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Ill-posedness of the basic equations of fluid dynamics in Besov spaces. (English) Zbl 1423.76085

Summary: We give a construction of a divergence-free vector field \(u_o\in H^s\cap B^{-1}_{\infty,\infty}\), for \(s<1/2\), 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 \(1\leq r\leq 2\). This includes the space \(B^{1/3}_{3,\infty}\), which is known to be critical for the energy conservation in ideal fluids.

MSC:

76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35Q31 Euler equations
35Q30 Navier-Stokes equations
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