisotropic perturbations.