Existence of Global Weak Solutions to a Hybrid Vlasov-MHD Model for Magnetized Plasmas
aa r X i v : . [ m a t h . A P ] M a y EXISTENCE OF GLOBAL WEAK SOLUTIONS TO AHYBRID VLASOV-MHD MODEL FOR MAGNETIZED PLASMAS
BIN CHENG, ENDRE S ¨ULI, AND CESARE TRONCI
Abstract.
We prove the global-in-time existence of large-data finite-energy weak solutions to an incompress-ible hybrid Vlasov-magnetohydrodynamic model in three space dimensions. The model couples three essentialingredients of magnetized plasmas: a transport equation for the probability density function, which modelsenergetic rarefied particles of one species; the incompressible Navier–Stokes system for the bulk fluid; and aparabolic evolution equation, involving magnetic diffusivity, for the magnetic field. The physical derivation ofour model is given. It is also shown that the weak solution, whose existence is established, has nonincreasingtotal energy, and that it satisfies a number of physically relevant properties, including conservation of thetotal momentum, conservation of the total mass, and nonnegativity of the probability density function forthe energetic particles. The proof is based on a one-level approximation scheme, which is carefully devisedto avoid increase of the total energy for the sequence of approximating solutions, in conjunction with a weakcompactness argument for the sequence of approximating solutions. The key technical challenges in the anal-ysis of the mathematical model are the nondissipative nature of the Vlasov-type particle equation and passageto the weak limits in the multilinear coupling terms. Introduction
In multiscale dynamics, hybrid kinetic-magnetohydrodynamic (MHD) theory offers the opportunityof a multi-physics modeling approach in which a macroscopic fluid flow is coupled to a kinetic equationincorporating the microscopic dynamics of a particle ensemble. Over the past decades, various hybridmodels were formulated for different purposes, ranging from combustion theory [32] to polymeric fluid flows(see, for example, [13], [1] and the references therein).In plasma physics, linear hybrid schemes have now been used over several decades to model the interac-tion of the MHD bulk fluid with a rarified ensemble of energetic particles, which in turn affect the overallenergy and momentum balance. For example, in tokamak devices, the fusion reactions produce energeticrarefied alpha particles that escape a fluid description and thus require a kinetic treatment. Although thelinear theory of hybrid kinetic-MHD schemes has been consolidated by computer simulations and analyticalstability results [12], its nonlinear counterpart poses several consistency questions, which have only beenapproached during the last few years [27, 30, 19]. In particular, while different hybrid schemes are currentlyused in computer simulations, many of them have been found to lack energy conservation [30], therebygenerating unphysical instabilities above a certain frequency range. The formulation and analysis of hybridkinetic-MHD models therefore represents a fascinating research area, in which the analysis of nonlinearfeatures of the kinetic-MHD coupling necessitates the use of powerful modern mathematical techniques.1.1.
Mathematical setup.
The physical derivation of the system of partial differential equations (PDE)studied in this paper is postponed to Section 2. In its original form, the system is stated in (2.21) whichis a model from the so-called current-coupling scheme (CCS), and is an incompressible, dissipative versionof [27, eqs. (52)–(55)]. Here, for simplicity of the exposition, we shall set all positive physical constantsappearing in (2.21), including the density, to unity, as their specific values do not affect our considerations.Suppose that
T > T = R / (2 π Z ). For t ∈ [0 , T ], x ∈ T and v ∈ R , we shall seek the 3-vector velocity U = U ( t, x ) of the bulk fluid, the 3-vector magnetic field B = B ( t, x ), and the scalarprobability density function f ( t, x , v ) ≥
0, which models energetic rarefied particles of one species. It willbe implicitly understood throughout the paper that all functions of the variable x ∈ T satisfy 2 π -periodicboundary conditions with respect to x , and this property will only be explicitly stated when it is necessaryto emphasize it. Mathematics Subject Classification.
Key words and phrases.
Global weak solution, Vlasov equation, magnetohydrodynamics.
The unknown functions U , B and f are then required to satisfythe following coupled system of nonlinearPDEs: ∂ t f + v · ∇ x f = (( U − v ) × B ) · ∇ v f∂ t U + ( U · ∇ x ) U + ∇ x P − ( ∇ x × B ) × B − ∆ x U = Z R ( U − v ) × B f d v (subject to ∇ x · U = 0) ,∂ t B − ∇ x × ( U × B ) = ∆ x B (subject to ∇ x · B = 0) , where the auxiliary variable P denotes the “pressure”. Since U and B are divergence-free, one can applythe identities (A.7) and (A.6) in Appendix A to rewrite( ∇ x × B ) × B = ( B · ∇ x ) B − ∇ x | B | , ∇ x × ( U × B ) = ( B · ∇ x ) U − ( U · ∇ x ) B . Then, by adding | B | to the pressure P , the incompressible Vlasov-MHD system can be restated as follows: ∂ t f + v · ∇ x f = (( U − v ) × B ) · ∇ v f, (1.1a) ∂ t U + ( U · ∇ x ) U − ( B · ∇ x ) B − ∆ x U = Z R ( U − v ) × B f d v − ∇ x P (subject to ∇ x · U = 0) , (1.1b) ∂ t B + ( U · ∇ x ) B − ( B · ∇ x ) U − ∆ x B = 0 (subject to ∇ x · B = 0) . (1.1c)It is this set of PDEs that we shall focus on on this article, subject to the initial conditions U (0 , x ) = ˚ U ( x ) , B (0 , x ) = ˚ B ( x ) , f (0 , x , v ) = ˚ f ( x , v ) ≥ , x ∈ T , v ∈ R , (1.2)together with 2 π -periodic boundary conditions with respect to x . The initial data ˚ U ( x ) and ˚ B ( x ) areassumed to be divergence-free and 2 π -periodic.We remark that ∇ x · B = 0 is an invariant of the evolutionary PDE system (1.1), at least when U and B have sufficiently many derivatives. Indeed, by taking the divergence of (1.1c) and using the identity P j =1 ( ∂ x j U · ∇ x ) B j − ( ∂ x j B · ∇ x ) U j ≡
0, we have that ∂ t ( ∇ x · B ) + ( U · ∇ x )( ∇ x · B ) − ( B · ∇ x )( ∇ x · U ) − ∆ x ( ∇ x · B ) = 0 , so that if ∇ x · U = 0 for all times and the initial magnetic field ˚ B is divergence free, then ∇ x · B = 0 forall times. A pressure-like term in the evolution equation for B would be trivially constant and is thereforeabsent from (1.1c). For weak solutions, which this article is concerned with, however, the divergence-freeinvariance of B is not immediate; we therefore retain the divergence-free condition in (1.1c) for clarity.Another reason for explicitly stressing this condition is that our proof of the existence of weak solutions tothe PDE system (1.1) involves a sequence of approximate PDE systems to (1.1), which do not possess thisdivergence-free invariance. Thus, in these approximate systems, we must require B to be divergence-freeand add an explicit pressure-like term in the magnetic equation.1.2. Main result.
Here, we summarize our main result, Theorem 5.8. It states the existence of global-in-time weak solutions to the PDE system (1.1) in a sense to be made precise in Definition 5.4. This definitiongives a weak formulation of (1.1) along with certain physically relevant properties satisfied by such weaksolutions.Consider the incompressible , current-coupling scheme of the resistive Vlasov-MHD system (1.1) for t ∈ [0 , ∞ ), with x contained in a three-dimensional torus, T , and v contained in the whole three-dimensionalspace, R . The given initial data are: the fluid velocity field U (0 , x ), the initial magnetic field B (0 , x ), whichare both divergence-free and (Lebesgue) square-integrable, and the probability density of particles f (0 , x , v ),which is pointwise nonnegative, (Lebesgue) integrable and essentially bounded. Suppose also that the initialenergy is finite, i.e.,12 Z T (cid:0)(cid:12)(cid:12) U (0 , x ) (cid:12)(cid:12) + (cid:12)(cid:12) B (0 , x ) (cid:12)(cid:12) (cid:1) d x + 12 Z T Z R f (0 , x , v ) | v | d v d x < ∞ . Then, there exists a finite-energy global-in-time large-data weak solution ( f ( t, x , v ) , U ( t, x ) , B ( t, x )) to thesystem (2.21) for t ∈ [0 , ∞ ) in the sense of Definition 5.4. In particular, the total energy does not exceed EAK SOLUTIONS TO A HYBRID VLASOV-MHD MODEL IN PLASMA PHYSICS 3 its initial value and the integrability properties assumed on the initial data as above are preserved in thecourse of evolution in time for all t >
Mathematical literature.
During the past decade several mathematical studies of PDE systemsof coupled Navier–Stokes–Vlasov type (with or without the Fokker-Planck term) have been undertaken;the reader is referred, for example, to [21, 7, 33, 31] and the references therein. The existence of globalweak solutions has been proved in these in several instances, using the key fact that the total energyis nonincreasing, in conjunction with weak compactness arguments based on moment-estimates for theprobability density function f . While these techniques have inspired the analysis performed in this paper,there is a significant difference in terms of the formulation: the coupling terms in the existing literature havealmost exclusively taken to be of (linear) drag-force type, which are proportional to ( U − v ) or (cid:0) U − K n (cid:1) ,whereas our model includes, instead, nonlinear coupling terms of Lorenz-force type, which is the naturalchoice from the point of view of plasma physics.There are also a number of results concerning the existence of weak and classical solutions to Vlasov–Maxwell equations, without coupling to fluid dynamics. The list is long and we shall only mention [14]for the existence of global weak solutions, [17] for the existence of classical solutions under the a prioriassumption that the plasma density vanishes for high velocities, and [16, Chapter 5], which includes anumber of additional references on the subject. We note that the v -advection term of the Vlasov–Maxwellsystem is a constant times ( v × B + E ) · ∇ v f , so that when the ideal Ohm’s law E + U × B = 0 is used, itcoincides with its counterpart in (1.1a).The proof of the existence of global-in-time classical solutions to our hybrid kinetic-MHD system (1.1),which nonlinearly couples the incompressible Navier–Stokes equations to evolution equations for the mag-netic field B and the probability density function f , involves significant technical difficulties, even if oneadopts an a priori assumption similar to that in [17]. Indeed, even for the source-free Navier–Stokes systemthe proof of the existence of global-in-time classical solutions for arbitrary smooth initial data is lacking,nor is there a counterexample to the breakdown of regularity of classical solutions. Our study of the hybridkinetic-MHD system (1.1) therefore concentrates here on the existence of global-in-time weak solutions.The rest of the article is organized as follows. In Section 2, we present a detailed physical derivation of thesystem (1.1) in the context of plasma physics and modeling. Then, we formally prove the key property thatthe total energy is nonincreasing in conjunction with rigorous proofs of moment-estimates in Section 3. Weconstruct in Section 4 a mollified system, using a single-level approximation, for which the desired regularityand energy bound are rigorously verified. The existence of a solution to this approximating system is shownby proving it to be a fixed point of a mollified mapping. We carefully devise this mapping so as to ensure thatits fixed point leads to a nonincreasing total energy. Finally in Section 5, we employ various compactnesstechniques to show that a subsequence of the sequence of approximating solutions converges weakly to aweak solution of the original PDE system. Since f is governed by a transport equation without any diffusion,its regularity needs to be studied with particular care. We also address the lack of L compactness, whichis, to some extent, alleviated by the assumption that the initial datum for f is in L r for all r >
1. It isworth mentioning that the spatial L integrability of ( U , B ) and the trilinear coupling terms in the PDErequire the moments n, K to have rather high integrability indices, which we believe necessitates the highintegrability indices of f .2. Plasma modeling and the physical derivation of the main PDE system
Many different hybrid models are available in the plasma physics literature [22], typically dependingon whether the “Hall term” is retained in Ohm’s Law – c.f. the comments above (2.18). In the absenceof a kinetic component, when the Hall term is neglected, the quasi-neutrality and the inertia-less electronassumptions lead to the most basic MHD fluid equations [15]. Then, one may or may not consider resistivityeffects, thereby obtaining a resistive or an ideal MHD model, respectively.When the kinetic description of energetic particles is included, the coupling of MHD to the kinetic com-ponent depends on the particular description that is adopted to model the energetic particles. Here, we shallfocus on Vlasov-MHD models that neglect the Hall term, since such a treatment is customary in the nuclearfusion and solar physics literature. In this class of models, two main kinetic-MHD coupling schemes are
BIN CHENG, ENDRE S ¨ULI, AND CESARE TRONCI discussed in the literature: the current-coupling scheme (CCS) [6, 23, 22, 26, 25] and the pressure-couplingscheme (PCS) [11, 23, 22], as they differ by the nature of the coupling terms in the fluid momentum equation.Upon adopting the Vlasov description for energetic particle kinetics, references [27, 19] showed that (in theideal limit) the CCS conserves energy exactly as a consequence of its variational/Hamiltonian structures,while the PCS (as it appears in the literature) lacks an energy balance, unless extra inertial force termsare added to the Vlasov equation. These last terms are produced naturally by the variational/Hamiltonianapproach and lead to an entirely new hybrid theory, which is currently under study and was shown toreproduce Landau damping. The Lyapunov stability of energy-conserving hybrid models has recently beenstudied in [28, 29], whereas previously proposed nonconservative models are known to exhibit unphysicalinstabilities [30].While the mathematical approach to kinetic-MHD theories is simplified by the use of the Vlasov equationfor the kinetic component, practical computer simulations [6, 11, 23, 22, 25, 26] employ the correspondingdrift-kinetic or gyrokinetic approximations [5]: these are low-frequency kinetic equations that are obtainedby sophisticated perturbation techniques to average out the fast Larmor gyration around the magnetic field.While these low-frequency options are the subject of current research in terms of geometric variationalmethods [8], here we shall consider the general case of full-orbit particle motion, thereby focusing on Vlasov-MHD models. As it was done in [27, 19, 30], we shall modify the standard CCS appearing in the plasmaphysics literature [23, 26, 22] by replacing the drift-kinetic equation with the Vlasov equation. In itsCCS variant, the set of partial differential equations (PDEs) of the Vlasov-MHD model (in the absence ofcollisional effects) reads in standard plasma physics notation as follows: ∂f∂t + v · ∇ x f + q h m h ( v − U ) × B · ∇ v f = 0 , (2.1a) ̺ (cid:18) ∂ U ∂t + ( U · ∇ x ) U (cid:19) = (cid:0) q h n U − q h K + µ − ∇ x × B (cid:1) × B − ∇ x P , (2.1b) ∂̺∂t + ∇ x · ( ̺ U ) = 0 , (2.1c) ∂ B ∂t = ∇ x × (cid:0) U × B (cid:1) , (2.1d)where the operators ∇ x and ∇ v are understood to be taken with respect to the x and v variables, respec-tively. The prognostic quantities are the probability density of the number of energetic particles, f ( t, x , v ),of dimension ( time ) / ( length ) ; B ( t, x ), denoting the magnetic field; and U ( t, x ) and ̺ ( t, x ), denoting thevelocity and density of the bulk fluid, respectively. The derived, diagnostic quantities (which are treatedas auxiliary variables throughout the article) are n ( t, x ) := R R f ( t, x , v ) d v of dimension 1 / ( length ) de-noting the total number of energetic particles per volume, and K ( t, x ) := R R v f ( t, x , v ) d v of dimension1 / (cid:2) ( length ) ( time ) (cid:3) denoting the sum of velocities of energetic particles per volume. Also, µ denotes themagnetic constant. Note that the pressure P is determined by an equation of state, which we shall assumeto be barotropic, so that P = P ( ̺ ), although the analysis in the current article is performed on an incom-pressible model, which is indifferent to the choice of the equation of state. Finally, the physical constantssubscripted with h (standing for “hot”) are all associated with intrinsic properties of the energetic particlespecies; in particular, q h , m h signify the charge and the mass of a single (energetic) particle, respectively,and a h := q h /m h denotes the charge-to-mass ratio.The total energy Hamiltonian, H ( f, U , B ) = 12 Z R ̺ | U | d x + m h Z R Z R f | v | d v d x + Z R U ( ̺ ) d x + 12 µ Z R | B | d x , (2.2)is conserved by the dynamics of (2.1). Here, we have assumed a barotropic pressure law P = P ( ̺ ), so thatthe internal energy per unit volume U depends only on the mass density ̺ and satisfies ̺ U ′ ( ̺ ) = P ( ̺ ).Its variational Euler–Poincar´e and Hamiltonian structures were characterized in [19], where conservationsof the magnetic helicity R R A · B d x and the cross helicity R R U · B d x were also verified explicitly.The conservative properties of (2.1) are no longer true upon the introduction of collisional effects into themodel. Such collisional effects are often incorporated in the plasma physics literature via a finite resistivity[25] that breaks the so-called “frozen-in condition” (2.1d) (as it is expressed in terms of Lie-dragging, this EAK SOLUTIONS TO A HYBRID VLASOV-MHD MODEL IN PLASMA PHYSICS 5 condition enforces fluid particles on the same magnetic field line to always remain on the same field line).In this paper, we adopt the same strategy in order to study the existence of weak solutions for a resistive variant of the system (2.1). More particularly, we shall insert a finite resistivity in the problem so thatthe total energy Hamiltonian (2.2) decreases in time. Although a complete physical treatment would alsorequire incorporating the collisional effects emerging from the energetic particle dynamics, we shall look ata mathematically more tractable case here by only considering collisional effects in the MHD part of themodel that are associated with the bulk fluid.Now we derive the main focus of this article: a member of the family of resistive Vlasov-MHD models inits CCS variant. This model has appeared in the work of Belova and collaborators [4, 2], who implementedit in the
