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

By Jean Bourgain, Natavsa Pavlovi'c. 2008

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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.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5821cc1e98e9f94d91355ce0d8201bb5c3bc43ce

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In Applied Mathematics

In Control and Systems Engineering

In Mathematical Physics

Citations in the corpus

Convex integration and phenomenologies in turbulence (2019, background): “1, if one considers the family of Banach spaces CtH x , then α∗ = αWP = 1/2 [84, 109, 113, 127, 11].”
The Incompressible Navier–Stokes Equations on Non-compact Manifolds (2014, background): “The largest critical space is Ḃ−1 ∞,∞, but the Cauchy problem is showed to be ill-posed by [7].”
An alternative approach to regularity for the Navier–Stokes equations in critical spaces (2009, neutral): “well-posedness in Ḃ −1+ 3 p p,∞ and BMO−1, and the recent ill-posedness result [4] for Ḃ−1 ∞,∞.”
Global regularity for some classes of large solutions to the Navier-Stokes equations (2008, background): “thus seems that the Koch-Tataru theorem is optimal for the wellposedness of the Navier-Stokes equations — note that it was very recently proved (see [14] and [1]) that the equations are in fact ill-posed in Ḃ−1 ∞,∞.”
The Navier-Stokes equations and weak Herz spaces (2011, background): “We remark that Bourgain and Pavlović [5] proved the ill-posedness of the NavierStokes equations in Ḃ−1∞,∞.”
Well‐posedness for fractional Navier–Stokes equations in the largest critical spaces Ḃ∞,∞−(2β−1)(Rn) (2012, background): “Later, Bourgain and Pavlović studied in [17] the Navier–Stokes equations itself and constructed a class of initial values whose P B 1 1,1.”