Abstract
We consider the incompressible Navier-Stokes equation with a fractional power $\alpha\in[1,\infty)$ of the Laplacian in the three dimensional case. We prove the existence of a smooth solution with arbitrarily small in $\dot{B}_{\infty,p}^{-\alpha}$ ($2 0$ in arbitrarily small time. This extends the result of Bourgain and Pavlovic for the classical Navier-Stokes equation which utilizes the fact that the energy transfer to low modes increases norms with negative smoothness indexes. It is remarkable that the space $\dot{B}_{\infty,\infty}^{-\alpha}$ is supercritical for $\alpha >1$. Moreover, the norm inflation occurs even in the case $\alpha \geq 5/4$ where the global regularity is known.
Highlights
In this paper we study the three dimensional incompressible magneto-hydrodynamic (MHD) system with fractional powers of the Laplacian: ut + (u · ∇)u − (b · ∇)b + ∇p = −ν(− )α1 u, bt + (u · ∇)b − (b · ∇)u = −μ(− )α2 b, (1.1)
The goal of this paper is to find natural norm inflation spaces for the generalized MHD
We recall the definitions of norms for the homogeneous and non-homogeneous Besov spaces B∞−s,∞ and B∞−s,∞
Summary
Cheskidov and Dai [4] showed the norm inflation in subcritical spaces B∞−s,∞ for all s α1, α1 1 This provides a wide range of spaces where a small initial data result is not expected. Note that the natural space for the norm inflation B∞−α,∞1 is only scaling invariant in the classical case α1 = 1, and is subcritical for α1 > 1. This explains why small initial data results are only available for α1 < 1. The goal of this paper is to find natural norm inflation spaces for the generalized MHD system (1.1) and show that in general they are not scaling invariant, even in the classical case.
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