HYbrid and MHD simulation code (HYM) to support fusion experiments.More particularly, we shall focus on obtaining a consistent Ohm’s law for the electric field E . Theresistive term(s) shall be derived via a standard procedure of adding collisional terms in the fluid momentumequations. We start with the full system of three sets of equations using the notations introduced betweenequations (2.1) and (2.2): • kinetic Vlasov equation for energetic particles: ∂f∂t + v · ∇ x f + a h ( E + v × B ) · ∇ v f = 0; (2.3) • fluid equations for ions ( s = 1) and electrons ( s = 2) with momentum exchange via a friction term R asthe macroscopic description of collisional effects: ̺ s ∂ U s ∂t + ̺ s ( U s · ∇ x ) U s = a s ̺ s ( E + U s × B ) − ∇ x P s + ( − s R , (2.4) ∂̺ s ∂t + ∇ x · ( ̺ s U s ) = 0; and (2.5) • Maxwell equations: ∂ B ∂t = −∇ x × E , (subject to ∇ x · B = 0) , (2.6) µ ǫ ∂ E ∂t = ∇ x × B − µ X s =1 a s ̺ s U s − µ q h K , (2.7) ǫ ∇ x · E = X s =1 a s ̺ s + q h n ; (2.8)where the physical constants q s , m s for s = 1 ,
2, just like their counterparts for energetic particles, denotethe charge and the mass, respectively, of a single ion ( s = 1) or a single electron ( s = 2) and a s := q s /m s denotes the charge-to-mass ratio. By the nature of all relevant physical settings, we have m h and m at thesame scale and m extremely small. Then, by the fact that the energetic particles are very rarefied, we havethe scaling regime m n/̺ ≪ . (2.9)Since q h , q , q are all at the same scale, the above relation implies q h n ≪ a ̺ , which will be directly usedlater.The opposite signs of the collisional/frictional term R in the two momentum equations ensure conser-vation of the total momentum. The detailed derivation of R starting from particle or kinetic description isbeyond the scope of this article, and we only refer to [9, (2.17)], [20, (3.105)] and state that it is of the form R = ν̺ ( U − U ) , ν = C R ̺ , where the positive parameter ν is the Maxwellian-averaged electron-ion collision frequency (despite itsname, ν is in fact the average momentum relaxation rate for the slowly changing Maxwellian distributionof electrons ). Also, the positive parameter C R can be well approximated by a constant for barotropic flowssince both [9, (2.17)] and [20, (3.105)] show that the electron-ion collision frequency ν is proportional tothe number of ions per unit volume, which is, apparently, proportional to the ion density ̺ . The factor BIN CHENG, ENDRE S ¨ULI, AND CESARE TRONCI ( U − U ) in the formula for R is also consistent with the fact that the (macroscopic) collisional effects aredetermined by the (macroscopic) drift velocity of the electrons relative to the ions.We will later insert kinematic viscosity as part of the standard Navier–Stokes equations, which accountsfor collisions amongst ions. Here, we have made the assumption that ions and electrons do not collide with theenergetic particles, which are themselves assumed to be also collisionless. Although this assumption may notbe completely justified, we shall pursue this direction to simplify the problem as much as possible. Further,we perform the same approximations as in standard MHD theory [15]. First, the enormous disparity inmasses m ≫ m allows us to approximate (2.4) with s = 2 by neglecting the left-hand side terms, resultingin = a ̺ ( E + U × B ) − ∇ x P + C R ̺ ̺ ( U − U ) . (2.10)Then, adding this to (2.4) with s = 1 produces ̺ ∂ U ∂t + ̺ ( U · ∇ x ) U = X s =1 a s ̺ s E + X s =1 a s ̺ s U s × B − ∇ x ( P + P ) , (2.11)where the collisional terms ± R cancel out. Next, upon assuming quasi-neutrality by the formal limit ǫ → a ̺ U + a ̺ U + q h K = 1 µ ∇ x × B (= J ) , (2.12)quasi-neutrality: a ̺ + a ̺ + q h n = 0 , (2.13)where J := µ − ∇ x × B denotes the electric current density in the system, and is henceforth always anauxiliary, diagnostic variable. Then, the ion momentum equation (2.11) becomes (with P = P + P ) ̺ ∂ U ∂t + ̺ ( U · ∇ x ) U = − q h n E + ( J − q h K ) × B − ∇ x P . (2.14)Thus we have now reduced the two-fluid model (2.4), (2.5) to a single-fluid model, for which we retain thecontinuity equation only for s = 1 as well. One can then combine these with the kinetic equation (2.3),Faraday’s law (2.6) and use the elementary identities listed in Appendix A to formally prove conservationof the total momentum: Z R (cid:16) ̺ U + Z R v f d v (cid:17) d x and can also formally deduce the rate of change of the total energy; indeed, by considering the Hamiltonian H defined in (2.2), we have thatdd t H ( f, U , B ) = − Z R (cid:16)(cid:0) J − q h K + q h n U ) · E + (cid:0) J − q h K (cid:1) · (cid:0) U × B (cid:1)(cid:17) d x = − Z R (cid:16)(cid:0) J − q h K + q h n U ) · (cid:0) E + U × B (cid:1)(cid:17) d x . (2.15)Next, in order to relate the electric field E to the prognostic unknowns and thus close the system, wecombine Amp`ere’s current balance (2.12) and quasi-neutrality (2.13) to obtain U = − a ̺ + q h n (cid:0) J − q h K − a ̺ U (cid:1) , (2.16)so that, by simple manipulation, U − U = 1 a ̺ + q h n ( J − q h K + q h n U ) . (2.17)On the other hand, by the identity E + U × B = ( U − U ) × B + ( E + U × B ) and the inertia-lesselectron momentum equation (2.10), E + U × B = ( U − U ) × B + 1 a ̺ ∇ x P − C R ̺ a ( U − U ) , EAK SOLUTIONS TO A HYBRID VLASOV-MHD MODEL IN PLASMA PHYSICS 7 and therefore, upon substituting (2.17), we have E + U × B = 1 a ̺ + q h n (cid:0) J − q h K + q h n U (cid:1) × B − a ̺ + q h n ∇ x P − C R ̺ a ( a ̺ + q h n ) ( J − q h K + q h n U ) . Then, we imitate the derivation of ideal MHD [15] and assume that the “Hall effect” J × B and electronpressure gradient ∇ x P are both negligible compared to the Lorentz force a ̺ U × B . This step leads tothe relation E + U × B = q h a ̺ + q h n (cid:0) n U − K (cid:1) × B − C R ̺ a ( a ̺ + q h n ) ( J + q h n U − q h K ) . (2.18)Then, thanks to the assumption (2.9) for energetic particles (so that a ̺ + q h n ≃ a ̺ ) and the fact that a , a have opposite signs, we are justified to consider the resistivity η := − C R / ( a a ) to be a positive constant.Compared to U × B on the left-hand side of (2.18), the term q h a ̺ + q h n ( n U ) × B on the right-hand side isnegligible because of (2.9). Analogously, upon introducing the average particle velocity W = K /n < ∼ U ,the term q h na ̺ + q h n (cid:0) − W (cid:1) × B is also seen to be negligible. Therefore, we are left with the following formula,which we shall refer to as “extended Ohm’s law” (c.f. [25]): E + U × B = η ( J + q h n U − q h K ) . (2.19)Although (2.19) consistently guarantees that the rate of change (2.15) for the total energy is non-positive,the complicated form of the extended Ohm’s law leads to significant difficulties in the mathematical analysisof the model. Thus, we have simplified the problem by invoking, once again, the assumption (2.9) forenergetic particles to obtain the usual form of Ohm’s law: E + U × B = η J . (2.20)However, a consistency issue emerges here: this approximation does not guarantee the nonpositivity of thetime rate (2.15) for the total energy. In order to progress further, we make one additional approximation:we neglect all resistive force terms in the ion momentum equation and kinetic equation, namely we use theideal Ohm’s law E + U × B = 0 in (2.14) and (2.3), but we use the usual Ohm’s law (2.20) in Faraday’s Law(2.6). As has been noticed in [3], this step is needed for momentum conservation and it amounts to definingan effective electric field given by E − η J , where − η J represents the collisional drag on the ions and the hotparticles. Then, this results in an approximation of the kinetic equation (2.3), Faraday’s law (2.6) and theion momentum equation (2.14) by the following current-coupling scheme of resistive Vlasov-MHD (where U = U , and we also incorporate the kinetic viscosity κ , incompressibility and the constant ion density ¯ ̺ ): ∂f∂t + v · ∇ x f + q h m h ( v − U ) × B · ∇ v f = 0 , (2.21a)¯ ̺ (cid:16) ∂ U ∂t + ( U · ∇ x ) U (cid:1) = q h n U × B + ( J − q h K ) × B + κ ∆ x U − ∇ x P , (subject to ∇ x · U = 0) , (2.21b) ∂ B ∂t = ∇ x × ( U × B ) + ηµ ∆ x B , (subject to ∇ x · B = 0) . (2.21c)Here the unknowns are U ( t, x ), f ( t, x , v ), B ( t, x ), and the auxiliary variables involved are P and, as wehave defined before, J = µ − ∇ x × B , n ( t, x ) = Z R f ( t, x , v ) d v , K ( t, x ) = Z R v f ( t, x , v ) d v . The symbols q h , κ, m h , η, µ denote positive physical constants. This quantity is either very low or at most comparable with the MHD fluid velocity U . This is consistent with thehypothesis of energetic particles, since the latter hypothesis involves the temperature rather than the mean velocity. Denotingthe temperatures of the hot and fluid components by T h and T f , respectively, we have T h ≫ T f (see [11]). With the definition ofthe temperature T h = ( m h / nk B ) R R | v − W | f d v (where k B denotes Boltzmann’s constant), the assumption on the energeticcomponent amounts to an assumption on the trace of the second-order moment of the Vlasov density with no assumption onthe mean velocity, which is actually low for hot particles close to isotropic equilibria. BIN CHENG, ENDRE S ¨ULI, AND CESARE TRONCI
This system of hybrid Vlasov-MHD equations is implemented in the HYM code, as has been recentlypresented in [2]. The remainder of this paper is devoted to an analytical study of its parameter-free version(1.1). This system exhibits conservation of total momentum and nonincreasing total energy, thanks toa calculation similar to the one leading to (2.15), which will be discussed in Section 3. These physicalproperties, in fact, play a crucial role in our mathematical analysis of the model.3.
Conservation properties and bounds on the moments
In this section, assuming sufficient regularity of the solution to system (1.1), and 2 π -periodic boundaryconditions with respect to x and suitably rapid decay of f as | v | → ∞ , we shall present formal proofsof various balance laws and energy inequalities. Although subsequently we shall study the system (1.1)only, corresponding to the incompressible case, it is instructive at this point to discuss the (formal) energyequality in the compressible case as well. The argument in the incompressible case will be made rigorouslater on in the paper by fixing the function spaces in which the unknown functions f , U and B are sought.The question of existence of a global weak solution to the compressible model will be studied elsewhere.We also show in Proposition 3.1 that, at any time t ≥
0, the L r ( T ) norm (with a suitable values of r , whose choice will be made clear below,) of the moments of f are bounded in terms of the total energyand the L ∞ ( T × R ) norm of f , both of which will later be rigorously shown not to exceed their respectiveinitial sizes.The equation (1.1a) is a transport equation with divergence-free “velocity fields” with respect to boththe x and v coordinates. That is to say, ∇ x · v = ∇ v · (( v − U ) × B ) = 0 . As a consequence, the L r ( T × R ) norm of f is constant in time for all r ∈ [1 , ∞ ). In addition, one can showby the method of characteristics that the minimum and maximum values of f are preserved in the courseof temporal evolution, and therefore the L ∞ ( T × R ) norm of f is also constant in time; consequently, fora nonnegative initial datum ˚ f the associated solution f remains nonnegative in the course of evolution intime.It is also straightforward to show (formally) the conservation of the total momentum Z T h(cid:16) Z R v f d v (cid:17) + U i d x using the elementary identities from Appendix A.There are three contributions to the total energy of the system: from energetic particles, from the kineticenergy of the bulk fluid, and from the magnetic field. The total energy is therefore defined as follows: E inctotal = E inctotal [ f, U , B ] := Z T (cid:20)(cid:18) Z R | v | f d v (cid:19) + 12 | U | + 12 | B | (cid:21) d x . Assuming that basic physical laws are obeyed, we must have conservation of the total energy when alldissipation terms are set to zero. In order to illustrate the energy budget and the energy exchange betweenthe equations in the system, we introduce the following energy conversion rates : R := Z T (cid:20) ( U × B ) · Z R v f d v (cid:21) d x (energy of the particles to kinetic energy of the fluid); R := Z T ( ∇ x × B ) · ( U × B ) d x (kinetic energy of the fluid to magnetic energy). The electric field E also stores energy, but with our scalings here its contribution is neglected. In order to justify this, wereturn to physical units and consider linear materials with homogeneous permittivity ε and permeability µ , so that D = ε E and H = B /µ . In MHD models, Faraday’s law ∂ t B + ∇ x × E = 0 is used, which implies the scaling law [ B ][ t ] ∼ [ E ][ x ] . Meanwhile,the MHD approximation adopts the zero displacement-current limit of the Maxwell–Amp´ere equation ∇ x × H − J = ∂ t D ≈ ,which implies another scaling law: [ H ][ x ] ≫ [ D ][ t ] . Multiplying these two scaling laws we obtain[ B ][ H ] ≫ [ E ][ D ] . Therefore, the contribution of the electric field to the total electromagnetic energy density, ( E · D + B · H ), is negligible. EAK SOLUTIONS TO A HYBRID VLASOV-MHD MODEL IN PLASMA PHYSICS 9
We shall now decompose dd t E inctotal into its three constituents in order to highlight the roles of these energyconversion rates.(i) Change in the total energy of energetic particles.
Integrating | v | · (1.1a) over T × R and performingintegration by parts yieldsdd t Z T × R | v | f d x d v = Z T × R | v | ∇ v · (( U − v ) × B f ) d x d v = − Z T × R v · (( U − v ) × B f ) d x d v = − Z T (cid:16) Z R v f d v (cid:17) · ( U × B ) d x = −R . (ii) Change in the kinetic energy of the bulk fluid.
Integrating (1.1b) · U over T and performing integrationby parts yieldsdd t Z T | U | d x = − Z T ( ∇ x × B ) · ( U × B ) d x − Z T |∇ x U | + ( ∇ x · U ) d x + Z T ( U × B ) · (cid:16)Z R v f d v (cid:17) d x = −R − Z T |∇ x U | d x + R . (iii) Change in the magnetic energy.
By integrating (1.1c) · B over T , and using the identity ∇ x · ( U ′ × B ) =( ∇ x × U ′ ) · B − ( ∇ x × B ) · U ′ with U ′ := U × B (cf. (A.4)) we obtain, after integrating by parts, thatdd t Z T | B | d x = Z T ( ∇ x × B ) · ( U × B ) d x − Z T |∇ x B | d x = R − Z T |∇ x B | d x . To conclude, for a smooth solution ( f, U , B ) to (1.1), with f decaying sufficiently rapidly as | v | → ∞ ,we have that E inctotal ( t ) = E inctotal (0) − Z t Z T (cid:0) |∇ x U | + |∇ x B | (cid:1) d x d s ∀ t ∈ [0 , T ] , (3.1)as well as k f ( t ) k L r ( T × R ) = k f k L r ( T × R ) ∀ t ∈ [0 , T ] , ∀ r ∈ [1 , ∞ ] . (3.2)These bounds on f then imply the relevant bounds on the moments of f in the following sense. Proposition 3.1.
