Free Boundary Problems (FBP) of Magnetohydrodynamics (MHD)

Magnetohydrodynamics (MHD) (magnetofluiddynamics or hydromagnetics) is the academic discipline which studies the dynamics of electrically conducting fluids. Examples of such fluids include plasmas, liquid metals, and salt water. The word magnetohydrodynamics (MHD) is derived from magneto- meaning magnetic field, and hydro- meaning liquid, and -dynamics meaning movement. The field of MHD was initiated by Hannes Alfvén[1], for which he received the Nobel Prize in Physics in 1970.

The idea of MHD is that magnetic fields can induce currents in a moving conductive fluid, which create forces on the fluid, and also change the magnetic field itself. The set of equations which describe MHD are a combination of the Navier-Stokes equations of fluid dynamics and Maxwell’s equations of electromagnetism. These differential equations have to be solved simultaneously, either analytically or numerically. MHD is a continuum theory and as such it cannot treat kinetic phenomena, i.e. those in which the existence of discrete particles or of a non-thermal velocities distribution are important.

Ideal and resistive MHD

The simplest form of MHD, Ideal MHD, assumes that the fluid has so little resistivity that it can be treated as a perfect conductor. This is the limit of infinite magnetic Reynolds number. In ideal MHD, Lenz’s law dictates that the fluid is in a sense tied to the magnetic field lines. To explain, in ideal MHD a small rope-like volume of fluid surrounding a field line will continue to lie along a magnetic field line, even as it is twisted and distorted by fluid flows in the system. The connection between magnetic field lines and fluid in ideal MHD fixes the topology of the magnetic field in the fluid—for example, if a set of magnetic field lines are tied into a knot, then they will remain so as long as the fluid/plasma has negligible resistivity. This difficulty in reconnecting magnetic field lines makes it possible to store energy by moving the fluid or the source of the magnetic field. The energy can then become available if the conditions for ideal MHD break down, allowing magnetic reconnection that releases the stored energy from the magnetic field. In the domain $\Omega_+(t)$: \begin{align} &\partial_t v+v\cdot\nabla v-\div T(v,p)=\div T_M(H), \\ &\partial_t H+v\cdot\nabla H-\frac{1}{\mu_1\alpha}\Delta H=H\cdot \nabla v, \\ &\div v=0,\quad \div H=0.\end{align} In the domain $\Omega_-(t)$: $$\text{curl } H=0,\quad \div H=0.$$ On the free interface: $$(T(v,p)+\ldbrack T_M(H)\rdbrack)N=\sigma N\bar{H},\quad V_N=v\cdot N,$$ $$\ldbrack \mu H\cdot N\rdbrack=0,\quad \ldbrack H_\tau\rdbrack=0.$$ On the fixed boudary: $$H\cdot N=0.$$ Here $T(v,p)=-pI+\nu S(v)$ it the viscous stress tensor, $S(v)=\nabla v+(\nabla v)^\top$ is the doubled rate-of-strain tensor, $\nu$ is the kinematic viscosity, $T_M(H)=\mu(H\otimes H-\frac{1}{2}|H|^2 I)$ is the magnetic stress tensor, $\mu$ is the piece-wise constant  magnetic permeabilities, $\ldbrack \cdot\rdbrack$ denotes the jump functions.

Ideal MHD Equations

The ideal MHD equations consist of the continuity equation, the momentum equation, and Ampere’s Law in the limit of no electric field and no electron diffusivity, and a temperature evolution equation. As with any fluid description to a kinetic system, a closure approximation must be applied to highest moment of the particle distribution equation. This is often accomplished with approximations to the heat flux through a condition of adiabaticity or isothermality.

