## Abstract

The equations of magnetohydrodynamics of partially ionized plasmas have been known for a long time, but rarely studied. Instead, several simplifications have been applied to different physical models, ranging from magnetic reconnection to ambipolar drift. The original system relates the electric field to kinetic magnitudes by means of a non-local law, so that the equations describing it involve partial differentials as well as functional operators. We prove a theorem of existence and uniqueness of solutions for a finite time by means of a fixed point argument in an appropriate functional setting.

## 1. Introduction

In contrast to the exalted state of the Navier–Stokes equation among fluid dynamics theorists, researchers in plasma theory are highly conscious of the radical character of the approximations needed to transform the Boltzmann equations into the single-fluid description of magnetohydrodynamics (Chen 1983). In fact, many physically relevant plasmas do not satisfy these requirements. One of the most important instances occurs when the plasma cannot be considered to be formed by a single species. The decoupling of the neutral plasma to ions and electrons near current sheets plays a basic role in the explanation of fast magnetic reconnection (Biskamp 2000); also, partially ionized plasmas are the rule rather than the exception in astrophysics, with important consequences such as the Hall effect (Shay *et al.* 2001), Alfvén wave damping (de Pontieu & Haerendel 1998) and ambipolar drift (Brandenburg & Zweibel 1994). We will study a three-component plasma formed by ions, electrons and neutrals, which is general enough to cover many cases. The first part of this paper will analyse the relevant equations, in particular Ohm's law, which we will show in general to be a non-local elliptic equation; however, the vanishing of certain terms will yield in succession the equations of two-fluid MHD, Hall MHD and classical MHD systems, or in the other direction the strong coupling approximation. Unfortunately, the coefficients to be neglected are consistently the higher order terms of the equation, with the result that their character changes in each of these modifications and an existence theorem proved for a certain version will not work for a simplified one. The aim of this paper is to prove the existence and uniqueness theorem for a collisional, viscous plasma with three species and electron inertia. For simplicity, we will assume each of the species to be incompressible. The resulting equations do not form a classical evolution system due to the presence of non-local terms, but nonetheless the problem is amenable to a fixed point argument. A word about notation: customarily three vectors are written in boldface character. Since we will be forced to deal with nine vectors as well, it would be impractical to use separate notations. Hence, we will omit the bold characters altogether; the context will make clear the dimension of each magnitude. Thus, all velocities plus the electric and magnetic fields will always be three-dimensional.

## 2. The main equations

As stated, we will assume that the plasma is formed by three species, electrons, ions and neutrals, denoted by sub-indices *e*, *i* and *n*, respectively. Their number densities *n*_{j} and particle masses *m*_{j} yield their mass densities *ρ*_{j}=*m*_{j}*n*_{j}, which we assume constant, so that the continuity equation for each species is simply ∇.*v*_{j}=0, where *v*_{j} is the respective velocity. We also assume that the whole fluid is neutral, so that if *e* is the electron charge and −*e* the ion charge, we have *n*_{e}=*n*_{i}. There exists dissipation due to collisions between species. The collision frequencies *ν*_{ei}, *ν*_{en} and *ν*_{in} (*ρ*_{a}*ν*_{ab}=*ρ*_{b}*ν*_{ba}) may be expressed in terms of fractional ionizations and plasma temperature (Draine *et al.* 1983). Since collisions are relevant, it is logical to admit the existence of viscosities *ν*_{j}, although its value is often low enough to be omitted in several (in fact most) studies (Pandey & Wardle 2008). Recall that *ν*_{j}=(absolute viscosity)/(density) represents the kinematic viscosity. A complete description, including possible anisotropic pressure, may be found in Braginsky (1965); the ideal, inviscid case is dealt with in Spitzer (1956). For a recent exposition, see Brandenburg & Subramanian (2005). We will take units so that the speed of light *c*=1. The momentum equations for electrons, ions and neutrals are Navier–Stokes ones plus the forcing term given by the Lorentz force, minus the collisional effects:(2.1)(2.2)(2.3)where *E* is the electric field; *B* is the magnetic field; and *P*_{j} is the kinetic pressure. Other effects not included here are the ionization and recombination ones that add to the forcing terms in the previous equations (see Brandenburg & Zweibel 1995), where steep magnetic gradients are shown to form in a case model. Faraday's equation yields(2.4)Since we have assumed charge neutrality, ∇.*E*=0. As usual in non-relativistic plasmas, we omit the displacement current from Ampère's law, which becomes(2.5)where *J* is the current density and *μ* is the magnetic permeability. To avoid dragging constants throughout the paper, we will take *μ*=1. The connection between the flow and the current is given by the fact that *J* may be considered as the flow of positive charges(2.6)Equations (2.1)–(2.6) form the original system.

