Non existence and strong ill-posedness in $C^k$ and Sobolev spaces for SQG

Diego Córdoba; Luis Martínez-Zoroa

Non existence and strong ill-posedness in $C^k$ and Sobolev spaces for SQG

Analysis of PDEs 2022-05-03 v3

Authors: Diego Córdoba , Luis Martínez-Zoroa

Abstract

We construct solutions in $\mathbb{R}^2$ with finite energy of the surface quasi-geostrophic equations (SQG) that initially are in $C^k$ ( $k\geq 2$ ) but that are not in $C^{k}$ for $t>0$ . We prove a similar result also for $H^{s}$ in the range $s\in(\frac32,2)$ . Moreover, we prove strong ill-posedness in the critical space $H^{2}$ .

Keywords

surface quasi-geostrophic equations non existence strong ill-posedness C^k spaces Sobolev spaces finite energy solutions

Cite

@article{arxiv.2107.07463,
  title  = {Non existence and strong ill-posedness in $C^k$ and Sobolev spaces for SQG},
  author = {Diego Córdoba and Luis Martínez-Zoroa},
  journal= {arXiv preprint arXiv:2107.07463},
  year   = {2022}
}

Comments

Some typos and minor errors fixed. Some extra explanations included

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