Ill-posedness of basic equations of fluid dynamics in Besov spaces

Details Ill-posedness of basic equations of fluid dynamics in Besov spaces Ill-posedness of basic equations of fluid dynamics in Besov spaces

2009-04-14

arxiv.org/abs/0904.2196v1

Mathematics

Analysis of PDEs

9 pages

Scientific paper

We give a construction of a divergence-free vector field $u_0 \in H^s \cap B^{-1}_{\infty,\infty}$, for all $s<1/2$, such that any Leray-Hopf solution to the Navier-Stokes equation starting from $u_0$ is discontinuous at $t=0$ in the metric of $B^{-1}_{\infty,\infty}$. For the Euler equation a similar result is proved in all Besov spaces $B^s_{r,\infty}$ where $s>0$ if $r>2$, and $s>n(2/r-1)$ if $1 \leq r \leq 2$.

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