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Ill-posedness of the Navier-Stokes equations in a critical space in 3D. (English) Zbl 1161.35037
Summary: We prove that the Cauchy problem for the three-dimensional Navier-Stokes equations is ill-posed in \({\dot B}^{-1,\infty}_{\infty}\) in the sense that a “norm inflation” happens in finite time. More precisely, we show that initial data in the Schwartz class \(\mathcal S\) that are arbitrarily small in \({\dot B}^{-1,\infty}_{\infty}\) can produce solutions arbitrarily large in \({\dot B}^{-1,\infty}_{\infty}\) after an arbitrarily short time. Such a result implies that the solution map itself is discontinuous in \({\dot B}^{-1,\infty}_{\infty}\) at the origin.

MSC:
35Q30 Navier-Stokes equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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