FBP of Incompressible Ideal MHD

Incomp. Ideal MHD (n-D)Bounded domain (~ ball)Bounded domain ($\mathbb{T}^{n-1}\times (a,b) $)Infinite depth domain
A priori estimates [HL14]：$n=2,3$, $H^{n+1}$;
[HY-b]: $n=3$, with surface tension 		[LZ21]: $n=3$, $(a,b)=(0,1)$, $H^{3.5}$, with surface tension
[LZ20]: $n=3$, $\mathbb{T}^{n-1}\times (0,\bar{\epsilon})$, $\bar{\epsilon}\ll 1$, $H^{2.5+}$, small volume bounded domain with proof sketch;
LWP, limit[HL21]: $n=3$, LWP, linearized equations[GW19]: $n=3$, $(a,b)=(0,1)$, LWP;
[GLZ22]: $n=3$, $(a,b)=(0,1)$, zero surface tension limit;
[GLZ23]: $n=3$,
[Z24]: $n=3$, Taylor-sign, Eulerian coordinates		[Lee17]: $n=3$, LWP as viscosity=diffusivity$\to 0$, with gravitational force
Ill-posedness, singularity, breakdown criterion[HL20]: $n=2$, disc, ill-posedness;
[FHYZ-a]: $n=3$, ball, breakdown criterion in $H^4$
Bounded Domain with Closed Surface (not being a graph)
  1. [HL14] Hao, Chengchun; Luo, Tao, A priori estimates for free boundary problem of incompressible inviscid magnetohydrodynamic flows. Arch. Ration. Mech. Anal.212(2014), no.3, 805–847. (2&3D a priori estimates)
  2. [HL20] Hao, Chengchun; Luo, Tao, Ill-posedness of free boundary problem of the incompressible ideal MHD. Comm. Math. Phys. 376 (2020), no. 1, 259–286. (2D, ill-posedness)
  3. [HL21] Hao, Chengchun; Luo, Tao, Well-posedness for the linearized free boundary problem of incompressible ideal magnetohydrodynamics equations. J. Differential Equations 299(2021), 542–601. (3D, linearized, well-posedness)
  4. [FHYZ-a] Fu, Jie; Hao, Chengchun; Yang, Siqi; Zhang, Wei, A Beale-Kato-Majda criterion for free boundary incompressible ideal magnetohydrodynamics. Journal of Mathematical Physics 64(9): 091505, 16pp, 2023. (3D, breakdown criterion in $H^4$)
  5. [HY-b] Hao, Chengchun; Yang, Siqi, On the motion of the closed free surface in three-dimensional incompressible ideal MHD with surface tension, arXiv: 2312.09473, 2023.

Other Bounded Dmain (boundary being a graph)
  1. [Z24]Zhao, Wenbin, Local well-posedness of the plasma-vacuum interface problem for the ideal incompressible MHD, Journal of Differential Equations 381 (2024) 151–184.
  2. [GLZ23] Gu, Xumin; Luo, Chenyun; Zhang, Junyan, Local well-posedness of the free-boundary incompressible magnetohydrodynamics with surface tension, J. Math. Pures Appl. (2023), doi: https://doi.org/10.1016/j.matpur.2023.12.009.
  3. [GLZ22] Gu, Xumin; Luo, Chenyun; Zhang, Junyan, Zero surface tension limit of the free-boundary problem in incompressible magnetohydrodynamics, Nonlinearity 35 (2022), no. 12, 6349–6398. (3D, domain $\mathbb{T}^2\times (0,1)$, zero surface tension limit)
  4. [GW19] Gu, Xumin; Wang, Yanjin, On the construction of solutions to the free-surface incompressible ideal magnetohydrodynamic equations. J. Math. Pures Appl. (9)128(2019), 1–41. (3D, domain $\mathbb{T}^2\times (0,1)$, LWP)
  5. [LZ21] Luo, Chenyun; Zhang, Junyan, A priori estimates for the incompressible free-boundary magnetohydrodynamics equations with surface tension. SIAM J. Math. Anal.53(2021), no.2, 2595–2630. (3D, a priori estimates in $H^{3.5}$, with surface tension)
  6. [LZ20] Luo, Chenyun; Zhang, Junyan, A regularity result for the incompressible magnetohydrodynamics equations with free surface boundary. Nonlinearity33(2020), no.4, 1499–1527. (3D, a priori estimates, minimal regularity $H^{2.5+}$, small volume domain)