Consider a measurable nonnegative function f : ( t, x , v ) ∈ [0 , T ] × T × R f ( t, x , v ) ∈ R such that k f ( t, · , · ) k L ∞ ( T × R ) < ∞ for t ∈ [0 , T ], and assume that E par [ f ]( t ) := Z T × R | v | f ( t, x , v ) d x d v < ∞ for t ∈ [0 , T ] . (3.3)Then, the following bounds on the zeroth, first and second moment of f hold for t ∈ [0 , T ]: (cid:13)(cid:13)(cid:13) Z R f ( t, · , v ) d v (cid:13)(cid:13)(cid:13) L ( T ) ≤ C k f ( t, · , · ) k L ∞ ( T × R ) (cid:16) E par [ f ]( t ) (cid:17) ; (3.4) (cid:13)(cid:13)(cid:13) Z R | v | f ( t, · , v ) d v (cid:13)(cid:13)(cid:13) L ( T ) ≤ C k f ( t, · , · ) k L ∞ ( T × R ) (cid:16) E par [ f ]( t ) (cid:17) ; (3.5) (cid:13)(cid:13)(cid:13) Z R | v | f ( t, · , v ) d v (cid:13)(cid:13)(cid:13) L ( T ) ≤ C E par [ f ]( t ) . (3.6)More generally, for any real number k ∈ [0 ,
2] and t ∈ [0 , T ], we have that (cid:13)(cid:13)(cid:13) Z R | v | k f ( t, · , v ) d v (cid:13)(cid:13)(cid:13) L k ( T ) ≤ C k f ( t, · , · ) k − k L ∞ ( T × R ) (cid:16) E par [ f ]( t ) (cid:17) k . (3.7)We note that the bound (3.4) is stronger than the L ( T × R ) integrability of f , and (3.5) is strongerthan the result of applying H¨older’s inequality to the product v √ f √ f over T × R . Proof.
Take any
N > C denote a generic positive constant, independent of N , whose value mayvary from line to line. Then, with ( t, x ) ∈ [0 , T ] × T fixed,0 ≤ Z R f ( t, x , v ) d v = Z | v |≤ N f ( t, x , v ) d v + Z | v | >N f ( t, x , v ) d v ≤ CN k f ( t, · , · ) k L ∞ ( T × R ) + N − Z R | v | f ( t, x , v ) d v . (3.8)Now, again with ( t, x ) ∈ [0 , T ] × T fixed, the right-hand side in the last inequality attains its minimum at N = N ( t, x ) = C (cid:16) Z R | v | f ( t, x , v ) d v / k f ( t, · , · ) k L ∞ ( T × R ) (cid:17) , and therefore 0 ≤ Z R f ( t, x , v ) d v ≤ C k f ( t, · , · ) k L ∞ ( T × R ) (cid:16) Z R | v | f ( t, x , v ) d v (cid:17) . Hence, (cid:13)(cid:13)(cid:13) Z R f ( t, · , v ) d v (cid:13)(cid:13)(cid:13) L ( T ) ≤ C k f ( t, · , · ) k L ∞ ( T × R ) Z T Z R | v | f ( t, x , v ) d v d x ≤ C k f ( t, · , · ) k L ∞ ( T × R ) E par [ f ]( t ) , which directly implies (3.4).The inequality (3.5) is a consequence of (3.4) and H¨older’s inequality (applied twice). That is, with( t, x ) ∈ [0 , T ] × T fixed, we have that (cid:12)(cid:12)(cid:12) Z R | v | f ( t, x , v ) d v (cid:12)(cid:12)(cid:12) ≤ (cid:16) Z R f ( t, x , v ) d v (cid:17) (cid:16) Z R | v | f ( t, x , v ) d v (cid:17) , and therefore, for t ∈ [0 , T ], (cid:13)(cid:13)(cid:13) Z R | v | f ( t, · , v ) d v (cid:13)(cid:13)(cid:13) L ( T ) ≤ Z T (cid:16) Z R f ( t, x , v ) d v (cid:17) (cid:16) Z R | v | f ( t, x , v ) d v (cid:17) d x ≤ (cid:20)Z T (cid:16) Z R f ( t, x , v ) d v (cid:17) · d x (cid:21) (cid:20)Z T (cid:16) Z R | v | f ( t, x , v ) d v (cid:17) · d x (cid:21) = (cid:13)(cid:13)(cid:13) Z R f ( t, · , v ) d v (cid:13)(cid:13)(cid:13) L ( T ) (cid:16) E par [ f ]( t ) (cid:17) . Substituting (3.4) into the right-hand side of the last inequality then yields (3.5).An alternative proof of (3.5) proceeds similarly to that of (3.4): for any
N > t, x ) ∈ [0 , T ] × T ,we have that (cid:12)(cid:12)(cid:12)(cid:12)Z R | v | f ( t, x , v ) d v (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z | v |≤ N | v | f ( t, x , v ) d v + Z | v | >N | v | f ( t, x , v ) d v ≤ CN k f ( t, · , · ) k L ∞ ( T × R ) + N − Z R | v | f ( t, x , v ) d v , which, upon choosing N so that the right-hand side of the last inequality attains its minimum, and thenconsidering the L ( T × R ) norm of the expression on the left-hand side of the resulting inequality, againyields (3.5). Either approach can be adapted to prove both (3.6) and the more general inequality (3.7), ofwhich (3.4)–(3.6) are special cases for k = 0 , ,
2, respectively. (cid:3)
Remark 3.2.
Similar estimates hold if we replace the L ∞ norm on the right-hand side of (3.7), with ageneral L r norm, but we shall not use bounds of this type in our proofs and we therefore omit the detailsof their derivation. EAK SOLUTIONS TO A HYBRID VLASOV-MHD MODEL IN PLASMA PHYSICS 11 Mollified PDE system: existence of solutions via a fixed point method
We shall assume throughout this section that the mollification parameter ε is fixed and 0 < ε ≪
1. Weconsider a nonnegative radially symmetric function θ ∈ C ∞ c ( R ) such that θ ( x ) = 0 for any | x | > , and Z | x |≤ θ ( x ) d x = 1 . The support of the function x ε − θ ( ε − x ) is then contained in the box domain (cid:8) x : | x | ∞ ≤ ε (cid:9) ; let θ ε x denote the 2 π -periodic extension of this function – we shall henceforth consider θ ε x ( x ) for x ∈ T only.The mollification of a 2 π -periodic locally integrable function v ( x ) is defined by convolution with θ ε x andis denoted by the superscript h ε i ; i.e., v h ε i ( x ) := Z T v ( y ) θ ε x ( x − y ) d y . We will use the following property of mollification, which is a consequence of the differentiation propertiesof convolution and H¨older’s inequality: k v h ε i k C m ( T ) ≤ C ε,r,m k v k L r ( T ) for 1 ≤ r ≤ ∞ , ≤ m < ∞ , (4.1)where C ε,r,m = k D m x θ ε x k L r ′ ( T ) , r + r ′ = 1.We also introduce the Banach space of (weakly) divergence-free, square-integrable vector field pairs L , inc ( T ; R ) := n ( U , B ) ∈ L ( T ; R ) (cid:12)(cid:12)(cid:12) Z T U · ∇ x φ d x = Z T B · ∇ x φ d x = 0 for all φ ∈ C ( T ; R ) o , which we equip with the usual L norm. For notational simplicity, when such functions appear within the L norm sign, the superscript “inc” will be omitted from our notation for the norm.4.1. Definition of the mollified mapping.
We proceed by defining a mapping F = F ˚ f, ˚ U , ˚ B : C ([0 , T ]; L , inc ( T ; R ))
7→ C ([0 , T ]; L , inc ( T ; R ))as follows. Note that F depends on the initial data (˚ f , ˚ U , ˚ B ), which are considered as being fixed throughoutSection 4; therefore the dependence of F on the initial data will be, for the sake of brevity, usually omittedfrom our notation.Given ( e U , e B ) ∈ C ([0 , T ]; L , inc ( T ; R )), which are 2 π -periodic with respect to x ,let ( U , B )( t, x ) = F ( e U , e B ) = F ˚ f, ˚ U , ˚ B ( e U , e B ) solve (4.3), (4.4) (4.2)in the sense that ∂ t f + v · ∇ x f = (( e U h ε i − v ) × e B h ε i ) · ∇ v f , (4.3a)with C ∞ c ( T × R ; R ) initial datum: f (cid:12)(cid:12)(cid:12) t =0 = ˚ f ≥ , (4.3b)where f is assumed to be 2 π -periodic with respect to x for all ( t, v ) ∈ [0 , T ] × R , while ˚ f is assumed to be2 π -periodic with respect to x and compactly supported in T × R ; ∂ t U + ( e U h ε i · ∇ x ) U − ( e B h ε i · ∇ x ) B − ∆ x U + ∇ x P = U × e B h ε i (cid:16) Z R f d v (cid:17) + (cid:16) e B h ε i × Z R v f d v (cid:17) h ε i (subject to ∇ x · U = 0); (4.4a) ∂ t B + ( e U h ε i · ∇ x ) B − ( e B h ε i · ∇ x ) U − ∆ x B + ∇ x P B = 0 (subject to ∇ x · B = 0); (4.4b)with 2 π -periodic divergence-free C ∞ ( T ; R ) initial data: ( U , B ) (cid:12)(cid:12)(cid:12) t =0 = (˚ U , ˚ B ) , (4.4c)and with ( U , B ) subject to 2 π -periodic boundary conditions with respect to x . This system may be solvedas follows: first we solve for f from the linear equation (4.3); we then treat f as given and solve for ( U , B )from the linear system (4.4). Once ( f , U , B ) is found (in a function space to be made precise in Lemma4.1 below), the auxiliary pressure variables P , P B can be recovered by a standard procedure (for example,taking divergence of (4.4a), applying ∇ x · U = 0 and inverting the Laplacian gives P uniquely up to aconstant), and are henceforth not considered as part of the solution. We note that no decay hypotheses need to be imposed on f when | v | → ∞ , since (as will be shownbelow) the assumption that ˚ f is compactly supported guarantees that f ( t, · , · ) is also compactly supportedfor all t ∈ (0 , T ].We also note that the mollification in the second product on the right-hand side of (4.4a) is intentional. Itis to ensure that one can conveniently estimate the energy exchange between (4.3) and (4.4), and eventuallyeliminate from the energy equality the terms representing energy exchanges; c.f. Remark 4.3 below. Lemma 4.1.
Consider any
T > that is independent of ε . Let ( e U , e B ) ∈ C ([0 , T ]; L , inc ( T ; R )) . Then,the system (4.3) , (4.4) , subject to the initial conditions specified therein, admits a solution ( f , U , B ) thatsatisfies f ∈ C ([0 , T ]; C m ( T × R )) and ( U , B ) ∈ C ([0 , T ]; C m ( T ; R )) for all m ≥ . (4.5) Moreover, • f ≥ , and there exists a scalar-valued mapping G ( · , · , · ) such that it is monotonically increasing withrespect to its first and third arguments, and f ( t, x , v ) = 0 for all | v | > G (cid:18) T, ˚ f , max [0 ,T ] × T (cid:8) | e U h ε i | , | e B h ε i | (cid:9)(cid:19) and all ( t, x ) ∈ [0 , T ] × T ; (4.6) • we have, for all t ∈ [0 , T ] , that k f ( t ) k L r ( T × R ) = k ˚ f k L r ( T × R ) , r ∈ [1 , ∞ ]; (4.7) • and the following energy equality holds for all t ∈ [0 , T ]:12 (cid:0) k U ( t ) k L ( T ) + k B ( t ) k L ( T ) (cid:1) + E par [ f ]( t ) + Z t (cid:0) k∇ x U ( s ) k L ( T ) + k∇ x B ( s ) k L ( T ) (cid:1) d s = 12 (cid:0) k ˚ U k L ( T ) + k ˚ B k L ( T ) (cid:1) + E par [˚ f ] + Z t R ( s ) d s, where R ( s ) := Z T ( U ( s, x ) − e U ( s, x )) h ε i · (cid:16) e B h ε i ( s, x ) × Z R v f ( s, x , v )d v (cid:17) d x , (4.8) with the notation E par [ · ] defined in (3.3) .Proof. First, by the properties of the mollifier (4.1), we have that( e U h ε i , e B h ε i ) ∈ C ([0 , T ]; C m ( T ; R )) for all m ≥ ( ∂ t X ( t ; x , v ) = V ( t ; x , v ) , X (0; x , v ) = x ,∂ t V ( t ; x , v ) = − ( e U h ε i ( t, X ( t ; x , v )) − V ( t ; x , v ) × e B h ε i ( t, X ( t ; x , v )) , V (0; x , v ) = v , (4.9)with ( X , V ) ∈ T × R . As ( e U h ε i , e B h ε i ) ∈ C ([0 , T ]; C m ( T ; R )) for all m ≥
0, the right-hand side of (4.9)and their D X , V , D X , V , . . . derivatives are continuous and grow at most linearly in | V | ; in fact, they arebounded by a | V | + b , where a, b only depend on suitable C ([0 , T ]; C m ( T ; R )) norms of ( e U h ε i , e B h ε i ). Then,by applying classical results from the theory of ordinary differential equations to the system (4.9) and to its D x , v , D x , v , . . . derivatives, we deduce that, for any initial data ( x , v ) ∈ T × R , (4.9) admits a uniquesolution, all of whose D x , v , D x , v , . . . derivatives are continuous functions defined in [0 , T ] × T × R (see, for example, [18, Corollary 4.1 on p.101]). Moreover, this reasoning is time-reversible, so that for any t ∈ [0 , T ], ( X ( t ; · , · ) , V ( t ; · , · )) has a unique inverse, which we denote by ( X − t ( · , · ) , V − t ( · , · )), all of whose D x , v , D x , v , . . . derivatives are continuous in [0 , T ] × T × R . All in all, f ( t, x , v ) := ˚ f (cid:0) X − t ( x , v ) , V − t ( x , v ) (cid:1) ≥ D mx,p f ∈ C ([0 , T ]; C ( T × R )) for all m ≥ f has been assumed to have compact support in T × R , there exists a positive realnumber C = C (˚ f ) such that f (0 , x , v ) = 0 for all | v | > C and all x ∈ T . (4.10) EAK SOLUTIONS TO A HYBRID VLASOV-MHD MODEL IN PLASMA PHYSICS 13
By the second equation of (4.9), we have, with C := max [0 ,T ] × T (cid:8) | e U h ε i | , | e B h ε i | (cid:9) ,12 dd t | V ( t ; x , v ) | = − V · (cid:16) ( e U h ε i × e B h ε i )( t, X ( t ; x , v )) (cid:17) ≤ | V | + 12 C , which then implies, for any v ∈ R such that | v | ≤ C , that | V ( t ; x , v ) | ≤ C e T + (e T − C =: G ∀ t ∈ [0 , T ] . Together with (4.10), this immediately implies that f ( t, x , v ) = 0 for all v ∈ R such that | v | > G and all( t, x ) ∈ [0 , T ] × T , as has been stated in (4.6). The monotonicity of G = G (cid:0) T, ˚ f , max [0 ,T ] × T (cid:8) | e U h ε i | , | e B h ε i | (cid:9)(cid:1) as stated above (4.6) also follows.Next we shall prove (4.7). The construction of f implies that the function t ∈ [0 , T ]
7→ k f ( t, · , · ) k L ∞ ( T × R ) is constant. For r ∈ [1 , ∞ ), the fact that t ∈ [0 , T ]
7→ k f ( t, · , · ) k L r ( T × R ) is constant follows from integrating | f | r − times (4.3a) over T × R and using that ∇ x · v = ∇ v · (( e U h ε i − v ) × e B h ε i ) = 0 together with thedivergence theorem, the compact support of f , and the 2 π -periodicity of f with respect to x .Next, we substitute ( f , e U h ε i , e B h ε i ) into the linear(ized) MHD system (4.4). Thanks to the smoothnessand compactness of the support of f ( t, · , · ), the hypotheses of Lemma B.1 are satisfied. Consequently, wehave that ( U , B ) ∈ C ([0 , T ]; C m ( T ; R )) for all m ≥ t ∈ [0 , T ],12 ( k U ( t ) k L ( T ) + k B ( t ) k L ( T ) ) + Z t (cid:0) k∇ x U ( s ) k L ( T ) + k∇ x B ( s ) k L ( T ) (cid:1) d s = 12 ( k U (0) k L ( T ) + k B (0) k L ( T ) ) + Z t Z T U h ε i · (cid:16) e B h ε i × Z R v f d v (cid:17) d x d s. (4.11)Note that in the last term we transferred the mollifier h ε i onto U .Also, by multiplying (4.3a) with | v | , integrating over T × R , and performing integration by parts(which is justified, since f ( t, x , · ) is compactly supported in R and f ( t, · , v ) is 2 π -periodic in x ), we obtaindd t Z T × R | v | f d v d x = Z T × R | v | ∇ v · (( e U h ε i − v ) × e B h ε i f ) d v d x = − Z T × R v · (( e U h ε i − v ) × e B h ε i f ) d v d x = − Z T (cid:18)Z R v f d v (cid:19) · ( e U h ε i × e B h ε i ) d x = − Z T e U h ε i · (cid:16) e B h ε i × Z R v f d v (cid:17) d x . We then integrate this equality from 0 to t ∈ (0 , T ] and add (4.11) to it to complete the proof of (4.8). (cid:3) Verification of the hypotheses of Schauder’s fixed point theorem.
