English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

A basic example of shear flow was introduced by DiPerna and Majda to studythe weak limit of oscillatory solutions of the Euler equations ofincompressible ideal fluids. In particular, they proved by means of thisexample that weak limit of solutions of Euler equations may, in some cases,fail to be a solution of Euler equations. We use this shear flow example toprovide non-generic, yet nontrivial, examples concerning the loss of smoothnessof solutions of the three-dimensional Euler equations, for initial data that donot belong to $C^{1,\alpha}$. Moreover, we show by means of this shear flowexample the existence of weak solutions for the three-dimensional Eulerequations with vorticity that is having a nontrivial density concentrated onnon-smooth surface. This is very different from what has been proven for thetwo-dimensional Kelvin-Helmholtz problem where a minimal regularity implies thereal analyticity of the interface. Eventually, we use this shear flow toprovide explicit examples of non-regular solutions of the three-dimensionalEuler equations that conserve the energy, an issue which is related to theOnsager conjecture.

Loss of smoothness and energy conserving rough weak solutions for the 3d Euler equations