On the Cauchy Problem for the Hall and Electron Magnetohydrodynamic Equations Without Resistivity I: Illposedness Near Degenerate Stationary Solutions

Abstract

In this article, we prove various illposedness results for the Cauchy problem for the incompressible Hall- and electron-magnetohydrodynamic (MHD) equations without resistivity. These PDEs are fluid descriptions of plasmas, where the effect of collisions is neglected (no resistivity), while the motion of the electrons relative to the ions (Hall current term) is taken into account. The Hall current term endows the magnetic field equation with a quasilinear dispersive character, which is key to our mechanism for illposedness. Perhaps the most striking conclusion of this article is that the Cauchy problems for the Hall-MHD (either viscous or inviscid) and the electron-MHD equations, under one translational symmetry, are ill-posed near the trivial solution in any sufficiently high regularity Sobolev space $H^{s}$ and even in any Gevrey spaces. This result holds despite obvious wellposedness of the linearized equations near the trivial solution, as well as conservation of the nonlinear energy, by which the $L^{2}$ norm (energy) of the solution stays constant in time. The core illposedness (or instability) mechanism is degeneration of certain high frequency wave packet solutions to the linearization around a class of linearly degenerate stationary solutions of these equations, which are essentially dispersive equations with degenerate principal symbols. The method developed in this work is sharp and robust, in that we also prove nonlinear $H^{s}$-illposedness (for s arbitrarily high) in the presence of fractional dissipation of any order less than 1, matching the previously known wellposedness results. The results in this article are complemented by a companion work, where we provide geometric conditions on the initial magnetic field that ensure wellposedness(!) of the Cauchy problems for the incompressible Hall and electron-MHD equations. In particular, in stark contrast to the results here, it is shown in the companion work that the nonlinear Cauchy problems are well-posed near any nonzero constant magnetic field.

Appendix A: Existence of an \(L^2\)-Solution for the Linearized Systems

In this section, we give a sketch of the proof of existence of an $L^2$-solution for the linearized Hall-MHD and electron-MHD systems, which are recalled here for convenience. In the case of Hall-MHD ($\nu \ge 0$), we seek a solution $(u,b) \in C_w(I;L^2)$ satisfying

$$\begin{aligned} \left\{ \begin{aligned}&\partial _{t} u - \nu \Delta u = {\mathbb {P}}((\nabla \times \mathring{\mathbf{B}}) \times b + (\nabla \times b) \times \mathring{\mathbf{B}}) \\&\partial _{t} b + \nabla \times (u \times \mathring{\mathbf{B}}) + \nabla \times ((\nabla \times b) \times \mathring{\mathbf{B}}) + \nabla \times ((\nabla \times \mathring{\mathbf{B}}) \times b) = 0, \\&\nabla \cdot u = \nabla \cdot b = 0, \end{aligned} \right. \end{aligned}$$

(A. 1)

in the sense of distributions with the extra requirement $u \in L^2_t(I;{\dot{H}}^1)$ in the case of $\nu > 0$, and in the electron-MHD case, we simply need $b \in C_w(I;L^2)$ to satisfy

$$\begin{aligned} \left\{ \begin{aligned}&\partial _{t} b + \nabla \times ((\nabla \times b) \times \mathring{\mathbf{B}}) + \nabla \times ((\nabla \times \mathring{\mathbf{B}}) \times b) = 0, \\&\nabla \cdot b = 0. \end{aligned} \right. \end{aligned}$$

(A. 2)

Proposition A. 1

Let $M = ({\mathbb {T}}, {\mathbb {R}})_{x} \times ({\mathbb {T}}, {\mathbb {R}})_{y} \times {\mathbb {T}}_{z}$ and $\mathring{\mathbf{B}}$ be a sufficiently smooth stationary magnetic field. For any divergence-free initial data $(u_0,b_0) \in L^2(M)$, there exists a solution $(u,b) \in C_w([0,\infty );L^2)$ to (A. 1) with initial data $(u_0,b_0)$ satisfying

