Local well-posedness for the two-and-a-half-dimensional EMHD system with split fractional dissipation

TL;DR

This paper establishes local well-posedness for a 2.5D EMHD system with split fractional dissipation on a torus, demonstrating that combined fractional dissipation controls the Hall nonlinearity.

Contribution

It proves local well-posedness for the 2.5D EMHD system with componentwise fractional dissipation, allowing for partial dissipation in each component.

Findings

Well-posedness holds for initial data in Sobolev spaces with s ≥ 2 - ε.

The combined fractional dissipation (α + β > 2) controls the nonlinear Hall effect.

The proof employs Littlewood--Paley energy estimates, commutator bounds, and cancellation techniques.

Abstract

We study the $2\frac12$-dimensional electron magnetohydrodynamics (EMHD) system on $\mathbb T^2$ with componentwise fractional dissipation: $\partial_t a+a_yb_x-a_xb_y=-\Lambda^\alpha a$ and $\partial_t b-a_y\Delta a_x+a_x\Delta a_y=-\Lambda^\beta b$, where $0<\alpha,\beta<2$. This system is a $2\frac12$-dimensional reduction of the magnetic equation in Hall--MHD/EMHD under the ansatz $B=\nabla\times(ae_z)+be_z$. We prove local well-posedness for initial data $(a_0,b_0)\in H^{s+1}(\mathbb T^2)\times H^s(\mathbb T^2)$ with $s\geq 2-\varepsilon$, provided that $\alpha+\beta>2$. Thus neither component is required to carry a full Laplacian dissipation; the smoothing effects of the two fractional dissipations can be combined to control the Hall nonlinearity. The proof is based on Littlewood--Paley energy estimates, commutator bounds, and cancellations between the leading low--high interactions.