Infinite or finite depth domain
  1. [Lee17] Lee, Donghyun, Uniform estimate of viscous free-boundary magnetohydrodynamics with zero vacuum magnetic field. SIAM J. Math. Anal.49(2017), no.4, 2710–2789. (3D infinite depth domain, LWP as both kinematic viscosity and magnetic diffusivity to zero with same speed, zero magnetic field condition on the free boundary, with gravitational force）

FBP of Incompressible Viscous or Resistive MHD

  1. [HY-a] Hao Chengchun; Yang Siqi, Splash singularity for the free boundary incompressible viscous MHD, Journal of Differential Equations 379, 26–103, 2024. (2D, splash singularity, viscous non-resistive MHD)
  2. [Lee18] Lee, Donghyun, Initial value problem for the free-boundary magnetohydrodynamics with zero magnetic boundary condition. Commun. Math. Sci. 16(2018), no.3, 589–615. (3D finite depth domain, LWP, viscous and resistive MHD, zero magnetic field condition on the free boundary, with gravitational force)

Incompressible Plasma-Vacuum Interface Problem

  1. [LX23b] Sicheng Liu, Zhouping Xin, On the free boundary problems for the ideal incompressible MHD equations, https://doi.org/10.48550/arXiv.2311.06581
  2. [WX21] Wang, Yanjin; Xin, Zhouping, Global well-posedness of free interface problems for the incompressible inviscid resistive MHD. Comm. Math. Phys. 388(2021), no.3, 1323–1401. (3D, domain $\mathbb{T}^2\times (-1,1)$, inviscid resistive MHD)
  3. [MTT14a] Morando, Alessandro; Trakhinin, Yuri; Trebeschi, Paola, Well-posedness of the linearized plasma-vacuum interface problem in ideal incompressible MHD. Quart. Appl. Math.72(2014), no.3, 549–587.
  4. [MTT14b] Morando, Alessandro; Trebeschi, Paola; Trakhinin, Yuri, The linearized plasma-vacuum interface problem in ideal incompressible MHD. Hyperbolic problems: theory, numerics, applications, 1007–1014. AIMS Ser. Appl. Math., 8 American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2014.
  5. [Gu19] Gu, Xumin, Well-posedness of axially symmetric incompressible ideal magnetohydrodynamic equations with vacuum under the non-collinearity condition. Commun. Pure Appl. Anal.18(2019), no.2, 569–602.
  6. [SWZ19] Sun, Yongzhong; Wang, Wei; Zhang, Zhifei, Well-posedness of the plasma-vacuum interface problem for ideal incompressible MHD. Arch. Ration. Mech. Anal.234(2019), no.1, 81–113.
  7. [Tra20] Trakhinin, Yuri, On violent instability of a plasma-vacuum interface for an incompressible plasma flow and nonzero displacement current in vacuum. Commun. Math. Sci.18(2020), no.2, 321–337.

Two-phase Flow Problem in Ideal Incompressible MHD

  1. [LX23a] Sicheng Liu, Zhouping Xin, Local Well-posedness of the Incompressible Current-Vortex Sheet Problems, https://doi.org/10.48550/arXiv.2309.03534
  2. [LL22] Li Changyan; Li Hui, Well-posedness of the free boundary problem in incompressible MHD with surface tension. Calculus of Variations and Partial Differential Equations  (2022)61, no: 191. ($\Omega={\mathbb {T}}^2\times [-1,1]$, LWP, zero surface tension limit under additional assumption that the Syrovatskij condition, two-phase MHD+MHD, with surface tension)

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