Motivated by an equation arising in magnetohydrodynamics, we address the well-posedness theroy for the non-diffusive magneto-geostrophic equation. Namely, an active scalar equation in which the divergence-free drift velocity is one derivative more singular than the active scalar. In Friedlander and Vicol (Nonlinearity 24(11)::3019–3042, 2011), the authors prove that the non-diffusive equation is ill-posed in the sense of Hadamard in Sobolev spaces, but locally well posed in spaces of analytic functions. Here, we give an example of a steady state that is nonlinearly stable for periodic perturbations with initial data localized in frequency straight lines crossing the origin. For such well-prepared data, the local existence and uniqueness of solutions can be obtained in Sobolev spaces and the global existence holds under a size condition over the \(H^{5/2^{+}}({{\mathbb {T}}}^3)\) norm of the perturbation.

根据磁流体动力学中产生的一个方程，我们解决了非扩散的磁地磁方程的正定性理论。即，其中无散度漂移速度比有源标量更奇异的一个导数的有源标量方程。在Friedlander和Vicol（非线性24（11）:: 3019–3042，2011）中，作者证明了在Sobolev空间中非扩散方程在Hadamard的意义上是不适的，但在解析函数的空间中是局部良好的。在这里，我们给出一个稳定的例子，该例子对于周期性扰动是非线性稳定的，其初始数据位于与原点相交的频率直线上。对于这样准备好的数据，可以在Sobolev空间中获得解的局部存在性和唯一性，并且在存在一定大小条件的情况下全局存在性成立。\（H ^ {5/2 ^ {+}}（{{\ mathbb {T}}} ^ 3）\）扰动的范数。