Non-unique solutions for electron MHD
Abstract
We consider the electron magnetohydrodynamics (MHD) equation on the 3D torus $\mathbb T^3$. For a given smooth vector field $H$ with zero mean and zero divergence, we can construct a weak solution $B$ to the electron MHD in the space $L^\gamma_tW^{1,p}_x$ for appropriate $(\gamma, p)$ such that $B$ is arbitrarily close to $H$ in this space. The parameters $\gamma$ and $p$ depend on the resistivity. As a consequence, non-uniqueness of weak solutions is obtained for the electron MHD with hyper-resistivity. In particular, non-Leray-Hopf solutions can be constructed. As a byproduct, we also show the existence of weak solutions to the electron MHD without resistivity.
- Publication:
-
arXiv e-prints
- Pub Date:
- May 2024
- DOI:
- 10.48550/arXiv.2405.14127
- arXiv:
- arXiv:2405.14127
- Bibcode:
- 2024arXiv240514127D
- Keywords:
-
- Mathematics - Analysis of PDEs;
- 35Q35;
- 76B03;
- 76D09;
- 76E25;
- 76W05
- E-Print:
- 26 pages