Ill-posedness of the Navier–Stokes equations in a critical space in 3D

Abstract

We prove that the Cauchy problem for the three-dimensional Navier-Stokes equations is ill-posed iṅ B −1,∞ ∞ in the sense that a "norm inflation" happens in finite time. More precisely, we show that initial data in the Schwartz class S that are arbitrarily small inḂ −1,∞ ∞ can produce solutions arbitrarily large inḂ −1,∞ ∞ after an arbitrarily short time. Such a result implies that the solution map itself is discontinuous inḂ −1,∞ ∞ at the origin.

Introduction to Regularity Structures

“…It is in some sense the largest space of distributions that is invariant under the scaling ϕ(·) → λ −α ϕ(λ −1 ·), see for example [BP08].…”

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

“…We recall that the homogeneous spaceẊ =Ḃ −1 ∞,∞ is invariant with respect to the natural scaling of the equation in R 3 . Moreover it is the largest such space (see [4]).…”

Dispersive Effect of the Coriolis Force and the Local Well-Posedness for the Navier-Stokes Equations in the Rotational Framework

“…In this direction, Hieber and Shibata [15] and Konieczny and Yoneda [18] obatained the uniform global solvability of (NSC) in the Sobolev space H 1=2 ðR 3 Þ and the Fourier-Besov space _ FB FB 2À3=p p; y ðR 3 Þ with 1 < p c y, respectively. For the global well-posedness for (NSC) with W ¼ 0 in the scaling invariant spaces, we refer to Fujita and Kato [9], Kato [16], Kozono and Yamazaki [19], Koch and Tataru [17], Germain [10], Bourgain and Pavlović [5] and Yoneda [22]. For the local existence of solutions to (NSC), Giga, Inui, Mahalov and Matsui [12] proved the uniform local solvability of (NSC) for large initial velocity in FM 0 .…”