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/

Global: Zendy Plus annual

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

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

For the 3D compressible isentropic Euler equations with an initialperturbation of size $\ve$ of a rest state, if the initial vorticity is of size$\dl$ with $0<\dl\le \ve$ and $\ve$ is small, we establish that the lifespan ofthe smooth solutions is $T_{\dl}=O(\min\{e^\frac{1}{\ve},\frac{1}{\delta}\})$for the polytropic gases, and $T_{\dl}=O(\frac{1}{\delta})$ for the Chaplygingases. For example, when $\dl=e^{-\f{1}{\ve^2}}$ is chosen, then$T_{\dl}=O(e^{\f{1}{\ve}})$ for the polytropic gases and$T_{\dl}=O(e^{\f{1}{\ve^2}})$ for the Chaplygin gases although theperturbations of the initial density and the divergence of the initial velocityare only of order $O(\ve)$. Our result illustrates that the time of existenceof smooth solutions depends crucially on the size of the vorticity of theinitial data, as long as the initial data is sufficiently close to a constant.The main ingredients in the paper are: introducing some suitably weightedenergies, deriving the pointwise space-time decay estimates of solutions,looking for the good unknown instead of the velocity, and establishing therequired weighted estimates on the vorticty.

Delayed singularity formation for the three dimensional compressible Euler equations with non-zero vorticity