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Sharp norm inflation for 3D Navier-Stokes equations in supercritical spaces
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abstract
We prove that the incompressible Navier-Stokes equations exhibit norm inflation in $\dot B^{s}_{p,q}(\mathbb{R}^3)$ with smooth, compactly supported initial data. Such norm inflation is shown in all supercritical $\dot B^{s}_{p,q} $ near the scaling-critical line $s = -1+ \frac{3}{p}$ except at $s=0$. The growth mechanism differs depending on the sign of the regularity index $s$: forward energy cascade driven by mixing for $s>0$ and backward energy cascade caused by un-mixing for $s<0$. The construction also demonstrates arbitrarily large, finite-time growth of the vorticity, the first of such examples for the Navier-Stokes equations.
Forward citations
Cited by 3 Pith papers
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Strong ill-posedness for the MHD system in the supercritical regime: inviscid and viscous
Arbitrarily small smooth data for 3D incompressible MHD produce norm inflation in every supercritical Sobolev space (e.g., H^s, 0<s<5/2, for ideal MHD), with the velocity staying bounded in the ideal case.
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Norm Inflation for Inviscid and Fully Dissipative Boussinesq Systems in Supercritical Spaces
Norm inflation is proven for 2D inviscid and fully dissipative Boussinesq systems in almost all supercritical Besov spaces, with the density, not the velocity, carrying the blow-up.
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Norm Inflation For The Critical SQG Equation
Critical SQG has H1 norm inflation from large smooth data and small-data norm inflation in supercritical W^{β,p} spaces.
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