Pressure Selection and Exponential Relaxation in a Periodic Two-Phase Magnetic Model

摘要: We study a two-phase magnetohydrodynamic model with a common velocity on
 the three-dimensional torus.  The transported mass fraction generates
 infinitely many integral invariants and permits non-constant equilibria
 with spatially constant total pressure.  The two densities therefore need
 not converge to their spatial means, even for small perturbations of a
 homogeneous state.

We select a reference pressure from the transported invariants and construct
 a normalized density defect with zero spatial mean.  An adapted relative
 internal energy, combined with a Bogovskii correction, yields exponential
 decay of this defect, the pressure perturbation, the velocity, and the
 magnetic field.  This estimate holds on every smooth existence interval
 under uniform positive lower and upper bounds for the phase densities,
 with zero total momentum and zero mean magnetic field.  It requires no
 smallness of the initial oscillation, and its constants are independent
 of the interval length.  Uniform higher Sobolev bounds then yield
 higher-order decay of these quantities and reconstruct the limiting
 density profiles through the limiting flow map.

We establish two global existence results for small modulated energy.
 For general transported composition and any fixed initial $H^3$ bound,
 we obtain a unique global strong solution, with an energy threshold
 depending on that bound and no separate smallness assumption on the
 composition amplitude.  For constant composition, fixed fractional
 Sobolev bounds on the initial velocity and magnetic field allow a
 threshold independent of all positive-order density norms.  In this
 class we obtain a unique global smooth solution and exponential decay
 of every spatial derivative.  The general-composition solutions relax
 to possibly non-constant constant-pressure profiles, whereas the
 constant-composition solutions converge to a homogeneous equilibrium.