Having shown that themapping F is correctly defined, we shall next apply the following version of Schauder’s fixed point theorem. Theorem 4.2 (Schauder’s fixed point theorem) . Suppose that K is a convex subset of a topological vectorspace and F is a continuous mapping of K into itself such that the image F ( K ) is contained in a compactsubset of K ; then, F has a fixed point. Here and below, compactness is in the strong sense, unless stated otherwise.
Remark 4.3.
When F admits a fixed point, i.e., ( e U , e B ) = ( U , B ) = F ( e U , e B ), the R term in (4.8) vanishes,and one recovers the usual energy law (3.1) for the mollified system. Our next objective is therefore to show,by applying Schauder’s fixed point theorem, that the mollified system has a solution. Once we have doneso, we shall pass to the limit ε → ε > G ( · , · , · ) specified therein.Together with the property of mollification k ( e U h ε i , e B h ε i ) k C ([0 ,T ]; C m ( T )) ≤ C ε,m k ( e U , e B ) k C ([0 ,T ]; L ( T )) , thisyields f ( t, x , v ) = 0 for | v | ≥ G (cid:16) T, ˚ f , C ε, k ( e U , e B ) k C ([0 ,T ]; L ( T )) (cid:17) , for any admissible ˚ f and t ∈ [0 , T ] . (4.12) The next three lemmas are concerned with verifying the hypotheses of Schauder’s fixed point theoremfor the mapping F defined in (4.2). We begin, in the next lemma, by proving the continuity of F . Lemma 4.4.
For any
T > that is independent of ε , the mapping F defined in (4.2) subject to fixed initialdata (˚ f , ˚ U , ˚ B ) is continuous from the Banach space C ([0 , T ]; L , inc ( T ; R )) into itself.Proof. For i = 1 ,
2, consider ( e U i , e B i ) ∈ C ([0 , T ]; L , inc ( T ; R )) and the associated ( U i , B i ) and f i . Clearly,Lemma 4.1 guarantees that, for i = 1 ,
2, ( U i , B i ) ∈ C ([0 , T ]; L , inc ( T ; R )) and f i , U i , B i are smooth.Suppose further that δ := (cid:13)(cid:13)(cid:13) ( e U − e U , e B − e B ) (cid:13)(cid:13)(cid:13) C ([0 ,T ]; L ( T )) ≪ . (4.13)Let ( e U , e B ) be fixed so that ( f , U , B ) are all fixed. Then, our goal is to show that ( U , B ) → ( U , B )strongly in C ([0 , T ]; L ( T ; R )) as δ → f := f − f and likewise for U , B . First, by (4.13) and the properties of the mollification (4.1), we have thatlim δ → ( e U h ε i , e B h ε i ) = ( e U h ε i , e B h ε i ) strongly in C ([0 , T ] × T ; R ) . (4.14)Next, by (4.3), the governing equation for f is ∂ t f + v · ∇ x f = (( e U h ε i − v ) × e B h ε i ) · ∇ v f + (cid:16) ( e U h ε i i − v ) × e B h ε i i (cid:17) (cid:12)(cid:12)(cid:12) i =1 i =2 · ∇ v f , with initial datum f (cid:12)(cid:12)(cid:12) t =0 = 0. By the method of characteristics, similarly to (4.9) and the argumentthereafter, one can show that k f k C ([0 ,T ] × T × R ) ≤ T (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) ( e U h ε i i − v ) × e B h ε i i (cid:17) (cid:12)(cid:12)(cid:12) i =1 i =2 · ∇ v f (cid:13)(cid:13)(cid:13)(cid:13) C ([0 ,T ] × T × R ) . Since f is fixed and smooth (c.f. (4.5)), thanks to the compactness of its support, as specified in (4.12), andthe above estimate together with f (cid:12)(cid:12)(cid:12) t =0 = 0, we have that f also has compact support in [0 , T ] × T × R that is independent of δ . Therefore, we combine the last estimate and (4.14) to obtainlim δ → Z R f d v = 0 and lim δ → Z R v f d v = , (4.15)strongly in C ([0 , T ] × T ) and C ([0 , T ] × T ; R ), respectively.We move on to (4.4) and write the governing equations for ( U , B ) as ∂ t U + e U h ε i · ∇ x U − e B h ε i · ∇ x B − ∆ x U − U × e B h ε i Z R f d v + ∇ x P = − (cid:16) e B h ε i i × Z R v f i d v (cid:12)(cid:12)(cid:12) i =1 i =2 (cid:17) h ε i − e U h ε i · ∇ x U + e B h ε i · ∇ x B + U × e B h ε i i Z R f i d v (cid:12)(cid:12)(cid:12) i =1 i =2 (4.16a)(subject to ∇ x · U = 0) ,∂ t B + e U h ε i · ∇ x B − e B h ε i · ∇ x U − ∆ x B + ∇ x ( P B ) = − e U h ε i · ∇ x B + e B h ε i · ∇ x U (4.16b)(subject to ∇ x · B = 0) , with initial data ( U , B ) (cid:12)(cid:12)(cid:12) t =0 = ( , ) . Since ( U , B ) is fixed and smooth, we use (4.14) and (4.15) to deduce thatthe right-hand sides of (4.16a), (4.16b) converge to strongly in C ([0 , T ] × T ; R ) as δ → . (4.17)Finally, we invoke (B.2) of Lemma B.1 and use the Cauchy–Schwarz inequality to bound the right-handside of (B.2) by T max [0 ,T ] (cid:0) k U k L ( T ) k h k L ( T ) + k B k L ( T ) k h k L ( T ) (cid:1) . We then combine this with (4.16) and(4.17) to finally deduce thatlim δ → ( U , B ) = ( U , B ) strongly in C ([0 , T ]; L ( T ; R )) . That completes the proof of the lemma. (cid:3)
EAK SOLUTIONS TO A HYBRID VLASOV-MHD MODEL IN PLASMA PHYSICS 15
Next, we will verify the endomorphism hypothesis in Schauder’s fixed point theorem, for which theenergy equality (4.8) plays a key role. For a >
0, we introduce the following convex set: K [ T, a ] := n ( U , B ) ∈ C ([0 , T ]; L , inc ( T ; R )) (cid:12)(cid:12)(cid:12) k ( U , B ) k C ([0 ,T ]; L ( T )) ≤ a o . We also recall the definition of E par [ · ] stated in (3.3), and the definition of E inctotal [ f, U , B ] at the start ofSection 3. For brevity, let ˚ E := E inctotal [˚ f , ˚ U , ˚ B ] . Lemma 4.5.
For a fixed ε > , there exists a constant C ε > that only depends on ε and such that, with T ♭ = T ♭ (cid:0) ε, | ˚ f | L ∞ ( T × R ) , ˚ E (cid:1) := C ε | ˚ f | − L ∞ ( T ) (2˚ E ) − , (4.18) the mapping F , defined in (4.2) subject to fixed initial data (˚ f , ˚ U , ˚ B ) , maps the convex set K [ T ♭ , E ] ⊂C ([0 , T ♭ ]; L , inc ( T ; R )) into itself. Furthermore, for any t ∈ [0 , T ♭ ] , (cid:0) k U ( t ) k L ( T ) + k B ( t ) k L ( T ) (cid:1) + E par [ f ]( t ) + Z t (cid:0) k∇ x U k L ( T ) + k∇ x B k L ( T ) (cid:1) d s ≤ E. (4.19) Proof.
In order to avoid the trivial case of zero initial data, we only consider T ♭ < ∞ . We begin by choosingany ( e U , e B ) ∈ K [ T ♭ , E ], i.e, a pair that satisfies the bound k ( e U , e B ) k C ([0 ,T ♭ ]; L ( T )) ≤ E. (4.20)Next, in the definition of R featuring in the energy equality (4.8), we transfer the first mollifier h ε i onto thesecond factor and apply the Cauchy–Schwarz inequality to obtain | R | ≤ k U − e U k L ( T ) (cid:13)(cid:13)(cid:13)(cid:2) e B h ε i × Z R v f d v (cid:3) h ε i (cid:13)(cid:13)(cid:13) L ( T ) . Then, to estimate the second factor, we apply twice the property of mollification stated in (4.1) togetherwith H¨older’s inequality, to deduce the existence of a constant C ε , which depends on ε , such that (cid:13)(cid:13)(cid:13)(cid:2) e B h ε i × Z R v f d v (cid:3) h ε i (cid:13)(cid:13)(cid:13) L ( T ) ≤ C ε k e B k L ( T ) (cid:13)(cid:13)(cid:13) Z R v f d v (cid:13)(cid:13)(cid:13) L ( T ) ≤ C ε k e B k L ( T ) E par [ f ] | ˚ f | L ∞ ( T ) , (4.21)where the last inequality follows from Proposition 3.1 and the invariance property (4.7). Therefore, | R ( s ) | ≤ C ε (cid:0) k U ( s ) k L ( T ) + k e U ( s ) k L ( T ) (cid:1) k e B ( s ) k L ( T ) [ E par [ f ]( s )] | ˚ f ( s ) | L ∞ ( T ) , s ∈ [0 , T ] . Substituting this bound on R into (4.8), noting (4.20), we arrive at12 (cid:0) k U ( t ) k L ( T ) + k B ( t ) k L ( T ) (cid:1) + E par [ f ]( t ) + Z t (cid:0) k∇ x U ( s ) k L ( T ) + k∇ x B ( s ) k L ( T ) (cid:1) d s ≤ ˚ E + Z t C ε (cid:0) k U ( s ) k L ( T ) + (4˚ E ) (cid:1) (4˚ E ) [ E par [ f ]( s )] | ˚ f ( s ) | L ∞ ( T ) d s, t ∈ [0 , T ] . We see that the expression on the left-hand side of inequality (4.19) is a continuous function of t , whosevalue at t = 0 is strictly less than E on the right-hand side of (4.19). For nontriviality, we only considerthe case when the left-hand side of (4.19) equals 2˚ E at least once, at a certain positive time, so that we candefine T ♯ > earliest time at which this happens. We then set t = T ♯ in the above estimate, whichmakes the left-hand side equal 2˚ E , i.e.,2˚ E ≤ ˚ E + Z T ♯ C ε (cid:0) k U ( s ) k L ( T ) + (4˚ E ) (cid:1) (4˚ E ) [ E par [ f ]( s )] | ˚ f | L ∞ ( T ) d s. The minimality of T ♯ also means that, for all t ∈ [0 , T ♯ ], the estimate k U ( t ) k L ( T ) + E par [ f ]( t ) ≤ E holds.Thus, continuing from the last inequality, we obtain2˚ E ≤ ˚ E + T ♯ C ε (cid:0) (4˚ E ) + (4˚ E ) (cid:1) (4˚ E ) (2˚ E ) | ˚ f | L ∞ ( T ) , which implies that T ♯ ≥ T ♭ , with T ♭ as defined in (4.18) (upon redefining C ε ). Since, T ♯ > t ∈ [0 , T ♭ ]. The proof is complete. (cid:3) Finally, to verify the compactness condition (in the strong topology) in Schauder’s fixed point theorem,we shall use the Aubin–Lions–Simon lemma (see, for example, [24, Corollary 4 on p.85]).
Theorem 4.6 (Aubin–Lions–Simon lemma) . Let X , X and X − be three Banach spaces with compactembedding X ֒ → ֒ → X and continuous embedding X ֒ → X − . For ≤ r, s ≤ ∞ and positive constants C , C , consider the set S := n u (cid:12)(cid:12) k u k L r ((0 ,T ); X ) ≤ C , k ∂ t u k L s ((0 ,T ); X − ) ≤ C o . Then, the following statements hold: (i) If r < ∞ , then S is compact in L r ((0 , T ); X ) ; (ii) If r = ∞ and s > , then S is compact in C ([0 , T ]; X ) . Lemma 4.7.
Suppose that ε > . With the same hypotheses and notations as in Lemma 4.5, the image ofthe convex set K := K h T ♭ , E i ⊂ C ([0 , T ♭ ]; L , inc ( T ; R )) under F is contained in a compact subset of K .Proof. By applying (B.3) of Lemma B.1 to the system (4.3), (4.4), we deduce that, for all t ∈ (0 , T ♭ ], k ( ∇ x U , ∇ x B ) k L ( T ) (cid:12)(cid:12)(cid:12) t + Z t k ( ∂ t U , ∂ t B ) k L ( T ) d s ≤ [0 ,t ] × T {| e U h ε i | , | e B h ε i | , | g | , } Z t k ( ∇ x U , ∇ x B , U , h ) k L ( T ) d s, (4.22)where g := e B h ε i Z R f d v , h := (cid:16) e B h ε i × Z R v f d v (cid:17) h ε i . One can then derive an upper bound on the right-hand side of (4.22) that only depends on ε and the initialdata as follows. First, combining property (4.1) and the estimates (4.19)–(4.21) in Lemma 4.5 and its proof,we establish bounds onmax [0 ,t ] × T {| e U h ε i | , | e B h ε i | } and Z t k ( ∇ x U , ∇ x B , U , h ) k L ( T ) d s, where 0 < t ≤ T ♭ . It remains to bound the maximum of | g | , which requires bounding the L ∞ norm of Z R f d v over [0 , T ♭ ] × T . This cannot be done using Proposition 3.1; instead, one can obtain the desired bound byrecalling the compact support (4.12) and the L ∞ x , v invariance property (4.7).All in all, we have obtained bounds on k ( U , B ) k L ∞ ((0 ,T ♭ ); H ( T )) and k ∂ t ( U , B ) k L ((0 ,T ♭ ); L ( T )) that onlydepend on ε and the initial data. Therefore, by item (ii) of Theorem 4.6 and recalling the compact Sobolevembedding H ( T ) ֒ → ֒ → L ( T ), we complete the proof of the lemma. (cid:3) Classical solution of the mollified system.
We have thus shown that F = F ˚ f, ˚ U , ˚ B : C ([0 , T ♭ ]; L , inc ( T ; R )) → C ([0 , T ♭ ]; L , inc ( T ; R )) , defined in (4.2), satisfies the three hypotheses of Theorem 4.2, which are: • continuity (by Lemma 4.4); • endomorphism (by Lemma 4.5); and • compactness (by Lemma 4.7).We therefore deduce from Theorem 4.2 that F has a fixed point ( U , B ) in the space C ([0 , T ♭ ]; L , inc ( T ; R )),where T ♭ is no less than (recalling (4.18)) T ♭ = T ♭ (cid:0) ε, | ˚ f | L ∞ ( T × R ) , ˚ E (cid:1) = C ε | ˚ f | − L ∞ ( T ) (2˚ E ) − > , EAK SOLUTIONS TO A HYBRID VLASOV-MHD MODEL IN PLASMA PHYSICS 17 and also ( U , B ) and the associated f satisfy the smoothness properties (4.5). By the left-continuity of themapping t ∈ [0 , T ♭ ] ( U ( t ) , B ( t )) ∈ L ( T ; R ), we can repeat the same argument inductively for the timeintervals [ nT ♭ , ( n + 1) T ♭ ] of equal length T ♭ , for n = 0 , , , . . . , [ T /T ♭ ] + 1, in order to reach the endpoint T of the time interval [0 , T ]. The success of this inductive process is guaranteed by the following facts, whichare independent of n : • f remains compactly supported thanks to (4.12); • the first argument ε of T ♭ ( · , · , · ) is fixed and the second argument | ˚ f | L ∞ ( T × R ) of T ♭ ( · , · , · ) is alsoconstant thanks to (4.7); and • the third argument ˚ E of T ♭ ( · , · , · ) is nonincreasing (as the induction step n increases) thanks to theenergy equality (4.8) with fixed point e U = U so that R ≡
0; therefore, the value of T ♭ is nondecreasing and we can thus fix it as its initial value at n = 0 without affecting the final conclusion.Having shown the existence of a fixed point in C ([0 , T ]; L , inc ( T ; R )) for the mapping F = F ˚ f, ˚ U , ˚ B , wecan set ( U , B ) = ( e U , e B ) = ( U ε , B ε ) and f = f ε , which also makes ( e U h ε i , e B h ε i ) = ( U h ε i ε , B h ε i ε ), in (4.3), (4.4)and associated results (especially in Lemma 4.1) to deduce the following main result of this section. Theorem 4.8.
