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

Abstract

We give a construction of a divergence-free vector field u 0 ∈ H s ∩ B ∞ , ∞ − 1 u_0 \in H^s \cap B^{-1}_{\infty ,\infty } , for all s > 1 / 2 s>1/2 , with arbitrarily small norm ‖ u 0 ‖ B ∞ , ∞ − 1 \|u_0\|_{B^{-1}_{\infty ,\infty }} such that any Leray-Hopf solution to the Navier-Stokes equation starting from u 0 u_0 is discontinuous at t = 0 t=0 in the metric of B ∞ , ∞ − 1 B^{-1}_{\infty ,\infty } . For the Euler equation a similar result is proved in all Besov spaces B r , ∞ s B^s_{r,\infty } where s > 0 s>0 if r > 2 r>2 , and s > n ( 2 / r − 1 ) s>n(2/r-1) if 1 ≤ r ≤ 2 1 \leq r \leq 2 . This includes the space B 3 , ∞ 1 / 3 B^{1/3}_{3,\infty } , which is known to be critical for the energy conservation in ideal fluids.