When the plasma is totally ionized, we may omit all the neutral terms. Substituting into the electron momentum equation, we find(2.7)Assuming the velocities small enough for all quadratic (advective) terms to be taken as zero, and comparing with the ion momentum equation, one finds(2.8)Since , by taking the quotient as zero, we obtain(2.9)which is the two-fluid Ohm equation. The valueis the plasma resistivity and (*en*_{e})^{−1} is the Hall coefficient. The pressure term is the cause of the so-called Biermann battery, which in the case that *n*_{e} is not constant may provide a thermal source for the magnetic field (Mestel & Roxburgh 1962). Using Faraday's law, this yields the two-fluid induction equation(2.10)The electron inertia term,is generally very small, although extremely important in the analysis of the system (Núñez 2005). If this is omitted, we get the Hall induction equation, and when the term in ∇×(*J*×*B*) is also suppressed, the classical MHD system is obtained. As stated, every one of these simplifications needs a different study.

In the three-species case, the common course is to cancel not only the advective and viscous terms but also the electron velocity term in (2.1), given the small mass of the electron. In this case, (2.1) becomes an equilibrium equation(2.11)which is analogous to (2.9), except by the absence of electron inertia and the presence of a new term. This may be estimated by assuming that the accelerations of ions and neutrals are similar, . Adding (2.2) and (2.3) to obtain the momentum equation for the mean velocity,and plugging (2.11) into it, we obtain a new induction equation(2.12)where *r*=*ρ*_{i}/(*ρ*_{i}+*ρ*_{n}). This approximation is useful when dealing with ambipolar diffusion (Indebetouw & Zweibel 2000). The new term ∇×((*J*×*B*)×*B*) provides a decrease in the size of the magnetic energy of the form , i.e. the square of the magnitude of the Lorentz force. This may be shown to imply, broadly speaking, that the ion and neutral velocities tend to converge and the magnetic field tends to a force-free state (Núñez 2006). An even more drastic simplification for weakly ionized plasmas yields the strong coupling approximationWhile all these approaches work fairly well in their respective spheres, there is something unsatisfactory in their deduction. First, one cancels one term, then a different one, and it is often not clear why one or the other may be neglected. It seems reasonable to return to the original system and prove a theorem of existence of solutions.

Let us consider appropriate boundary conditions. We will deal with a smooth bounded domain *Ω*, and since the fluid is viscous, we assume a Dirichlet boundary condition for the velocity: for *j*=*e*, *i*, *n*. If we assume that the electric field vanishes outside *Ω* and there is no surface charge at the boundary, also ; this implies that the normal component of ∇×*E* also vanishes at ∂*Ω*, so that if *B*(0) is parallel to the boundary, the magnetic field remains parallel for all time; this is the continuity condition when the field vanishes outside *Ω*. None of these conditions is absolutely necessary, and they could be modified in many ways; thus, periodic boundary conditions would work as well. Let us see how to reduce (2.1)–(2.6) to a working evolution equation. Differentiating (2.6)(2.13)and taking this value to the difference between (2.1) and (2.2)(2.14)On the other hand, since ∇.*E*=0,(2.15)so that(2.16)This is the unsimplified law of Ohm. We see that *E* is not a local function of the kinetic variables, but the solution of an elliptic problem. Now we will plug this relationship into (2.1) and (2.2), which jointly with (2.3) and (2.4) will form a closed system in the variables *v*_{e}, *v*_{i}, *v*_{n} and *B*.