Consider the following mollified hybrid Vlasov-MHD system, with fixed ε > : ∂ t f ε + v · ∇ x f ε = (cid:0) ( U h ε i ε − v ) × B h ε i ε (cid:1) · ∇ v f ε , (4.23a) with initial datum f ε (cid:12)(cid:12)(cid:12) t =0 = ˚ f ε ∈ C ∞ c ( T × R ) , (4.23b) where ˚ f ε is compactly supported in T × R ; and ∂ t U ε + ( U h ε i ε · ∇ x ) U ε − ( B h ε i ε · ∇ x ) B ε − ∆ x U ε + ∇ x P = U ε × B h ε i ε (cid:16) Z R f ε d v (cid:17) + (cid:16) B h ε i ε × Z R v f ε d v (cid:17) h ε i (subject to ∇ x · U ε = 0) , (4.24a) ∂ t B ε + ( U h ε i ε · ∇ x ) B ε − ( B h ε i ε · ∇ x ) U ε − ∆ x B ε + ∇ x P B = 0 (subject to ∇ x · B ε = 0) , (4.24b) with π -periodic divergence-free initial data ( U ε , B ε ) (cid:12)(cid:12)(cid:12) t =0 = (˚ U ε , ˚ B ε ) ∈ C ∞ ( T ) . (4.24c) Then, for any
T > , the above system admits a classical solution f ε ∈ C ([0 , T ]; C m ( T × R )) and ( U ε , B ε ) ∈ C ([0 , T ]; C m ( T ; R )) for all m ≥ .Moreover, for all t ∈ (0 , T ] , we have the invariance k f ε ( t ) k L r ( T × R ) = k ˚ f ε k L r ( T × R ) , r ∈ [1 , ∞ ]; (4.25) and the following energy equality holds: E inctotal [ f ε , U ε , B ε ]( t ) + k ( ∇ x U ε , ∇ x B ε ) k L ([0 ,t ]; L ( T )) = E inctotal [˚ f ε , ˚ U ε , ˚ B ε ] , (4.26) where E inctotal [ f, U , B ]( t ) := 12 (cid:0) k U ( t ) k L ( T ) + k B ( t ) k L ( T ) (cid:1) + E par [ f ]( t ) . In particular, the remainder term R in (4.8) vanishes from the energy equality (4.26) thanks to the fixedpoint property U = e U = U ε . Also, due to the same reasoning as below (4.4), the auxiliary variables P , P B are not considered to be part of the solution.5. Proof of the main result: global existence of weak solutions
In this section we prove the main result of this article, Theorem 5.8, by showing that, as ε →
0, asubsequence of { ( f ε , U ε , B ε ) } ε> that solves the mollified system formulated in the previous section convergesto a weak solution ( f, U , B ) that solves the original incompressible hybrid Vlasov-MHD model (1.1), andthat this weak solution exists globally in time, i.e., for all nonnegative times.Throughout this section, the initial data of the mollified system (4.23), (4.24) will be constructed fromthe original initial data, (1.2), as follows:˚ U ε = ˚ U h ε i , ˚ B ε = ˚ B h ε i , ˚ f ε = (cid:0) ˚ f · χ ( | v |≤ /ε ) ( v ) (cid:1) ∗ ( θ ε x θ ε v ) . (5.1) Here, χ ( | v |≤ /ε ) is the cut-off function taking the value 1 in the ball of radius 1 /ε in R and 0 otherwise, andthe mollifier θ ε v = θ ε v ( v ) := ε − θ ( ε − v ) for v ∈ R where θ (and also θ ε x ) has been defined at the start ofSection 4.We will work with the weak formulation of the mollified system (4.23), (4.24), which is constructed asfollows. We fix any T >
0. The test functions used in this section are: scalar-valued, compactly supportedfunctions g ∈ C c ([ − , T + 1] × T × R ), and R -valued functions V ∈ C ([ − , T + 1] × T ; R ) satisfying ∇ x · V = 0.For any t ∈ (0 , T ], we multiply (4.23a) by g and integrate over [0 , t ] × T × R , and we take the dotproduct of (4.24a) and (4.24b) with V , and then integrate both over [0 , t ] × T . By performing integrationsby parts, noting that ∇ v · (cid:0) ( U h ε i ε − v ) × B h ε i ε (cid:1) = ∇ x · U h ε i ε = ∇ x · B h ε i ε = 0 , and observing that, because of the periodic boundary conditions with respect to x and thanks to thecompactness of the support of f ε with respect to v , all “boundary terms” arising in the course of the partialintegrations are annihilated, we obtain the following weak formulation of the system (4.23), (4.24), where“:” denotes the scalar product in R × : Z T × R f ε ( t, x , v ) g ( t, x , v ) d x d v − Z T × R ˚ f ε ( x , v ) g (0 , x , v ) d x d v − Z t Z T × R f ε ∂ t g d x d v d s = Z t Z T × R ( v f ε ) · ∇ x g − (cid:0) ( U h ε i ε − v ) × B h ε i ε f ε (cid:1) · ∇ v g d x d v d s (5.2a)for all g ∈ C c ([ − , T + 1] × T × R ); Z T U ε ( t, x ) · V ( t, x ) d x − Z T ˚ U ε ( x ) · V (0 , x ) d x − Z t Z T U ε · ∂ t V d x d s = Z t Z T (cid:0) U h ε i ε ⊗ U ε − B h ε i ε ⊗ B ε (cid:1) : ∇ x V d x d s − Z t Z T ∇ x U ε : ∇ x V d x d s + Z t Z T (cid:0) U ε × B h ε i ε (cid:1)(cid:16) Z R f ε d v (cid:17) · V + (cid:16) B h ε i ε × Z R v f ε d v (cid:17) h ε i · V d x d s (5.2b)for all V ∈ C ([ − , T + 1] × T ; R ) satisfying ∇ x · V = 0; and Z T B ε ( t, x ) · V ( t, x ) d x − Z T ˚ B ε ( x ) · V (0 , x ) d x − Z t Z T B ε · ∂ t V d x d s = Z t Z T (cid:0) U h ε i ε ⊗ B ε − B h ε i ε ⊗ U ε (cid:1) : ∇ x V d x d s − Z t Z T ∇ x B ε : ∇ x V d x d s (5.2c)for all V ∈ C ([ − , T + 1] × T ; R ) satisfying ∇ x · V = 0.As a classical solution, whose existence is guaranteed by Theorem 4.8, is thereby automatically a weaksolution (in the sense of (5.2a)–(5.2c)), we directly deduce the existence of a triple ( f ε , U ε , B ε ) satisfying(5.2a)–(5.2c) for the initial data (5.1) under consideration.Next, we summarize, without proof, some standard properties of mollifiers, which will be extensivelyused in the course of the discussion that follows. Lemma 5.1.
Suppose that Ω is one of T , R or T × R , and let θ ε denote one of θ ε x , θ ε v or θ ε x θ ε v ,respectively, with ε > . Let X denote one of L r (Ω) or L s ((0 , T ); L r (Ω)) , where r, s ∈ [1 , ∞ ) and T > .Then, for any w ∈ X , lim ε → k w ∗ θ ε − w k X = 0 and k w ∗ θ ε k X ≤ k w k X ; (5.3a) for any w ∈ X , { w n } n ≥ ⊂ X , and any sequence { ε n } n ≥ of positive real numbers,if lim n →∞ ε n = 0 and lim n →∞ k w n − w k X = 0 , then lim n →∞ k w n ∗ θ ε n − w k X = 0 . (5.3b)Here, (5.3b) follows from (5.3a) and the triangle inequality k w n ∗ θ ε n − w k X ≤ k w n ∗ θ ε n − w ∗ θ ε n k X + k w ∗ θ ε n − w k X . EAK SOLUTIONS TO A HYBRID VLASOV-MHD MODEL IN PLASMA PHYSICS 19
It then follows thatlim ε → (cid:13)(cid:13) (˚ U ε , ˚ B ε ) − (˚ U , ˚ B ) (cid:13)(cid:13) L ( T ) = 0 and for r ∈ [1 , ∞ ) , lim ε → k ˚ f ε − ˚ f k L r ( T × R ) = 0 . (5.4)Also, k ˚ f ε k L r ( T × R ) ≤ k ˚ f k L r ( T × R ) , for all r ∈ [1 , ∞ ], including the L ∞ norm. Thus, thanks to the L r invariance property (4.25), we have the uniform L r bounds k f ε ( t ) k L r ( T × R ) ≤ k ˚ f k L r ( T × R ) for all t ∈ [0 , T ] , r ∈ [1 , ∞ ] . (5.5)Concerning the initial energy of particles E par [˚ f ε ], by shifting the mollifier under the integral sign, wehave that E par [˚ f ε ] = Z T × R (cid:0) | v | ∗ ( θ ε x θ ε v ) (cid:1) (cid:0) ˚ f · χ ( | v |≤ /ε ) (cid:1) d v d x ≤ Z T × R (cid:0) | v | ∗ ( θ ε x θ ε v ) (cid:1) ˚ f d v d x . Since θ ε v is an even function with unit integral over R , we have | v | ∗ ( θ ε x θ ε v ) = Z R Z T | v − w | θ ε v ( w ) θ ε x ( y ) d y d w = Z R ( | v | + | w | ) θ ε v ( w ) d w = | v | + C θ ε , where C θ := R R | x | θ ( x ) d x < θ ( · ) at the start of Section 4). Hence we deduce that E par [˚ f ε ] ≤ E par [˚ f ] + ε | ˚ f | L ( T × R ) . Combining this with (5.3a) and recalling the definition of E inctotal we have that E inctotal [˚ f ε , ˚ U ε , ˚ B ε ] ≤ E inctotal [˚ f , ˚ U , ˚ B ] + ε | ˚ f | L ( T × R ) . (5.6)By further considering the energy equality (4.26) and the uniform L ∞ bound in (5.5), we obtain E inctotal [ f ε , U ε , B ε ]( t ) + k ( ∇ x U ε , ∇ x B ε ) k L ((0 ,T ); L ( T )) + k f ε ( t ) k C ( T × R ) ≤ ˚ F , (5.7)for every t ∈ [0 , T ] and ε ∈ (0 , F := E inctotal [˚ f , ˚ U , ˚ B ] + | ˚ f | L ∞ ( T × R ) + | ˚ f | L ( T × R ) . Then, by Proposition 3.1 we also have that (cid:13)(cid:13)(cid:13) Z R f ε d v (cid:13)(cid:13)(cid:13) C ([0 ,T ]; L ( T )) + (cid:13)(cid:13)(cid:13) Z R v f ε d v (cid:13)(cid:13)(cid:13) C ([0 ,T ]; L ( T )) ≤ C ˚ F . (5.8)Here and henceforth C will signify a generic positive constant that is independent of ε .5.1. Time regularity and compactness of the sequence { ( U ε , B ε ) } ε> . We would like to apply theAubin–Lions–Simon compactness result stated in Theorem 4.6 to the sequence { ( U ε , B ε ) } ε> to deduce itsstrong convergence in a suitable norm, and to this end an ε -uniform bound on { ( ∂ t U ε , ∂ t B ε ) } ε> is needed.Since the left-hand sides of (5.2b) and (5.2c) are equal to, respectively Z t Z T ∂ t U ε · V d x d s and Z t Z T ∂ t B ε · V d x d s, we will focus on bounding the right-hand sides of (5.2b) and (5.2c), and in particular the trilinear andquadrilinear terms. Note that an ε -uniform bound on { ∂ t f ε } ε> is not sought here, since the weak andweak* compactness of the sequence { f ε } ε> will suffice for our purposes.(i) To estimate the first integrals on the right-hand sides of (5.2b), (5.2c), we employ Ladyzhenskaya’sinequality (which is a special case of the, more general, Gagliardo–Nirenberg inequality) to deduce that k U ε ( t ) k L ( T ) ≤ C k U ε ( t ) k L ( T ) k∇ x U ε ( t ) k L ( T ) + C k U ε ( t ) k L ( T ) , and likewise for B ε ( t ) , for any t ∈ [0 , T ]. Therefore, by noting the uniform energy bound (5.7), we deduce that (cid:13)(cid:13) ( U ε , B ε ) (cid:13)(cid:13) L ((0 ,T ); L ( T )) ≤ C T ˚ F .
Here and henceforth C T will signify a generic positive constant that may depend on T but is independentof the mollification parameter ε . Hence, by H¨older’s inequality (applied twice) and the fact that mollification does not increase Sobolevnorms, we have that (cid:12)(cid:12)(cid:12) the first integral on the right-hand side of (5.2b) and (5.2c) (cid:12)(cid:12)(cid:12) ≤ C Z T k ( U ε , B ε ) k L ( T ) k∇ x V k L ( T ) d t ≤ C (cid:16) Z T k ( U ε , B ε ) k · L ( T ) d t (cid:17) (cid:16) Z T k∇ x V k L ( T ) d t (cid:17) ≤ C T ˚ F k V k L ((0 ,T ); H ( T )) . (ii) The second integral on the right-hand side of (5.2b) and (5.2c) is bounded by C ˚ F k V k L ((0 ,T ); H ( T )) .(iii) It remains to bound the last integral of (5.2b). We invoke the Gagliardo–Nirenberg inequality, k U ε k L ( T ) ≤ C k U ε k L ( T ) k∇ x U ε k H ( T ) + C k U ε k L ( T ) , and likewise for B ε , for any t ∈ [0 , T ], and combine it with the uniform energy bound (5.7) to obtain (cid:13)(cid:13) ( U ε , B ε ) (cid:13)(cid:13) L ((0 ,T ); L ( T )) ≤ C T ˚ F .
Combining this with the uniform bounds on the moments in (5.8) and applying H¨older’s inequality(twice) together with the fact that mollification does not increase Sobolev norms, we get that (cid:12)(cid:12)(cid:12) the last integral of (5.2b) (cid:12)(cid:12)(cid:12) ≤ C ˚ F Z T k U ε k L ( T ) k B ε k L ( T ) k V k C ( T ) d t + C ˚ F Z T k B ε k L ( T ) k V k C ( T ) d t ≤ C ˚ F (cid:13)(cid:13) U ε (cid:13)(cid:13) L ((0 ,T ); L ( T )) (cid:13)(cid:13) B ε (cid:13)(cid:13) L ((0 ,T ); L ( T )) k V k L ((0 ,T ); C ( T )) + C ˚ F (cid:13)(cid:13) B ε (cid:13)(cid:13) L ((0 ,T ); L ( T )) k V k L ((0 ,T ); C ( T )) ≤ C T ( ˚ F + ˚ F ) k V k L ((0 ,T ); H ( T )) (by Sobolev inequalities applied to V ) . By combining the bounds established in (i), (ii), (iii) with (5.2b), (5.2c) we deduce that, (cid:12)(cid:12)(cid:12) Z T Z T ∂ t U ε · V d x d s (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) Z T Z T ∂ t B ε · V d x d s (cid:12)(cid:12)(cid:12) ≤ C T ( ˚ F + ˚ F + ˚ F ) k V k L ((0 ,T ); H ( T )) . This estimate now implies that, for fixed
T >
0, any subsequence of { ( ∂ t U ε , ∂ t B ε ) } ε> is uniformly boundedin L (cid:0) (0 , T ); (cid:0) H ( T ; R ) (cid:1) ∗ (cid:1) , where (cid:0) H ( T ; R ) (cid:1) ∗ is the dual space of H ( T ; R ). Together with theuniform energy bound (5.7), this bound allows us to apply Theorem 4.6 to the sequence { ( U ε , B ε ) } ε> with r = 2 , s = 109 , X = H ( T ; R ) , X = L ( T ; R ) , X − = (cid:0) H ( T ; R ) (cid:1) ∗ . Indeed, using the L ( T ) inner product for the duality pairing, we have that the continuous embedding H ( T ) ֒ → L ( T ) implies the continuous embedding L ( T ) = ( L ( T )) ∗ ֒ → ( H ( T )) ∗ , and thereforewe have X continuously embedded in X − . Also, by the Rellich–Kondrashov theorem, we have compactembedding of X into X . Therefore, the hypotheses of Theorem 4.6 are satisfied. Lemma 5.2.
Let (˚ U , ˚ B ) ∈ L ( T ; R ) , ˚ f ∈ L ∞ ( T × R ) ∩ L ( T × R ) , T > , and consider, for t ∈ (0 , T ] ,the family of solutions to (4.23) , (4.24) with mollified initial data (5.1) . Then, there exist a sequence ofpositive real numbers { ε n } n ≥ satisfying lim n →∞ ε n = 0 and a limit solution ( U , B ) ∈ L ((0 , T ); H ( T ; R )) ∩ L ∞ ((0 , T ); L ( T ; R )) , such that ( U ε n , B ε n ) → ( U , B ) strongly in L ((0 , T ); L ( T ; R )) , weakly in L ((0 , T ); H ( T ; R )) , and weak* in L ∞ ((0 , T ); L ( T ; R )) as n → ∞ . Here, the weak and weak* convergence results are direct consequences of the uniform energy bound(5.7), the reflexivity of L ((0 , T ); H ( T ; R )) and the Banach–Alaoglu theorem. The strong convergenceresult in L ((0 , T ); L ( T ; R )) asserted in Lemma 5.2 will play an important role later on, in passing to thelimit in the trilinear and quadrilinear terms in (5.2b) that involve the moments of f ε . EAK SOLUTIONS TO A HYBRID VLASOV-MHD MODEL IN PLASMA PHYSICS 21
Weak* convergence of the sequence { f ε } ε> and its moments. The aim of this section is toestablish the following lemma, concerning weak* convergence of the sequence { f ε } ε> and of its moments. Lemma 5.3.
