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

We study the $2\frac{1}{2}$-dimensional electron magnetohydrodynamics (EMHD) system on ${\mathbb{T}}^2$ with componentwise fractional dissipation: ${\mathrm{\partial }}_{t}a+a_y{b}_x-a_x{b}_y=-{\mathrm{\Lambda }}^{\alpha }a$ and ${\mathrm{\partial }}_{t}b-a_y\mathrm{\Delta }{a}_x+a_x\mathrm{\Delta }{a}_y=-{\mathrm{\Lambda }}^{\beta }b$, where $0<\alpha ,\beta <2$. This system is a $2\frac{1}{2}$-dimensional reduction of the magnetic equation in Hall–MHD/EMHD under the ansatz $B=\mathrm{\nabla }\times (a{e}_z)+b{e}_z$. 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 $\alpha +\beta >2$. 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.