Conversely, if we have such a solution, we define *E* by solving (2.8). Then (2.4) implies , so that the ‘real’ electric field *E*_{1} must satisfy ∇×*E*=∇×*E*_{1}. In a simply connected set, they differ in a gradient ∇*F*. Since we assume that there are no free charges, ∇.*E*_{1}*=*0. Our proof method will also find *E* within a space of solenoidal fields, so that ∇*F*=0, which in conjunction with the homogeneous boundary condition for both fields yields *E*=*E*_{1}. On the other hand, since ∂*J*/∂*t*=Δ*E* and this value is known from (2.16), taking it back to (2.1) and (2.2), we find again (2.13). Hence, provided the initial conditions satisfy *J*(0)=∇×*B*(0)=*en*_{e}(*v*_{i}(0)−*ν*_{e}(0)), (2.6) holds for all the time.

## 3. Existence theorems: preliminaries

The basic Hilbert spaces used in the proof will be will denote the orthogonal projection(3.1)where *A* is a positive elliptic operator. Therefore,is a bijection, whose inverse is bounded with the *L*^{2} norm in *H* and the *H*^{2} one in *D*(*A*). Thus, if we call *E* this inverse (recall that at present *E* is not a function, but an operator), *AE* is a continuous endomorphism of *H*, and *EA* is an endomorphism of *D*(*A*) that may be extended to endomorphisms *V*→*V* and *H*→*H*, continuous with the respective norms.

To avoid the use of sub-indices as far as possible, we will denote the *L*^{2} norm by | | and the *H*^{1} norm by . The inequality of Poincaré asserts that we can choose this norm as . We will also handle function spaces , , and will denote the supremum norm, respectively, by | |_{∞,T}, . The norm in the space will be denoted by | |_{2,T}, so that will be (equivalent to) the norm in .

For the nine-dimensional vector given by the three velocities, we will use the space *H*_{3}, identical to *H*^{3} but with a weighted product: , the same for *V*_{3} and *V*^{3}. Obviously, the norms of *H*_{3} and *V*_{3} are equivalent to their respective norms in *H*^{3} and *V*^{3}. Thus, we will also denote the norm in *H*_{3} by | | and the one in *V*_{3} by ; there is no danger of confusion, since in one case we are dealing with three-dimensional vectors and in the other one with nine-dimensional vectors. The same thing may be said for the function spaces taking values in *H*_{3}, *V*_{3}, etc., such as . Obviously, all the Sobolev inequalities valid for *H* and *V* hold for *H*_{3} and *V*_{3}; essentially, it all boils down to each component of these vectors. We define (3.2)so thatNote that the norms in *v*_{j} are norms in *V*. Therefore,(3.3)Now the norm of *v* is the one of *V*_{3}. We will denote and . Further notations are(3.4)For one 3-vector ** v** and 1-vector

**,(3.5)and for a three-vector**

*h***(3.6)There is no need to project now, since all the terms are in**

*v**H*. Note that the condition implies that (adding the spaces as sub-indices for clarity)(3.7)which does not happen if we make the product in

*H*

^{3}. Contracted (three-dimensional) versions of

*Φ*,

*Ψ*and

*Λ*will be(3.8)(3.9)(3.10)Note that

*P*

_{H}kills all the gradients, which will be useful to eliminate the pressures. Since

*E*∈

*H*anyway, by projecting in

*H*(2.10) may be written(3.11)Let us turn now to the Sobolev inequalities we will use. For a three-dimensional domain

*Ω*, and

*s*>3/2, . We take, for example,

*s*=5/3 and conclude that for

*u*∈

*D*(

*A*), the interpolation inequality(3.12)holds.