Under the hypotheses of Lemma 5.2, there exist a subsequence of { ε n } n ≥ , still denoted by { ε n } n ≥ satisfying lim n →∞ ε n = 0 , and a limit function f = f ( t, x , v ) , such that, as n → ∞ , f ε n → f weak* in L ∞ ((0 , T ) × T × R ) , (5.9) f ≥ everywhere on [0 , T ] × T × R , (5.10) f ε n → f weak* in L ∞ ((0 , T ); L r ( T × R )) for all r ∈ (1 , ∞ ) , (5.11) Z R v f ε n ( · , · , v ) d v → Z R v f ( · , · , v ) d v weak* in L ∞ ((0 , T ); L ( T )) , (5.12) Z R f ε n ( · , · , v ) d v → Z R f ( · , · , v ) d v weak* in L ∞ ((0 , T ); L ( T )) , (5.13) k f k L ∞ ((0 ,T ); L r ( T × R )) ≤ k ˚ f k L r ( T × R ) for all r ∈ [1 , ∞ ] . (5.14) Proof.
For the sake of simplicity of the notation, repeatedly extracted subsequences involved in the proofwill all be denoted by { f ε n } n ≥ . As before, χ ( | v |≤ N ) will denote the cut-off function taking the value 1 inthe ball of radius N in R centred at the origin, and equal to 0 otherwise; let χ ( | v | >N ) = 1 − χ ( | v |≤ N ) .The uniform L ∞ bound on { f ε } ε> established in (5.5) and the Banach–Alaoglou theorem imply (5.9).Then, the nonnegativity of f on [0 , T ] × T × R stated in (5.10) follows from the nonnegativity of thecontinuous functions f ε n by Lemma 4.1. Indeed, for any nonnegative function η ∈ L ((0 , T ) × T × R ), theweak* convergence (5.9) implies Z T Z T × R f η d x d v d t = lim n →∞ Z T Z T × R f ε n η d x d v d t ≥ . By choosing η = χ ( f ≤− N ) · χ ( | v |
0, we then deduce that f ≥ , T ) × T × R , andwe can then modify f on a subset of [0 , T ] × T × R with zero Lebesgue measure to ensure its nonnegativityeverywhere. The already proven weak* convergence result (5.9) is not affected by such an alteration on aset of zero Lebesgue measure.The uniform L r bound on { f ε } ε> for r ∈ (1 , ∞ ) in (5.5) and the Banach–Alaoglou theorem imply(5.11), but only for a fixed r . Proving that there exists a subsequence { f ε n } n ≥ that simultaneously weak*converges in L ∞ ((0 , T ); L r ( T × R )) for all r ∈ (1 , ∞ ) requires a subtle argument because of the lack ofcompactness for r = 1. We begin by finding a sequence of nested subsequences (starting with the one usedfor (5.9)): { f ε n } n ≥ ⊃ S ⊃ S ⊃ S ⊃ · · · such that the subsequence S n makes (5.11) true for r = 1 + 2 k with k = ( − n ⌊ n ⌋ . By a test-function argument, all these (countably many) weak* limits can be takento be the same f as in (5.11). Then, by a diagonal argument, we construct the subsequence { f ε n } n ≥ suchthat f ε n is the n -th element of S n , which makes (5.11) simultaneously true for all r = 1 + 2 k where k ∈ Z .By the uniform L r bound (5.5) and the weak* lower-semicontinuity of the norm of any Banach space, wehave k f k L ∞ ((0 ,T ); L r ( T × R )) ≤ k ˚ f k L r ( T × R ) for all r = 1 + 2 k where k ∈ Z . Then, we use H¨older’s inequalityto interpolate these L k norms and obtain k ( f, f ε n ) k L ∞ ((0 ,T ); L r ( T × R )) ≤ (cid:0) k ˚ f k L ∞ ( T × R ) + k ˚ f k L ( T × R ) (cid:1) =: C for all r ∈ (1 , ∞ ) . (5.15)Next, for any r ∈ (1 + 2 k − , k ), consider the conjugate, r ′ := r/ ( r − g in thefunction space L ((0 , T ); L r ′ ( T × R )), which is the predual of the space L ∞ ((0 , T ); L r ( T × R )). The sizeof g for large values of | v | can then be made sufficiently small, in the sense thatlim N →∞ k g χ ( | v | >N ) k L ((0 ,T ); L r ′ ( T × R )) = 0 . (5.16)By fixing N , applying H¨older’s inequality and using the uniform estimate (5.15), we have (cid:12)(cid:12)(cid:12)(cid:12)Z T Z T × R ( f ε n − f ) g χ ( | v | >N ) d x d v d t (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k g χ ( | v | >N ) k L ((0 ,T ); L r ′ ( T × R )) . (5.17)Also, since r ′ > (1+2 k ) / (1+2 k −
1) = 1+2 − k and g χ ( | v |≤ N ) ∈ L ((0 , T ; L r ′ ( T × R )) is compactly supported,we must have g χ ( | v |≤ N ) ∈ L ((0 , T ); L − k ( T × R )), whose dual is L ∞ ((0 , T ); L k ( T × R )). Recall that (5.11) for r = 1 + 2 k has been established, and thuslim n →∞ Z T Z T × R ( f ε n − f ) g χ ( | v |≤ N ) d x d v d t = 0 . Combining this with the uniform limit (5.16) and the uniform estimate (5.17), we have thatlim n →∞ Z T Z T × R ( f ε n − f ) g d x d v d t = 0 for any g ∈ L ((0 , T ); L r ′ ( T × R ))and therefore the subsequence { f ε n } n ≥ and the weak* limit f we have constructed so far make (5.11) truefor any r ∈ (1 + 2 k − , k ) for all k ≥
1. The proof of (5.11) is therefore complete.To show (5.12), we fix any φ ∈ L ∞ ((0 , T ) × T ). Then, we have v +1 χ ( | v |≤ N ) φ ∈ L ((0 , T ) × T × R )for any N >
0, where v is the first coordinate of v = ( v , v , v ) and v +1 := max { v , } . We apply (5.9) todeduce that Z T Z T × R v +1 f χ ( | v |≤ N ) φ ( t, x ) d x d v d t − lim n →∞ Z T Z T × R v +1 f ε n χ ( | v |≤ N ) φ ( t, x ) d x d v d t = 0 . (5.18)By the uniform energy bound (5.7) and letting K := ess sup n ∈ N , ( t, x ) ∈ [0 ,T ] × T (cid:8) | φ ( t, x ) | , E par [ f ε n ]( t ) (cid:9) < ∞ ,we obtain the following estimate concerning large values of | v | : (cid:12)(cid:12)(cid:12) Z T Z T × R v +1 f ε n χ ( | v | >N ) φ ( t, x ) d x d v d t (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) Z T Z T × R | v | N f ε n K d x d v d t (cid:12)(cid:12)(cid:12) ≤ T K N . (5.19)As the expression on the left-hand side of (5.19) is a bounded sequence in R (with respect to n ), it has aconvergent subsequence (not indicated). Thus, taking the limit n → ∞ in (5.19) over this subsequence andsubtracting the resulting inequality from (5.18) we deduce that − T K N ≤ Z T Z T × R v +1 f χ ( | v |≤ N ) φ ( t, x ) d x d v d t − lim n →∞ Z T Z T × R v +1 f ε n φ ( t, x ) d x d v d t ≤ T K N .
Next, we shall pass to the limit N → ∞ in this inequality; to this end, we shall suppose that φ ( t, x ) ≥ f , we can apply the monotone convergence theorem to the limitlim N →∞ Z T Z T × R v +1 f χ ( | v |≤ N ) φ ( t, x ) d x d v d t = Z T Z T × R v +1 f φ ( t, x ) d x d v d t and hence combine the last two estimates/limits to deduce that, for any φ such that 0 ≤ φ ∈ L ∞ ((0 , T ) × T ), Z T Z T × R v +1 f φ ( t, x ) d x d v d t = lim n →∞ Z T Z T × R v +1 f ε n φ ( t, x ) d x d v d t. Since the uniform estimate (5.8) guarantees that both sides are finite and hence all integrands (which arenonnegative) are Lebesgue-integrable, we can apply Fubini’s theorem to deduce that Z T Z T (cid:16) Z R v +1 f d v (cid:17) φ ( t, x ) d x d t = lim n →∞ Z T Z T (cid:16) Z R v +1 f ε n d v (cid:17) φ ( t, x ) d x d t for any φ ≥ φ ∈ L ∞ ((0 , T ) × T ).We then repeat the same procedure for any 0 ≥ φ ∈ L ∞ ((0 , T ) × T × R ), following which we repeatthe reasoning for v − , v , v , to finally deduce that Z T Z T (cid:16) Z R v f d v (cid:17) φ ( t, x ) d x d t = lim n →∞ Z T Z T (cid:16) Z R v f ε n d v (cid:17) φ ( t, x ) d x d t for all φ ∈ L ∞ ((0 , T ) × T ).On the other hand, thanks to the uniform estimate (5.8) and the Banach–Alaoglou theorem, there existsan M ∈ L ∞ ((0 , T ); L ( T )) such that, upon extraction of a subsequence (not indicated), Z R v f ε n ( · , · , v ) d v → M weak* in L ∞ ((0 , T ); L ( T )) as n → ∞ . Combining these two limits we deduce that Z T Z T (cid:16) Z R v f ( t, x , v ) d v (cid:17) φ ( t, x ) d x d t = Z T Z T M ( t, x ) φ ( t, x ) d x d t for all φ ∈ L ∞ ((0 , T ) × T ) , EAK SOLUTIONS TO A HYBRID VLASOV-MHD MODEL IN PLASMA PHYSICS 23 which directly implies (for example by du Bois-Reymond’s lemma) that Z R v f ( t, x , v ) d v = M ( t, x ) a.e. on (0 , T ) × T , and thus (5.12) has been proved. The proof of (5.13) proceeds analogously and is therefore omitted.Finally, the bound (5.14) for r ∈ (1 , ∞ ] follows from the uniform L r bound (5.5), the weak* convergence(5.9) and (5.11), and the weak* lower-semicontinuity of the norm of a Banach space. Note that covering thecase of r = ∞ requires particular care. In fact, for any nonnegative, measurable function f defined on theset (0 , T ) × T × R , we have k f k L ∞ ((0 ,T ) × T × R ) = k f k L ∞ ((0 ,T ); L ∞ ( T × R )) thanks to the following argument:with A := k f k L ∞ ((0 ,T ) × T × R ) and A := k f k L ∞ ((0 ,T ); L ∞ ( T × R )) , we deduce that A ≤ A by applyingFubini’s theorem to R T R | v |
0, andwe then prove that A ≥ A by applying Fubini’s theorem to R T R T × R χ ( f ≥ A ) d x d v d t .It remains to prove (5.14) for r = 1, which does not, in fact, rely on (5.11) but on (5.13). Consider aunivariate integrable function g ∈ L (0 , T ). Hence, automatically, g ∈ L ((0 , T ); L ( T )), whose dual spaceis L ∞ ((0 , T ); L ( T )). Therefore, by (5.13), Z T Z T (cid:16) Z R f ( t, x , v ) d v (cid:17) g ( t ) d x d t = lim n →∞ Z T Z T (cid:16) Z R f ε n ( t, x , v ) d v (cid:17) g ( t ) d x d t, where all integrands are Lebesgue integrable. We then apply Fubini’s theorem and use the fact that both f and f ε n are nonnegative to deduce that Z T k f ( t ) k L ( T × R ) g ( t ) d t ≤ lim inf n →∞ Z T k f ε n ( t ) k L ( T × R ) g ( t ) d t. By the uniform L r bound (5.5), we have that the right-hand side is dominated by k ˚ f k L ( T × R ) k g k L (0 ,T ) .Hence, by taking the supremum over all g such that k g k L (0 ,T ) = 1 gives (cid:16) k f ( t ) k L ( T × R ) (cid:17) ( L (0 ,T )) ∗ ≤ k ˚ f k L ( T × R ) . Since the norm in the dual space ( L (0 , T )) ∗ is identical to the L ∞ (0 , T ) norm, we have proved (5.14) for r = 1 as well. That completes the proof. (cid:3) With these convergence results in place, we are now ready to pass to the limit ε → The limit solves the weak form of the PDE.
It remains to prove that the limits identified inLemmas 5.2, 5.3 satisfy the weak form of the original (nonmollified) PDE (1.1), in a sense that will be madeprecise in the next definition.
Definition 5.4.
All functions in this definition are understood to be 2 π -periodic with respect to the inde-pendent variable x . Suppose that the initial datum ˚ f ∈ L ( T × R ) ∩ L ∞ ( T × R ) is such that E par [˚ f ] isfinite, and consider divergence-free initial data (˚ U , ˚ B ) ∈ L , inc ( T ; R ) and T >
0. We call( U , B ) ∈ L ∞ ((0 , T ); L , inc ( T ; R )) ∩ L ((0 , T ); H ( T ; R )) , (5.20a)with ∇ x · U = 0, ∇ x · B = 0 a.e. on (0 , T ) × T × R , and f ∈ L ∞ (cid:0) (0 , T ); L ( T × R ) ∩ L ∞ ( T × R )) , (5.20b)a weak solution to the hybrid incompressible Vlasov-MHD system (1.1), if, for every t ∈ [0 , T ], the followingare true: Z T × R f ( t, x , v ) g ( t, x , v ) d x d v − Z T × R ˚ f ( x , v ) g (0 , x , v ) d x d v − Z t Z T × R f ∂ t g d x d v d s = Z t Z T × R ( v f ) · ∇ x g − (cid:0) ( U − v ) × B f (cid:1) · ∇ v g d x d v d s (5.21a) for any compactly supported, scalar-valued test function g ∈ C c ([ − , T + 1] × T × R ); Z T U ( t, x ) · V ( t, x ) d x − Z T ˚ U ( x ) · V (0 , x ) d x − Z t Z T U · ∂ t V d x d s = Z t Z T (cid:0) U ⊗ U − B ⊗ B (cid:1) : ∇ x V d x d s − Z t Z T ∇ x U : ∇ x V d x d s + Z t Z T (cid:18) U × B (cid:18) Z R f d v (cid:19) − B × Z R v f d v (cid:19) · V d x d s (5.21b)for any R -valued test function V ∈ C ([ − , T + 1] × T ; R ) with ∇ x · V = 0; and Z T B ( t, x ) · V ( t, x ) d x − Z T ˚ B ( x ) · V (0 , x ) d x − Z t Z T B · ∂ t V d x d s = Z t Z T (cid:0) U ⊗ B − B ⊗ U (cid:1) : ∇ x V d x d s − Z t Z T ∇ x B : ∇ x V d x d s (5.21c)for any R -valued test function V ∈ C ([ − , T + 1] × T ; R ) with ∇ x · V = 0. Moreover, for almost every t ∈ [0 , T ], including t = 0, f ( t, · , · ) ≥ , k f ( t, · , · ) k L r ( T × R )) ≤ k ˚ f k L r ( T × R ) for all r ∈ [1 , ∞ ] , (5.22)12 k ( U ( t ) , B ( t )) k L ( T ) + E par [ f ]( t ) + k ( ∇ x U , ∇ x B ) k L ((0 ,t ); L ( T )) ≤ k (˚ U , ˚ B ) k L ( T ) + E par [˚ f ] . (5.23)By setting t = 0 in (5.21), we have ( f, U , B ) (cid:12)(cid:12) t =0 = (˚ f , ˚ U , ˚ B ) by the du Bois-Reymond lemma, ensuring thatthe initial conditions at t = 0 are satisfied. Remark 5.5.
We emphasize that (5.21) is valid for every t ∈ [0 , T ], although one normally sees “almostevery t ” in the literature. The rationale seems to be lack of uniform-in-time convergence and the lack of(strong) compactness for f ε . We will remedy this by using a version of the Lebesgue differentiation theoremin time, and then redefining the solution at the exceptional times, which form of a set of zero Lebesguemeasure in (0 , T ), using the weak formulation. The only adverse effect of such a redefinition is that theeverywhere nonnegativity of f in (5.10) is weakened to f ( t, · , · ) ≥ almost every t ∈ [0 , T ]. Remark 5.6.
Then, the validity of (5.21) for every t ∈ [0 , T ] also implies that any weak solution ( f, U , B )is right-continuous at t = 0 when regarded as a continuous linear functional over the space C c ( T × R ) ×C ( T ; R ) ×C ( T ; R ). In fact, more is true: as the expressions appearing on the right-hand sides of (5.21a),(5.21b) and (5.21c) are absolutely continuous functions of t ∈ [0 , T ], the same is true of the expressions ontheir left-hand sides, for any admissible choice of the test functions g and V . By considering in particularadmissible test functions g and V such that g ( t, x , v ) = g ( t ) g ( x , v ) where g ( t ) ≡ t ∈ [0 , T ], and V ( t, x ) = v ( t ) V ( x ) such that v ( t ) ≡ t ∈ [0 , T ], we deduce that t ∈ [0 , T ] Z T × R f ( t, x , v ) g ( x , v ) d x d v ,t ∈ [0 , T ] Z T U ( t, x ) · V ( x ) d x ,t ∈ [0 , T ] Z T B ( t, x ) · V ( x ) d x are absolutely continuous for any scalar-valued g ∈ C c ( T × R ), and any R -valued V ∈ C ( T ) satisfying ∇ x · V = 0. Remark 5.7.
