# 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} }

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

### Ill-posedness of the Navier-Stokes and magneto-hydrodynamics systems

- 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…

### On the well-posedness of strong solution to ideal magnetohydrodynamic equations

- MathematicsInt. 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.

### Unique weak solutions of the non-resistive magnetohydrodynamic equations with fractional dissipation

- MathematicsCommunications 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.…

### Norm inflation for the Boussinesq system

- MathematicsDiscrete & 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}…

### Well/ill-posedness for the dissipative Navier–Stokes system in generalized Carleson measure spaces

- 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 , ∞ )…

### Regularity criteria for the 3D Navier-Stokes and MHD equations

- 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…

### Existence and regularity of solutions for a 3D coupled elliptic-parabolic equations related to magnetic relaxation

- MathematicsJournal of Mathematical Analysis and Applications
- 2022

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