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\left( \Vert u(t)\Vert _{L^2(M)}^2 + \Vert b(t)\Vert _{L^2(M)}^2\right) + \nu \Vert u\Vert _{L^2([0,t];{\dot{H}}^1)}^2\\&\quad \le \frac{1}{2}\left( \Vert u_0\Vert _{L^2(M)}^2 + \Vert b_0\Vert _{L^2(M)}^2 \right) e^{ Ct\Vert \nabla \mathring{\mathbf{B}}\Vert _{C^1(M)}} \end{aligned} \end{aligned}$$

for all $t > 0$. In the case of (A. 2), there is a solution $b \in C_w([0,\infty );L^2)$ corresponding to any divergence-free data $b_0 \in L^2(M)$ satisfying

$$\begin{aligned} \begin{aligned} \frac{1}{2} \Vert b(t)\Vert _{L^2(M)}^2 \le \frac{1}{2} \Vert b_0\Vert _{L^2(M)}^2 e^{ Ct\Vert \nabla ^2\mathring{\mathbf{B}}\Vert _{L^\infty (M)}} \end{aligned} \end{aligned}$$

for all $t > 0$.

Proof

The proof is standard; see for instance [41, 42]. An alternative way is to mollify the equations as well as the data by truncating high frequencies while preserving the necessary structure for energy estimates, as it is done in [15]. We consider viscous regularizations of (A. 1) for $\epsilon > 0$, solve the regularized system

$$\begin{aligned} \left\{ \begin{aligned}&\partial _{t} u^{(\epsilon )} - \nu \Delta u^{(\epsilon )} = {\mathbb {P}}((\nabla \times \mathring{\mathbf{B}}) \times b^{(\epsilon )} + (\nabla \times b^{(\epsilon )}) \times \mathring{\mathbf{B}}) - \epsilon \Delta ^2 u^{(\epsilon )} \\&\partial _{t} b^{(\epsilon )} + \nabla \times (u^{(\epsilon )} \times \mathring{\mathbf{B}}) + \nabla \times ((\nabla \times b^{(\epsilon )}) \times \mathring{\mathbf{B}}) + \nabla \times ((\nabla \times \mathring{\mathbf{B}}) \times b^{(\epsilon )}) = -\epsilon \Delta ^2 b^{(\epsilon )}, \end{aligned} \right. \nonumber \\ \end{aligned}$$

(A. 3)

with the same initial data $(u_0,b_0)$, subject to divergence-free conditions. For each fixed $\epsilon > 0$, there is a unique global solution $(u^{(\epsilon )},b^{(\epsilon )})$ to (A. 3) with initial data $(u_0,b_0) \in L^2$, which is smooth once $t > 0$. The energy identity (1.4) with the extra term $\epsilon \Vert \Delta u^{(\epsilon )}\Vert _{L^2(M)}$ on the LHS can be justified for this solution. This shows that the sequence of solutions $\{ (u^{(\epsilon )},b^{(\epsilon )}) \}_{\epsilon >0}$ is uniformly bounded in $C_t(I;L^2)$ and $u^{(\epsilon )}$ is bounded uniformly in $L^2_t(I;{\dot{H}}^1)$ in the case of $\nu > 0$ for any fixed finite time interval $I = [0,T]$ with $T > 0$. In the same vein, the solution sequence is uniformly bounded in $\mathrm {Lip}_t(I;H^{-4})$. Applying the Aubin-Lions lemma (see [6, Theorem II.5.16] for a proof), we can extract a subsequence (still denoted by $\{(u^{(\epsilon )},b^{(\epsilon )}) \}$) which converges to some (u, b) in $C^0(I;H^{-s})$ for all $s < 0$. Since the space $L^\infty (I;L^2)$ is weak-* compact, we can guarantee that $(u,b) \in L^\infty (I;L^2)$ as well.

Clearly we have $(u,b)|_{t = 0} = (u_0,b_0)$, and the fact that (u, b) is a solution of (A. 1) and weakly continuous in time follows readily from strong convergence in $C^0(I;H^{-s})$.

The case of (A. 2) is only simpler and we omit the proof. $\square$

Remark A. 2

We observe that when the stationary magnetic field $\mathring{\mathbf{B}}$ and the initial data enjoy a set of symmetries respected by the Hall-MHD system (or electron-MHD system), the above proof actually guarantees existence of a solution satisfying the same set of symmetries as well.