Similarly as in the case of the three-dimensional incompressible Navier–Stokes equations, itis unclear whether the energy inequality (5.23) can be an equality, and whether it holds for every, ratherthan almost every, t ∈ [0 , T ].We are now ready to prove our main result: the existence of large-data finite-energy global weak solutionsto the hybrid Vlasov-MHD system. EAK SOLUTIONS TO A HYBRID VLASOV-MHD MODEL IN PLASMA PHYSICS 25
Theorem 5.8.
For any (˚ U , ˚ B ) ∈ L , inc ( T ; R ) , both of which are divergence-free in the sense of distribu-tions, any pointwise nonnegative ˚ f ∈ L ∞ ( T × R ) ∩ L ( T × R ) with finite E par [˚ f ] , and any T > , thereexists a weak solution ( f, U , B ) to (1.1) in the sense of Definition 5.4.Proof. We have already found a sequence { ( f ε n , U ε n , B ε n ) } n ≥ satisfying the assertions of Lemmas 5.2 and5.3. This leads to the regularity and integrability properties of the limit ( f, U , B ), as required in (5.20).The divergence-free property of ( U , B ) required by Definition 5.4 is the consequence of the weak con-vergence in L ((0 , T ); H ( T ; R )), as shown in Lemma 5.2 (so that (0 ,
0) = ( ∇ x · U ε n , ∇ x · B ε n ) convergesweakly to ( ∇ x · U , ∇ x · B ) in L ((0 , T ); L ( T ; R )) as well).In the rest of the proof, the strong convergence properties of mollifiers asserted in Lemma 5.1 areimplicitly used without specific referencing.We consider any compactly supported scalar-valued function g ∈ C c ([ − , T + 1] × T × R ) and any R -valued function V ∈ C ([ − , T + 1] × T ) with ∇ x · V = 0. We shall now proceed to confirm that as n → ∞ (and therefore ε n →
0) the limit of each term in the mollified weak formulation (5.2) is equal to itscounterpart in the weak formulation (5.21). • To prove the convergence towards the right-hand side of (5.21a) we proceed as follows. Thanks tothe strong convergence result in the function space L ∞ ((0 , T ); L ( T ; R )) stated in Lemma 5.2, wehave U h ε n i ε n × B h ε n i ε n → U × B strongly in L ((0 , T ); L ( T ; R )) . We then recall that g is, by hypothesis, a smooth function and has compact support; therefore, both R R |∇ v g | d v and R R | v × ∇ v g | d v are uniformly bounded in [0 , T ] × T as functions of t and x .Thus we have that, as n → ∞ , (cid:0) ( U h ε n i ε n − v ) × B h ε n i ε n (cid:1) · ∇ v g → (cid:0) ( U − v ) × B (cid:1) · ∇ v g strongly in L ((0 , T ) × T × R ) , which, together with the weak* convergence result (5.9) for f ε n , implies thatlim n →∞ (the right-hand side of (5.2a)) = (the right-hand side of (5.21a)) . • To prove the convergence towards the last integral on the right-hand side of (5.21b), we first rewritethe last triple product term appearing in (5.2b) as (cid:0) V h ε i × B h ε i ε (cid:1) · Z R v f ε d v ; then by the strongconvergence result stated in Lemma 5.2, we have that, as n → ∞ , V h ε i × B h ε n i ε n → V × B strongly in L ((0 , T ); L ( T ; R )), and( U ε n × B h ε n i ε n ) · V → ( U × B ) · V strongly in L ((0 , T ); L ( T )) . Consequently, by the weak* convergence results (5.12), (5.13) for the moments of f ε n and notingthat H¨older’s inequality can be applied to the difference between the last integrals in (5.2b) and(5.21b) (note that + = + + = 1), we have thatlim n →∞ (last integral of (5.2b)) = (last integral of (5.21b)) . • The rest of right-hand side of (5.21b) and the entire right-hand side of (5.21c) involve terms whichare of one of the following two types:1. For terms involving the operation ⊗ , the convergence of their counterparts in (5.2) follows fromthe strong L ∞ ((0 , T ); L ( T ; R )) convergence stated in Lemma 5.2.2. For the terms including ( ∇ x U : ∇ x V ) and ( ∇ x B : ∇ x V ), the convergence of their counterpartsin (5.2) follows from the weak convergence in L ((0 , T ); H ( T ; R )), as stated in Lemma 5.2. • To prove the convergence towards the terms appearing on the left-hand side of (5.21), we proceedby combining the strong convergence of the mollified initial data, as in (5.4) and Lemmas 5.2 and5.3. We then obtain all terms on the left-hand side of (5.21) , except the first, as limits of theircounterparts in (5.2) (recall the hypothesis that g is compactly supported).To summarize our conclusions so far, we havefor any t ∈ [0 , T ] and test functions g, v as specified above,lim n →∞ ((5.2a), (5.2b), (5.2c)) = ((5.21a), (5.21b), (5.21c)), except the first term in each equation. (5.24) Thus we shall now focus on the first term in each of (5.2a), (5.2b), (5.2c). First, thanks to the strong L ((0 , T ); L ( T ; R )) convergence stated in Lemma 5.2, there exists a subset Z ( U , B ) ⊂ [0 , T ] of zero Lebesguemeasure and a subsequence of { ε n } n ≥ (not indicated), such thatfor any t ∈ [0 , T ] \Z ( U , B ) , lim n →∞ (cid:13)(cid:13) ( U ε n ( t ) , B ε n ( t )) − ( U ( t ) , B ( t )) (cid:13)(cid:13) L ( T ) = 0 , and hence, (5.25)for t ∈ [0 , T ] \Z ( U , B ) , lim n →∞ (first terms of (5.2b), (5.2c)) = (first terms of (5.21b), (5.21c)) . (5.26)It then remains to consider the first term in (5.21a). In the absence of a strong convergence result for thesequence { f ε n } n ≥ the argument in this case is more delicate.To this end, we take any τ ∈ [0 , T ) and an arbitrarily small δ ∈ (0 , T − τ ), and consider the integralaverage of (5.2a) over [ τ, τ + δ ]:1 δ Z τ + δτ h Z T × R (cid:0) f ε n g (cid:1) ( t, x , v ) d x d v i d t − Z T × R ˚ f ε n ( x , v ) g (0 , x , v ) d x d v = 1 δ Z τ + δτ n Z t Z T × R f ε n ∂ t g + ( v f ε n ) · ∇ x g − (cid:0) ( U h ε n i ε n − v ) × B h ε n i ε n f ε n (cid:1) · ∇ v g d x d v d s o d t. (5.27)The first term on the left-hand side here involves integration with respect the t, x , v variables, so the weak*convergence (5.9) applies and we thus obtain the desired limit as n → ∞ (and therefore as ε n → f for f . Concerning the right-hand side, the expression in the curly brackets converges pointwise, for every t ∈ [ τ, τ + δ ], as stated in (5.24); it is dominated by a constant that is independent of ε n , thanks to theuniform energy bound (5.7) and the fact that g is smooth and compactly supported. As the constantfunction is, trivially, integrable over [ τ, τ + δ ], by Lebesgue’s dominated convergence theorem the right-handside above also has the desired limit as n → ∞ (and ε n → δ Z τ + δτ h Z T × R ( f g )( t, x , v ) d x d v i d t − Z T × R ˚ f ( x , v ) g (0 , x , v ) d x d v = 1 δ Z τ + δτ n Z t Z T × R f ∂ t g + ( v f ) · ∇ x g − (cid:0) ( U − v ) × B f (cid:1) · ∇ v g d x d v d s o d t =: 1 δ Z τ + δτ n Z t F ( s ) d s o . (5.28)At the start of the proof we showed the integrability properties of the limit ( f, U , B ), as required in(5.20) of Definition 5.4. The newly defined function F ( s ) is therefore in L (0 , T ), which immediately impliesthe absolute continuity of the mapping t ∈ [0 , T ] R t F ( s ) d s appearing in the right-hand side of the aboveequation. Thus, lim δ → δ Z τ + δτ n Z t F ( s ) d s o = Z τ F ( s ) d s for every τ ∈ [0 , T ) . To pass to the limit δ → { g n } n ≥ of C c ([ − , T + 1] × T × R ) consisting of compactly supported, scalar-valued test functions. Notethat we only need this countable subset to be dense with respect to the C ([ − , T + 1] × T × R ) norm; theexistence of such a sequence { g n } n ≥ follows from Proposition C.4.Then, for a fixed n , by the integrability of f g n and Lebesgue’s differentiation theorem,lim δ → δ Z τ + δτ h Z T × R ( f g n )( t, x , v ) d x d v i d t = Z T × R ( f g n )( τ, x , v ) for a.e. τ ∈ [0 , T ) . Since the countable union of sets of zero Lebesgue measure is a set of zero Lebesgue measure, we deducefrom the countability of { g n } n ≥ the existence of a set Z f ⊂ [0 , T ) of zero Lebesgue measure such thatlim δ → δ Z τ + δτ h Z T × R ( f g n (cid:1) ( t, x , v ) d x d v i d t = Z T × R (cid:0) f g n (cid:1) ( τ, x , v ) d x d v ∀ n ∈ N , ∀ τ ∈ [0 , T ) \Z f . (5.29) EAK SOLUTIONS TO A HYBRID VLASOV-MHD MODEL IN PLASMA PHYSICS 27
Recall that { g n } n ≥ is a dense (with respect to the C norm) subset of C c ([ − , T +1] × T × R ). Consequently,for each fixed g in the latter space, we can extract a subsequence { g n j } j ≥ from { g n } n ≥ such that k g n j − g k C ([ − ,T +1] × T × R ) ≤ j . This implies that, for any positive integer j and any τ ∈ [0 , T ) \ Z f , (cid:12)(cid:12)(cid:12)(cid:12) δ Z τ + δτ h Z T × R ( f g )( t, x , v ) d x d v i d t − Z T × R ( f g )( τ, x , v ) d x d v (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) δ Z τ + δτ h Z T × R ( f g n j )( t, x , v ) d x d v i d t − Z T × R ( f g n j )( τ, x , v ) d x d v (cid:12)(cid:12)(cid:12)(cid:12) + 2 j k f k L ∞ ((0 ,T ); L ( T × R )) . By (5.29) we then have, for any positive integer j ∈ N and all τ ∈ [0 , T ) \Z f , thatlim sup δ → (cid:12)(cid:12)(cid:12)(cid:12) δ Z τ + δτ h Z T × R ( f g )( t, x , v ) d x d v i d t − Z T × R ( f g )( τ, x , v ) d x d v (cid:12)(cid:12)(cid:12)(cid:12) ≤ j k f k L ∞ ((0 ,T ); L ( T × R )) . Passing to the limit j → ∞ , we deduce thatlim δ → δ Z τ + δτ h Z T × R ( f g )( t, x , v ) d x d v i d t = Z T × R ( f g )( τ, x , v ) d x d v for all τ ∈ [0 , T ) \Z f .Combining this with the convergence shown below (5.28), we have proved that the δ → t = τ ∈ [0 , T ) \Z f and a fixed g . As the set Z f that is chosen in the line above(5.29) is independent of g , we have that (5.21a) holds on [0 , T ) \ Z f for all admissible test functions g .By combining this assertion with (5.24), (5.26) we finally deduce that(5.21) holds for every τ ∈ [0 , T ) \ ( Z ( U , B ) ∪ Z f );in other words, (5.21) hold a.e. on [0 , T ].Regarding the “exceptional times”, at every t ∈ Z f (resp. t ∈ Z ( U , B ) ), we use (5.21a) (resp. (5.21b) and(5.21c)) to redefine f (resp. U and B ) as an element in the dual space of C c ( T × R ) (resp. C ( T ; R )).In the final part of the proof, we show the physically relevant properties (5.22), (5.23) for a.e. t ∈ [0 , T ];we note that they hold at t = 0, since the initial conditions are precisely satisfied, which is shown immediatelyafter the inequality (5.23).The nonnegativity of f stated in (5.22) follows from (5.10) and the fact that we have only redefined f ( t, · , · ) on a subset of [0 , T ] of zero Lebesgue measure. The estimate in (5.22) is just a duplication of (5.14),which is unaffected by the redefinition procedure.To show the energy inequality (5.23), we choose the test function in (5.21a) from the sequence ofnonnegative, ( t, x )-independent functions { g k } k ≥ ⊂ C c ( R ), which are nondecreasing pointwise (i.e., 0 ≤ g ( v ) ≤ g ( v ) ≤ · · · ) and converge to | v | pointwise. Since we have established that each term but the firstone in (5.21a) is the limit of its counterpart in (5.2a) as n → ∞ , then so is the first term in (5.21a); thus,for a.e. t ∈ [0 , T ] we have Z T × R f ( t, x , v ) g k ( v ) d x d v = lim n →∞ Z T × R f ε n ( t, x , v ) g k ( v ) d x d v ≤ lim inf n →∞ Z T × R f ε n ( t, x , v ) | v | d x d v . Hence, by the monotone convergence theorem (noting the nonnegativity of f in (5.22)), for a.e. t ∈ [0 , T ], Z T × R f ( t, x , v ) | v | d x d v ≤ lim inf n →∞ Z T × R f ε n ( t, x , v ) | v | d x d v . Combining this with the mollified version of the energy equality (4.26), the bound on the initial energy(5.6), the convergence of the fluid energy, which follows from (5.25) and H¨older’s inequality, and the weak L ((0 , T ); H ( T ; R )) convergence result from Lemma 5.2, we deduce the energy inequality (5.23) for a.e. t ∈ [0 , T ]. (cid:3) Additional properties of weak solutions.
By relying on Definition 5.4 only , it is possible to showthat weak solutions possess the following additional properties.
Proposition 5.9.
Let ( f, U , B ) be a weak solution in the sense of Definition 5.4. Then, the followingproperties hold:(a) t ∈ [0 , T ] f ( t, · , · ) ∈ (cid:0) C c ( T × R ) (cid:1) ∗ and t ∈ [0 , T ] ( U , B )( t, · , · ) ∈ (cid:0) C ( T ; R ) (cid:1) ∗ are absolutelycontinuous mappings.(b) The zeroth and first moment of f satisfy, respectively, Z R f ( · , · , v ) d v ∈ L ∞ ((0 , T ); L ( T )) and Z R | v | f ( · , · , v ) d v ∈ L ∞ ((0 , T ); L ( T )) . (5.30)(c) The total momentum is conserved, i.e., for almost every t ∈ [0 , T ], Z T × R v f ( t, x , v ) d x d v + Z T U ( t, x ) d x = Z T × R v ˚ f ( x , v ) d x d v + Z T ˚ U ( x ) d x . (5.31)(d) The total mass of energetic particles is conserved, i.e., for almost every t ∈ [0 , T ], Z T × R f ( t, x , v ) d x d v = Z T × R ˚ f ( x , v ) p d x d v . (5.32) Proof.
We prove the assertions item by item.(a) The stated absolute continuity properties directly follow from Remark 5.6.(b) (5.30) is a consequence of Proposition 3.1, Definition 5.4 and the energy inequality (5.23).(c) We choose the test function appearing in (5.21a) from the sequence of ( t, x )-independent functions { ˜ g k } k ≥ ⊂ C c ( R ) so that { ˜ g k } k ≥ converges to v (the first coordinate of v = ( v , v , v )) pointwise, {∇ v ˜ g k } k ≥ converges to (1 , , T pointwise, and | ˜ g k ( v ) | ≤ | v | , |∇ v ˜ g k ( v ) | ≤ . (5.33)Then, by Definition 5.4, for a.e. t ∈ [0 , T ], the measurable function f ( t, · , · ) satisfies Z T × R f ( t, x , v ) ˜ g k ( v ) d x d v − Z T × R ˚ f ( x , v ) ˜ g k ( v ) d x d v = − Z t Z T × R (cid:0) ( U − v ) × B f (cid:1) · ∇ v ˜ g k d x d v d s. By (5.20), (5.30), (5.33), each of the integrands is uniformly bounded by an integrable function, sothat by taking the k → ∞ limit and applying Lebesgue’s dominated convergence theorem we have Z T × R f ( t, x , v ) v d x d v − Z T × R ˚ f ( x , v ) v d x d v = − Z t Z T × R (cid:0) ( U − v ) × B f (cid:1) · ∇ v v d x d v d s. We then choose the test function v = (1 , , T in (5.21b) and add the resulting equality to the oneabove to deduce, for a.e. t ∈ [0 , T ], that Z T f ( t, x , v ) v d x d v + Z T U ( t, x ) d x = Z T × R ˚ f ( x , v ) v d x d v + Z T ˚ U ( x ) d x , where U is the first component of the velocity vector U = ( U , U , U ) T . Likewise, we can show theconservation of the second and third components of the total momentum, hence proving (5.31).(d) To show the conservation of the mass of energetic particles, as stated in (5.32), we change the sequence { ˜ g k } k ≥ ⊂ C c ( T × R ), used in the proof of item (c) above, so that its elements are dominated by,and converge to, 1 pointwise, with ∇ v ˜ g k dominated by 1 and converging to 0 pointwise. We skipthe remaining steps, as they are an easier version of the proof of item (c) above.That completes the proof of the proposition. (cid:3) EAK SOLUTIONS TO A HYBRID VLASOV-MHD MODEL IN PLASMA PHYSICS 29
Acknowledgments.