*c*is a constant that depends only on

*Ω*. Since we will use many dozens of such constants, notation would be uncontrollable with a different name for every one: hence, all of them will be

*c*. Thus, for example, . Equation (3.6) implies(3.13)(3.14)Therefore,(3.15)Also,(3.16)(with the same bound for

*ψ*), and(3.17)and the same for

*λ*. Classical elliptic inequalities for

*E*in (2.21) yield(3.18)(3.19)Note that the norm of the second term is an

*L*

^{2}one in both cases. Finally, defining(3.20)which obviously satisfies the same bounds, we may write (2.1–2.6) after projection in

*H*

_{3}as(3.21)(3.22)The incompressibility of the solutions is given by the fact that they are all in

*H*, and the stationary character of the flow at ∂

*Ω*will be a consequence of our posterior proof that

*v*∈

*V*

_{3}. The boundary condition for

*E*is a consequence of (3.11):

*E*is a solution within

*D*(

*A*).

Finally, we will use the following classical inequalities: Young's, , for 0<*α*<1, in the form(3.23)for *x*, *y*≥0, *γ*>1. *ϵ* may be taken as small as desired: in our case it will be a fraction of the infimum of viscosities *μ*. The other inequality is Hölder's, in the form,(3.24)for 0<*α*<2.

## 4. Existence theorems

We will prove local existence of solutions by considering separately the equations(4.1)(4.2)where *v*^{*} and *B*^{*} are the data. We will show that both (3.19) and (3.20) have unique solutions in a certain interval [0,*T*], and that the operator is contractive for appropriate norms. The fixed point of *S* will be the unique solution to the problem. Let us prove the first existence for (3.19) and estimate the size of the solution.

*Let* *and* . *Then there exists T*_{2}≤*T*_{1} *depending only on* , *and* *such that for every initial condition v*(0)∈*V*_{3}, *B*(0)∈*H*, *there exists a unique solution of* (*4.1*) *in* [0,*T*_{2}]. *This solution satisfies**Also,*

We have(4.3)Sincethe following identity holds:(4.4)Since ,(4.5)Finally, (3.1) implies (Λ*v*, *v*)≥0. These types of systems have already been studied (see Temam 1988, pp. 377–381), and we conclude that there exists a unique solution of (3.19) for a certain time interval [0,*T*_{3}]. We always take *T*_{3}≤1. Since the independent term is not constant as in Temam (1988), we must reconstruct the bounds on *v*. From (3.9)–(3.14),(4.6)and from Cauchy–Schwarz's inequality (recall that *c* represents any universal constant)(4.7)The kinetic energy estimates are easy: multiplying (4.1) by *v*(4.8)which implies, using Poincaré's and Young's inequalities,(4.9)By first omitting the terms in ,(4.10)which implies(4.11)Note that this is equally valid for any period smaller than *T*_{3}. This represents an *a priori* bound on by plus one term of the form , where *M* is a constant depending only on the size of (*v*^{*}, *B*^{*}) in the spaces occurring in the statement of the theorem.

Nevertheless, we need to bound *v* in the norm of *V* and *D*(*A*_{3}). To this end, we multiply (3.19) by *A*_{3}*v*(4.12)We haveandTherefore, an inequality of the form(4.13)will yield, by integration,(4.14)Let us first bound the remaining terms in (4.6), i.e. the function *G* in (4.7). Using Young's inequality,(4.15)We also have(4.16)and(4.17)Finally,(4.18)Cancelling all the terms in on the r.h.s. of (4.6) with half of the same term on the l.h.s., we find that satisfies an integral inequality of the type(4.19)Let us first forget about the term in ; the inequality certainly holds. The term in 1 proceeds from the term in , which we know already is bounded by (4.5). The only non-universal quantity within the coefficients of is , which is bounded *a priori* by the constant . The integral inequality (4.13) proves that the solution of (3.19) in a certain time interval [0,*T*_{4}], *T*_{4}≤*T*_{3}, is bounded by a constant *M* depending only on and the integral of in [0,*T*_{3}]. In this interval, we bound by *M*^{3} on the r.h.s. of (4.13), and we are left with a linear inequality. This implies(4.20)The exponential may be included in the constant. By our previous estimate on the integral in (4.5), we have(4.21)Since all the r.h.s. magnitudes are bounded *a priori*,(4.22)Recovering the term in in (4.13),(4.23)which, jointly with (4.15) and (4.16), yields an analogous estimate(4.24)Let us consider now the magnetic field. Existence of solutions to (3.20) is trivial: the solution is simply(4.25)To estimate the size of *B*, we use (3.13)(4.26)Therefore,(4.27)Integrating as before in [0,*T*_{4}] and applying Hölder's inequality,(4.28)i.e.(4.29)Inequalities (4.16), (4.18) and (4.23) clearly imply that for any given constants *M*_{1}, *M*_{2} and *M*_{3}, we can take *T*_{2}, depending on *M*_{j}, such that the mapping takesintoIt is enough to set the constants *M*_{j} equal to the radii of the last balls. ▪

Note that the same result is valid for all *T*≤*T*_{2}. We will denote this set, closed by *S*, as .

