Nonuniqueness and existence of continuous, globally dissipative Euler flows

2017/10/30 by Philip Isett, Isett, Philip · 3 citations
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.1710.11186

openalex publication_date 2017/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that Hölder continuous incompressible Euler flows that satisfy the local energy inequality ("globally dissipative" solutions) exhibit nonuniqueness and contain examples that strictly dissipate kinetic energy. The collection of such solutions emanating from a fixed initial data may have positive Hausdorff dimension in the energy space even if the local energy equality is imposed, and the set of initial data giving rise to such an infinite family of solutions is C⁰ dense in the space of continuous, divergence free vector fields on the torus \mathbb T³. The construction of these solutions involves a new and explicit convex integration approach mirroring Kraichnan's LDIA theory of turbulent energy cascades that overcomes the limitations of previous schemes, which had been restricted to bounded measurable solutions or to continuous solutions that dissipate total kinetic energy.

Cited by

• Dissipative solutions to randomly forced 3D Euler equations
• The Rayleigh-Taylor instability with local energy dissipation
• Global regularity for the Navier-Stokes equations with application to global solvability for the Euler equations

Related

• Dissipative continuous Euler flows in two and three dimensions
• Dissipative Euler Flows and Onsager's Conjecture
• The Rayleigh-Taylor instability with local energy dissipation
• Dissipative Euler flows with Onsager-critical spatial regularity
• The global Cauchy problem for compressible Euler equations with a nonlocal dissipation