# Norm inflation for generalized magneto-hydrodynamic system

@article{Cheskidov2014NormIF,
title={Norm inflation for generalized magneto-hydrodynamic system},
author={Alexey Cheskidov and Mimi Dai},
journal={Nonlinearity},
year={2014},
volume={28},
pages={129 - 142}
}
• Published 8 February 2014
• Mathematics
• Nonlinearity
We consider the incompressible magneto-hydrodynamic system with fractional powers of the Laplacian in the three-dimensional case. We discover a wide range of spaces where the norm inflation occurs and hence small initial data results are out of reach. The norm inflation occurs not only in scaling invariant (critical) spaces, but also in supercritical and, surprisingly, subcritical ones.
8 Citations
• Mathematics
• 2015
We demonstrate that the three dimensional incompressible magneto-hydrodynamics (MHD) system is ill-posed due to the discontinuity of weak solutions in a wide range of spaces. Specifically, we
• Mathematics
Int. J. Comput. Math.
• 2017
It is proved that the strong solution to the MHD equations is unique and depends continuously on the initial data in the spaces and the existence of the strong solutions is obtained by Galerkin method.
• Mathematics
Communications in Mathematical Sciences
• 2020
This paper examines the uniqueness of weak solutions to the d-dimensional magnetohydrodynamic (MHD) equations with the fractional dissipation $(-\Delta)^\alpha u$ and without the magnetic diffusion.
• Mathematics
Discrete & Continuous Dynamical Systems - B
• 2020
We prove the norm inflation phenomena for the Boussinesq system on $\mathbb T^3$. For arbitrarily small initial data $(u_0,\rho_0)$ in the negative-order Besov spaces $\dot{B}^{-1}_{\infty, \infty} • Mathematics • 2017 Abstract As an essential extension of the well known case β ∈ ( 1 2 , 1 ] {\beta\kern-1.0pt\in\kern-1.0pt({\frac{1}{2}},1]} to the hyper-dissipative case β ∈ ( 1 , ∞ ) • Mathematics • 2015 We prove that a solution to the 3D Navier-Stokes or MHD equations does not blow up at$t=T$provided$\displaystyle \limsup_{q \to \infty} \int_{\mathcal{T}_q}^T \|\Delta_q(\nabla \times u)\|_\infty

## References

SHOWING 1-10 OF 18 REFERENCES

• Economics
• 2011
Based on the construction of Bourgain and Pavlović [1] we show that the solutions to the Cauchy problem for the three-dimensional incompressible magneto-hydrodynamics (MHD) system can develop
This paper derives regularity criteria for the generalized magnetohydrodynamics (MHD) equations, a system of equations resulting from replacing the Laplacian −Δ in the usual MHD equations by a
It remains unknown whether or not smooth solutions of the 3D incompressible MHD equations can develop finite-time singularities. One major difficulty is due to the fact that the dissipation given by
• Mathematics
• 2009
We give a construction of a divergence-free vector field uo ∈ H s ∩ B -1 ∞,∞ for all s 0 if r > 2, and s > n(2/r - 1) if 1 ≤ r ≤ 2. This includes the space B 1/3 3,∞ , which is known to be critical
• Mathematics
• 2012
We study the incompressible Navier-Stokes equations with a fractional Laplacian and prove the existence of discontinuous Leray-Hopf solutions in the largest critical space with arbitrarily small
• Mathematics
• 2012
This note studies the well‐posedness of the fractional Navier–Stokes equations in some supercritical Besov spaces as well as in the largest critical spaces Ḃ∞,∞−(2β−1)(Rn) for β ∈ (1/2,1). Meanwhile,
• Mathematics, Computer Science
• 2012
The existence of a smooth solution with arbitrarily small in $\dot{B}_{\infty,p}^{-\alpha}$ ($2 0$ in arbitrarily small time) is proved.