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

Abstract

We study the $2\frac{1}{2}$-dimensional electron magnetohydrodynamics (EMHD) system on ${\mathbb{T}}^2$ with componentwise fractional dissipation: ${\partial }_{t}a+a_y{b}_x-a_x{b}_y=-\hat{I}{›}^{\hat{I}}\pm a$ and ${\partial }_{t}b-a_y\hat{I}”a_x+a_x\hat{I}”a_y=-\hat{I}{›}^{2}b$, where $0<\hat{I}\pm ,{\hat{I}}^2<2$. This system is a $2\frac{1}{2}$-dimensional reduction of the magnetic equation in Hall--MHD/EMHD under the ansatz $B=\nabla \times (a{e}_z)+b{e}_z$. 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 $s\ge 2-\epsilon$, provided that $\hat{I}\pm +{\hat{I}}^2>2$. 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.