*There exists T*_{0}≤*T*_{2} *such that S is a contractive mapping in the complete metric space* .

Consider the solutions and associated with and , with the same initial condition. Then *v*_{1}−*v*_{2} satisfies(4.30)We will use analogous bounds to the previous ones for the difference of these bilinear functions. Thus,(4.31)for *j*=*e*, *i*, *n*. The same bound holds therefore for *Φ*(4.32)An identical estimate holds for , adding asterisks to the velocities. Also,(4.33)Hence,(4.34)The same expression holds for , except for the presence of asterisks in the velocity. Finally, both *Λ* and *λ* are linear, so(4.35)and the same for *λ*. Therefore,(4.36)Multiplying (4.24) by and using the same bounds of the previous theorem(4.37)All the terms on the r.h.s. involving have a power of the order of at most 5/3. For those of the form , we use Young's inequality in the form(4.38)and for the rest of them(4.39)Since there are seven terms, by adding all of them and compensating with the l.h.s., we find(4.40)Now we have the advantage over the previous theorem that we know *a priori* that all the terms in and are bounded. If we incorporate all of them in *c*, omit for the moment the term in from the l.h.s. and integrate in time, recalling that the initial condition is zero(4.41)This is a linear inequality, and therefore it can be solved explicitly. The exponential term involvesso that for all *T*_{5}≤*T*_{2},(4.42)We can abbreviate this to(4.43)Recovering now the term in in (4.33), and integrating again (4.34), we reach the same conclusion(4.44)Finally,(4.45)We have(4.46)Recalling (4.29), now without taking squares(4.47)Using exactly the same estimates as in the previous cases, we get(4.48)All the bounds (4.35), (4.36) and (4.39) are valid for all *T*≤*T*_{5} and *c* does not depend on the period. Hence, it is possible to take *T*_{0}≤*T*_{5} such that the constants add at most to 1/2(4.49)This proves that in the operator *S* is contractive. ▪

We state now our main theorem.

*Let* , , *such that in a weak sense* . *Then the system (2.1–2.6) admits a unique solution* (*v*, *B*), *with**The value of T*_{0} *depends only on* *and* .

The solution is the unique fixed point of . ▪

## 5. Conclusions

The equations describing the behaviour of a partially ionized plasma in the magnetohydrodynamic regime have been known for a long time, but so far no attempt has been made to prove at least the existence and uniqueness of solutions for a certain time interval. This may be due to the fact that the full equations are seldom used; instead different terms have been assumed to vanish and the resulting systems adapted to describe a wide range of plasma phenomena, ranging from magnetic reconnection to molecular clouds. These simplified equations include the two-fluid, electron and Hall magnetohydrodynamics, as well as the ambipolar drift equations of weakly ionized plasmas. While these models have been fairly successful in their respective ranges, the reasons for considering a specific term irrelevant are mostly ad hoc and somewhat unsatisfactory from a theoretical viewpoint. Moreover, these cancelled terms are usually the highest order ones, so that the character of the system changes radically with each simplification and no general theorem is available for all of them. We consider the original equations and show that Ohm's law becomes an elliptic system satisfied by the electric field, in contrast to other classical local descriptions, so that the resulting dynamical system involves functional partial differential equations. Nevertheless, persistent use of several analytic inequalities makes the problem amenable to a fixed point argument in certain function spaces, which will yield the desired theorem of existence and uniqueness.

## Footnotes

- Received October 9, 2007.
- Accepted February 7, 2008.

- © 2008 The Royal Society