Transporting microstructure and dissipative Euler flows

Abstract

Recently the second and third author developed an iterative scheme for obtaining rough solutions of the 3D incompressible Euler equations in Hölder spaces (arXiv:1202.1751 and arXiv:1205.3626 (2012)). The motivation comes from Onsager's conjecture. The construction involves a superposition of weakly interacting perturbed Beltrami flows on infinitely many scales. An obstruction to better regularity arises from the errors in the linear transport of a fast periodic flow by a slow velocity field. In a recent paper P. Isett (arXiv:1211.4065) has improved upon our methods, introducing some novel ideas on how to deal with this obstruction, thereby reaching a better Hölder exponent - albeit below the one conjectured by Onsager. In this paper we give a shorter proof of Isett's final result, adhering more to the original scheme and introducing some new devices. More precisely we show that for any positive $\epsilon$ there exist periodic solutions of the 3D incompressible Euler equations which dissipate the total kinetic energy and belong to the Hölder class $C^{\frac{1}{5}-\epsilon }$.

35 pages; minor corrections relating to scales of parameters; results unchanged

References in corpus (2)

• Dissipative continuous Euler flows
• Cauchy problem for dissipative Hölder solutions to the incompressible Euler equations

Cited by in corpus (9)

• Global ill-posedness of the isentropic system of gas dynamics
• Ill-posedness results in critical spaces for some equations arising in hydrodynamics
• On Non-uniqueness of Hölder continuous globally dissipative Euler flows
• Fluids, Elasticity, Geometry, and the Existence of Wrinkled Solutions
• On Non-uniqueness of continuous entropy solutions to the isentropic compressible Euler equations
• Magnetic helicity and subsolutions in ideal MHD
• Weak solutions to the stationary incompressible Euler equations
• Analyticity of Lagrangian trajectories for well posed inviscid incompressible fluid models
• Non-conservative $H^{\frac{1}{2}-}$ weak solutions of the incompressible 3D Euler equations