C.T. acknowledges financial support from the Leverhulme Trust Research ProjectGrant No. 2014-112, and by the London Mathematical Society Grant No. 31439 (Applied GeometricMechanics Network). The authors thank the anonymous referees for their insightful comments and carefulproofreading, which greatly helped to improve the structure and rigor of this paper.
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Appendix A. Product rules
The following identities are useful variants of the product rule, where f and g are scalar-valued functionswhile ~u and ~v are R -valued functions: ∇ ( f g ) = f ∇ g + g ∇ f ; (A.1) ∇ ( u · v ) = u × ( ∇ × v ) + v × ( ∇ × u ) + ( u · ∇ x ) v + ( v · ∇ ) u ; (A.2) ∇ · ( f v ) = f ( ∇ · v ) + v · ( ∇ f ); (A.3) ∇ · ( u × v ) = v · ( ∇ × u ) − u · ( ∇ × v ); (A.4) ∇ × ( f v ) = ( ∇ f ) × v + f ( ∇ × v ); (A.5) ∇ × ( u × v ) = u ( ∇ · v ) − v ( ∇ · u ) + ( v · ∇ ) u − ( u · ∇ ) v ; (A.6)( ∇ × u ) × u = ( u · ∇ ) u − ∇| u | . (A.7) Appendix B. The linear(ized) incompressible MHD system
Let h· , ·i denote the usual L ( T ) inner product of scalar-valued or R -valued functions. We shall provethe following result. Lemma B.1.
Given any
T > , consider the following forced, linear(ized) incompressible MHD system overthe domain [0 , T ] × T : ( ∂ t U + a · ∇ x U − b · ∇ x B − ∆ x U = U × g + h − ∇ x P (subject to ∇ x · U = 0) ,∂ t B + a · ∇ x B − b · ∇ x U − ∆ x B = h − ∇ x P B (subject to ∇ x · B = 0) , (B.1)with divergence-free initial data ( U , B ) (cid:12)(cid:12)(cid:12) t =0 = (˚ U , ˚ B ) ∈ C ∞ ( T ; R ) , where a , b , g , h , h ∈ T m ≥ C ([0 , T ]; C m ( T ; R )) are given R -valued functions with ∇ x · a = ∇ x · b = 0 .Then, this system admits a classical solution ( U , B ) ∈ \ m ≥ C ([0 , T ]; C m ( T ; R )) that satisfies k ( U , B ) k L ( T ) (cid:12)(cid:12)(cid:12) T + 2 Z T k ( ∇ x U , ∇ x B ) k L ( T ) d t = 2 Z T h U , h i + h B , h i d t, (B.2) k ( ∇ x U , ∇ x B ) k L ( T ) (cid:12)(cid:12)(cid:12) T + Z T k ( ∂ t U , ∂ t B ) k L ( T ) d t ≤ [0 ,T ] × T {| a | , | b | , | g | , } Z T k ( ∇ x U , ∇ x B , U , h , h ) k L ( T ) d t. (B.3) Proof.
Let { W j } j ≥ ⊂ C ∞ ( T ; R ) be an orthonormal basis of eigenfunctions of ∆ x in the space of divergence-free R -valued vector fields in L ( T ; R ). For any, not necessarily divergence-free, vector field V ∈ C ( T ; R )and an integer N ≥
1, we define the projection P N [ V ] := N X j =1 h V , W j i W j . EAK SOLUTIONS TO A HYBRID VLASOV-MHD MODEL IN PLASMA PHYSICS 31
Further, we define an approximation of ( U , B ) by (cid:18) U N ( t, x ) B N ( t, x ) (cid:19) = N X j =1 (cid:18) u j ( t ) B j ( t ) (cid:19) W j ( x ) , t ∈ [0 , T ] , x ∈ T , that satisfies, for i = 1 , , . . . , N , dd t u i ( t ) = N X j =1 [ h W i , a · ∇ x W j i u j − h W i , b · ∇ x W j i B j + h W i , ∆ x W j i u j + h W i , W j × g i u j ] + h W i , h i , dd t B i ( t ) = N X j =1 [ h W i , a · ∇ x W j i B j − h W i , b · ∇ x W j i u j + h W i , ∆ x W j i B j ] + h W i , h i , with initial data ( U N , B N ) (cid:12)(cid:12)(cid:12) t =0 = P N (˚ U , ˚ B ) . This is a closed, 2 N -by-2 N system of linear ordinary differential equations for the unknowns { u j ( t ) , B j ( t ) } Nj =1 with coefficients depending on a , b , g , h , h , which are in C ([0 , T ]; C m ( T ; R )) for any m ≥
0. Therefore, itadmits a solution satisfying( U N , B N ) ∈ C ([0 , T ]; C m ( T ; R )) , ∂ t ( U N , B N ) ∈ C ([0 , T ]; C m ( T ; R )) , for all m ≥ . (B.4)We note that the above system of linear ordinary differential equations is equivalent to ( ∂ t U N = P N [ − a · ∇ x U N + b · ∇ x B N + ∆ x U N + U N × g + h ] ,∂ t B N = P N [ − a · ∇ x B N + b · ∇ x U N + ∆ x B N + h ] . (B.5)The regularity of ( U N , B N ) in (B.4) and the W j ’s being eigenfunctions of ∆ x allow us to apply theenergy method to (B.5) and integrate by parts (noting that ∇ x · a = ∇ x · b = 0) to obtaindd t (cid:0) k U N k L ( T ) + k B N k L ( T ) (cid:1) = − (cid:0) k∇ x U N k L ( T ) + k∇ x B N k L ( T ) (cid:1) + 2 h U N , h i + 2 h B N , h i . We then apply the energy method to the action of the differential operator ∆ m x on (B.5), noting that spatialdifferential operators commute with the projector P N thanks to the periodic boundary conditions withrespect to x , and we integrate by parts and apply the Cauchy–Schwarz inequality to obtaindd t (cid:18) k ∆ m x U N k L ( T ) + k ∆ m x B N k L ( T ) (cid:19) ≤ C k ( a , b ) k H m ( T ) k ( U N , B N ) k H m ( T ) + C k U N k H m ( T ) k h k H m ( T ) + C k B N k H m ( T ) k h k H m ( T ) , where m ≥ C denotes a generic constant that may depend on m but is independent of N . Since a , b ∈ C m ( T ; R ) and ( U , B ) (cid:12)(cid:12) t =0 ∈ C m ( T ; R ) for all m ≥
0, we can integrate the sum of the above equalityand inequality in time to obtain (cid:13)(cid:13) ( U N , B N ) (cid:13)(cid:13) C ([0 ,T ] ,H m ( T ; R )) ≤ F (cid:16) T, ( U , B ) (cid:12)(cid:12) t =0 , a , b , g , h , h , m (cid:17) ∀ N ∈ N . Here and below, F , F are functions that are independent of N . Also, take the L ( T ; R ) inner product of ∂ t U N and ∂ t B N with the respective equations of (B.5), apply the Cauchy–Schwarz inequality and combinethe resulting bound with the previous estimate to deduce that (cid:13)(cid:13) ∂ t ( U N , B N ) (cid:13)(cid:13) C ([0 ,T ] ,H m − ( T )) ≤ F (cid:16) T, ( U , B ) (cid:12)(cid:12) t =0 , a , b , g , h , h , m (cid:17) ∀ N ∈ N . Thus, these two uniform-in- N estimates imply that the family { ( U N , B N ) } N ≥ , viewed as a sequence ofcontinuous mappings from [0 , T ] into H m − ( T ; R ), is equicontinuous. Therefore, by the Arzel`a–Ascolitheorem, there exists a pair ( U , B ) ∈ C ([0 , T ]; C m − ( T ; R ))such that upon subtracting a subsequence and using the embedding H m − ( T ; R ) ⊂ C m − ( T ; R ),lim N →∞ ( U N , B N ) = ( U , B ) strongly and uniformly in C ([0 , T ]; C m − ( T ; R )) . (B.6) By recalling that { W j } j ≥ are, by definition, divergence-free, we deduce that the above limit ( U , B ) alsosatisfies the divergence-free condition for all t ∈ [0 , T ]. Moreover, thanks to the choice of initial data( U N , B N ) (cid:12)(cid:12)(cid:12) t =0 = P N (˚ U , ˚ B ) and the completeness of the basis { W j } j ≥ , we have ( U , B ) (cid:12)(cid:12)(cid:12) t =0 = (˚ U , ˚ B ).In order to show that ( U , B ), found in this way, does indeed satisfy the equation (B.1), we only take B for example. We integrate the second equation of (B.5) with respect to t from 0 to τ ≤ T , apply theprojection P N and use P N P N = P min { N,N } to deduce that P N B (cid:12)(cid:12)(cid:12) τ = Z τ P N [ − a · ∇ x B N + b · ∇ x U N + ∆ x B N + h ] d t for any N ≥ N . By holding N ≥ N → ∞ , we have (thanks to (B.6)), P N B (cid:12)(cid:12)(cid:12) τ = Z τ P N [ − a · ∇ x B + b · ∇ x U + ∆ x B + h ] d t for any N ≥ τ ∈ [0 , T ], and therefore B is a solution to the second equation in (B.1).The energy equality (B.2) is simply a consequence of taking the L ( T ; R ) inner product of the first andsecond equation in (B.1) with U and B , respectively, adding up, performing integrations by parts (thanksto the spatial regularity of every term), and cancellation using ∇ x · U = ∇ x · B = 0.Finally, for (B.3), we use the first equation (which is the more difficult one) of (B.1) as an example. Theregularity of each term allows us to take its L ( T ; R ) inner product with ∂ t U and perform integrations byparts to arrive at k ∂ t U k L ( T ) + 12 dd t k∇ x U k L ( T ) = h ∂ t U , − a · ∇ x U + b · ∇ x B + U × g + h i≤ k ∂ t U k L ( T ) k ( ∇ x U , ∇ x B , U , h ) k L ( T ) sup {| a | , | b | , | g | , }≤ k ∂ t U k L ( T ) + 12 k ( ∇ x U , ∇ x B , U , h ) k L ( T ) sup {| a | , | b | , | g | , } . We treat the second equation of (B.1) similarly and add the resulting inequality to the one above, and finallyintegrate the resulting sum in time to deduce (B.3). (cid:3)
Appendix C. Separability of C c ([ T , T ] × T × R )For T < T , denote by C c ([ T , T ] × T × R ) the set of all real-valued, compactly supported functionsin C ([ T , T ] × R × R ), which are 2 π -periodic with respect to their second argument, x , for all ( t, v ) ∈ [ T , T ] × R . Our goal is to show that C c ([ T , T ] × T × R ) is separable with respect to the C norm. Weshall rely on the classical Stone–Weierstrass theorem. Theorem C.1 (Stone–Weierstrass theorem) . Suppose that U is a compact Hausdorff space and A is asubalgebra of the space of real-valued continuous functions C ( U ; R ) , which contains a nonzero constantfunction. Then A is dense in C ( U ; R ) if, and only if, it separates points. A set A ⊂ C ( U ; R ) is said to separate points in U if, for any u = u ′ ∈ U , there exists at least one element g ∈ A such that g ( u ) = g ( u ′ ). The set A may or may not be countable.Now, for any positive integer N , let B R ( , N ) denote the open ball in R centered at the origin and ofradius N , and define U N := ( T × B R ( , N )) ⊂ ( T × R ). We then have the following natural consequenceof the Stone–Weierstrass theorem. Proposition C.2.
Let C ( U N ) denote the space of all continuous functions that are defined on the compactdomain U N . Then, C ( U N ) is separable with respect to the C norm. Proof.
Our proof consists of two steps.
Step 1.
In this step we construct an uncountable dense subset A of C ( U N ). Clearly, C ( U N ) is analgebra over the field R . By the Stone–Weierstrass theorem, we need at least one nonzero constant function1 ∈ A , and some other elements to “separate” point-pairs in U N . To this end, consider two different points u = ( x , x , x , v , v , v ), u ′ = ( x ′ , x ′ , x ′ , v ′ , v ′ , v ′ ). If v = v ′ , then obviously g ( u ) = v “separates” u and u ′ . So we include v ∈ A , and likewise for v and v . If x = x ′ as elements of T = R / (2 π Z ), then sin( x ) EAK SOLUTIONS TO A HYBRID VLASOV-MHD MODEL IN PLASMA PHYSICS 33 and cos( x ) together “separate” u and u ′ . Otherwise, having both sin( x ) = sin( x ′ ) and cos( x ) = cos( x ′ )would imply that 0 = (sin( x ) − sin( x ′ )) + (cos( x ) − cos( x ′ )) = 2 − x − x ′ ) , which would mean that x and x ′ are identical elements of T , thus contradicting the assumption that x = x ′ as elements of T . Thus, we include sin( x ) and cos( x ) in A , and likewise for the x and x coordinates.In summary, the smallest subalgebra A of C ( U N ) that contains { , v , v , v , sin( x ) , cos( x ) , sin( x ) , cos( x ) , sin( x ) , cos( x ) } is dense in C ( U N ). However, this algebra is over the field R , and therefore it is not countable. Step 2.
Let A Q therefore be the smallest algebra over the field Q of rational numbers that contains theabove 10 functions. Since Q is dense in R , and the above 10 functions are clearly bounded over the compact domain U N , we have that A Q is dense in A and thus also dense in C ( U N ) . Therefore, we have shown that C ( U N ) is separable with respect to the C norm and hence the proof is complete. (cid:3) We also need the following proposition.
Proposition C.3.
Given a separable metric space X , any set S ⊂ X has a countable subset that is densein S with respect to the X metric. Proof.
To prove this, let d ( · , · ) be the metric of X , and let { a n } n ≥ be dense in X . Let B ( a n , k ) be the openball centered at a n of radius k in the X metric. We then have countably many sets of the form S ∩ B (cid:18) a n , k (cid:19) , n, k ∈ N . For any nonempty such set, we pick a “representative” a n,k ∈ S ∩ B ( a n , k ). We claim that the countable set (cid:26) a n,k ∈ S ∩ B (cid:18) a n , k (cid:19) (cid:12)(cid:12)(cid:12) S ∩ B (cid:18) a n , k (cid:19) = ∅ , n, k ∈ N (cid:27) ⊂ S is dense in S . Indeed, given any k ∈ N and b ∈ S , by the density of { a n } n ≥ , we can find an a n such that d ( a n , b ) < k = ⇒ b ∈ B (cid:18) a n , k (cid:19) . Hence, b ∈ S ∩ B ( a n , k ) = ∅ , so the “representative” a n,k ∈ S ∩ B ( a n , k ) exists. Now, since b and a n,k areboth contained in the ball B ( a n , k ), it follows that d ( a n,k , b ) < /k . Thus, by the arbitrariness of k , wecomplete the proof. (cid:3) We are ready to state and prove the main proposition of this appendix.
Proposition C.4.
For any T < T , the space C c ([ T , T ] × T × R ) is separable with respect to the C norm. Proof.
By Proposition C.3, it suffices to prove the separability of C c ( R × T × R ). Since for functions definedon R × T × R the proof is identical to the one in the case of functions defined on T × R , for the sake ofsimplicity we shall only show that the space C c ( T × R ) is separable with respect to the C norm. To thisend, according to Proposition C.3 again, it suffices to show that:The space C c ( T × R ) is separable with respect to the C norm. (C.1)For any positive integer N , the space C c ( U N ) can be regarded as a subspace of C ( U N ). Then, bycombining Propositions C.2 and C.3, we obtain that C c ( U N ) has a countable subset that is dense in C c ( U N )with respect to the C norm.On the other hand, any element of C c ( U N ) can be naturally extended from U N to the whole of T × R and it can be therefore viewed as an element of C c ( T × R ). Hence, C c ( T × R ) = [ N ≥ C c ( U N ) . By a countability argument and the previous step we then deduce (C.1). (cid:3)
Department of Mathematics, University of Surrey, Guildford, GU2 7XH, UK
E-mail address : [email protected] Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, UK
E-mail address : [email protected] Department of Mathematics, University of Surrey, Guildford, GU2 7XH, UK
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