On compressible magnetic relaxation in planar symmetry

1. Problem Statement

The paper investigates the Compressible Magnetic Relaxation Equations (MRE) on the three-dimensional torus $\mathbb{T}^3$. The MRE system is derived from compressible Magnetohydrodynamics (MHD) by replacing the inertial acceleration term $(\partial_t + u \cdot \nabla)u$ with a Darcy-type friction term ($-\nabla p + (\nabla \times B) \times B = \rho u$).

The primary mathematical challenges addressed are:

1. Local Well-posedness: Proving the existence and uniqueness of smooth solutions for the compressible system, which lacks explicit diffusion and preserves magnetic topology.
2. Magnetic Relaxation: Determining whether the system converges to a steady state (specifically, a magnetohydrostatic equilibrium where velocity $u \to 0$) as $t \to \infty$.
3. Vacuum Avoidance: Establishing that the density $\rho$ does not vanish (no vacuum states) or implode prior to any potential singularity, which is a critical hurdle in compressible fluid dynamics.

The analysis is conducted under planar symmetry, reducing the 3D system to a 1D system on the torus $\mathbb{T}$.

2. Mathematical Formulation

Under the planar symmetry ansatz:
$$B = (B_0, 0, B(t, x)), \quad u = (u_x, u_y, u_z), \quad \rho = \rho(t, x)$$
where $B_0 \neq 0$ is a constant background magnetic field, the system (2) reduces to the following coupled parabolic-hyperbolic system (Equation 4):
$$\begin{cases} \partial_t \rho = \partial_x^2 \left( \frac{1}{\gamma} \rho^\gamma + \frac{1}{2}B^2 \right), \\ \partial_t B = \partial_x \left( \frac{B}{\rho} \partial_x \left( \frac{1}{\gamma} \rho^\gamma + \frac{1}{2}B^2 \right) + \frac{B_0^2}{\rho} \partial_x B \right). \end{cases}$$
Here, $\gamma > 1$ is the adiabatic constant. The system features mixed diffusion: the density equation diffuses based on pressure and magnetic energy, while the magnetic field equation diffuses based on density gradients and magnetic gradients. This coupling makes standard maximum principles difficult to apply directly.

3. Methodology

A. Local Well-posedness and Vacuum Avoidance (Theorem 1)

The core difficulty is that at the point where density $\rho$ is minimized, the sign of the second derivative of the magnetic field $\partial_x^2 B$ is unknown, preventing direct application of the maximum principle to prevent vacuum formation ($\rho \to 0$).

Key Innovation: Diagonalization via New Variables
To overcome the mixed diffusion, the author constructs a diagonalizing variable $F(\rho, B)$ such that the evolution equation for $F$ takes the form of a standard heat equation with a positive diffusion coefficient:
$$\partial_t F = c(\rho, B) \partial_x^2 F + \text{lower order terms}.$$
The author derives two specific variables, $W(\rho, B)$ and $Z(\rho, B)$, defined via a complex change of variables involving integrals of a function $f(\rho, B)$ (derived from the characteristic equation of the system).

• $W$ and $Z$ satisfy parabolic inequalities of the form $\partial_t W - \alpha \partial_x^2 W \leq 0$ and $\partial_t Z - \beta \partial_x^2 Z \leq 0$ with positive coefficients $\alpha, \beta$.
• By applying the Maximum Principle to $W$ and $Z$, the author proves that these variables remain bounded.
• Crucially, the level sets of $W$ and $Z$ are shown to be bounded away from $\rho=0$ and $\rho, B \to \infty$. This establishes that $\rho$ remains strictly positive and bounded, and $B$ remains bounded, preventing vacuum formation and implosion.

Existence and Uniqueness:
Using these a priori estimates, the author employs a parabolic regularization argument (adding $-\epsilon \partial_x^4$ terms) to construct smooth solutions. The uniform bounds on $\rho$ and $B$ allow passing to the limit $\epsilon \to 0$, proving local well-posedness for smooth initial data. Uniqueness is proven via an energy estimate on the difference of two solutions.

B. Magnetic Relaxation (Theorem 2)

The second part of the paper addresses the long-time behavior of solutions near constant steady states $(\rho, B) = (\bar{\rho}, \bar{B})$.

Key Innovation: Coupled Energy Decay
For the relaxation proof, the author linearizes the system and identifies the correct coupled modes that decay exponentially.

• Case $\bar{B} = 0$: Standard energy estimates on $\rho$ and $B$ separately show decay.
• Case $\bar{B} \neq 0$: The modes are coupled. The author defines new variables $\zeta_1 \rho + B$ and $\zeta_2 \rho + B$, where $\zeta_{1,2}$ are roots of a quadratic equation derived from the linearized system's symbol.
• By deriving energy estimates for these coupled variables, the author proves that the $H^s$ norms of the perturbations decay exponentially in time (specifically in $H^{s-2}$), while the $H^s$ norms remain bounded. This implies global existence and convergence to the steady state.

4. Key Contributions and Results

Theorem 1 (Local Well-posedness & No Vacuum):

• For smooth initial data with $1 < \gamma < 2$ and $B_0 \neq 0$, there exists a unique smooth solution on a maximal time interval $[0, T)$.
• Absence of Vacuum: The density $\rho$ is bounded away from zero ($\rho \geq M^{-1} > 0$) and bounded from above. Consequently, no vacuum states or implosions occur before a potential singularity.
• The solution blows up if and only if the $H^3$ norm of $(\rho, B)$ diverges.

Theorem 2 (Magnetic Relaxation):

• For initial data sufficiently close to a constant steady state $(\bar{\rho}, \bar{B})$ in $H^s$ (with $s \geq 6$), the solution exists globally in time ($T = \infty$).
• The solution converges to the steady state as $t \to \infty$:
  $$(\rho(t), B(t)) \to (\bar{\rho}, \bar{B}) \quad \text{in } H^m(\mathbb{T})$$
  for any $m < s$.
• The velocity field $u$ (recovered from $\rho$ and $B$) converges to zero.

5. Significance

1. Rigorous Justification of Numerical Schemes: The paper provides the first rigorous mathematical proof that the compressible Magnetic Relaxation Equations (often used in numerical codes like SIESTA and PIES to find MHS equilibria) actually converge to steady states. This validates the use of compressible formulations to avoid the computational cost of solving the incompressibility constraint (inverse Laplacian) at every step.
2. Handling Mixed Diffusion: The construction of the diagonalizing variables $W$ and $Z$ is a novel analytical technique for systems with mixed diffusion terms where standard maximum principles fail. This approach could be applicable to other coupled hyperbolic-parabolic systems in fluid dynamics.
3. Compressible MHD Theory: While most existing results on MHD relaxation focus on incompressible flows or viscous systems, this work addresses the ideal, inviscid, compressible case. It demonstrates that even without explicit viscosity, the Darcy friction combined with the magnetic field structure is sufficient to drive the system to equilibrium and prevent vacuum formation under planar symmetry.
4. Topological Hydrodynamics: The results support the broader program of using magnetic relaxation to find steady states with specific topological properties, extending the theory from incompressible to compressible regimes.

In summary, the paper successfully bridges the gap between numerical practice and mathematical theory for compressible magnetic relaxation, proving that the system is well-posed and relaxes to equilibrium without forming singularities like vacuum states.