Nonuniqueness and Existence of Continuous, Globally Dissipative Euler Flows

We show that Hölder continuous incompressible Euler flows satisfy the local energy inequality (“globally dissipative” solutions) exhibit nonuniqueness and contain examples strictly dissipate kinetic energy. The collection of such solutions emanating from a fixed initial data may have positive Hausdorff dimension in space even if equality is imposed, set giving rise to an infinite family $$C^0$$ dense continuous, divergence free vector fields on torus $${{\mathbb {T}}}^3$$ . construction these involves new explicit convex integration approach mirroring Kraichnan’s LDIA theory turbulent cascades overcomes limitations previous schemes, which had been restricted bounded measurable or total