分野別セミナー

We consider the quasi-geostrophic equation with its principal part
$(-\Delta)^{\alpha}$ for $\alpha >0$ in ${\mathbb R}^n$ with $n \ge 2$.
We show that for every initial data $\theta_0 \in \dot B^{1-2\alpha + \frac{n}{r}}_{r, q}$
with $1< r < \infty$ and $1 \le q \le \infty$, there exists a unique solution
$\theta$ in the class of maximal Lorentz-Besov regularity theorem
$\partial_t\theta, (-\Delta)^\alpha \theta \in L^{\gamma, q}(0, T; \dot B^s_{p, 1})$
for $2\alpha/\gamma + n/p -s =4\alpha -1$ with $n/p \le n/r < 2\alpha/\gamma + n/p$ and
$s>\max\{-1, 1-4\alpha + n/r\}$, where $0 < T \le \infty$.
If $\theta_0$ is sufficiently small,then we may take $T=\infty$.
Notice that both classes of initial data and solutions are scaling invariant.
This is the joint work with Peer C. Kunstmann(Karlsruhe) and Senjo Shimizu(Kyoto).

概要のpdfファイル版は下記でご覧いただけます．
https://archive.iii.kyushu-u.ac.jp/public/jOJCwpaIrQ5UQxwkCLD3wopd5BtdqifngmApyKTYp04i