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 Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We study a generalization due to De Gregorio and Wunsch et.al. of theConstantin-Lax-Majda equation (gCLM) on the real line \[ \omega_t + a u\omega_x = u_x \omega - \nu \Lambda^{\gamma} \omega, \quad u_x = H \omega , \]where $H$ is the Hilbert transform and $\Lambda = (-\partial_{xx})^{1/2}$. Weuse the method in \cite{chen2019finite} to prove finite time self-similarblowup for $a$ close to $\frac{1}{2}$ and $\gamma=2$ by establishing nonlinearstability of an approximate self-similar profile. For $a>-1$, we discussseveral classes of initial data and establish global well-posedness and anone-point blowup criterion for different initial data. For $a\leq-1$, we proveglobal well-posedness for gCLM with critical and supercritical dissipation.

Singularity Formation and Global Well-Posedness for the Generalized Constantin-Lax-Majda Equation with Dissipation