On the energy conservation by weak solutions of the relativistic Vlasov-Maxwell system
aa r X i v : . [ m a t h . A P ] D ec ON THE ENERGY CONSERVATION BY WEAK SOLUTIONS OFTHE RELATIVISTIC VLASOV-MAXWELL SYSTEM
REINEL SOSPEDRA-ALFONSO
Abstract.
We show that weak solutions of the relativistic Vlasov-Maxwellsystem preserve the total energy provided that the electromagnetic field islocally of bounded variation and, for any λ >
0, the one-particle distributionfunction has a square integrable λ -moment in the momentum variable. Introduction
Consider an ensemble of relativistic charged particles that interact through theirself-induced electromagnetic field. If collisions among the particles are so improba-ble that they can be neglected, then the ensemble can be modeled by the so-calledrelativistic Vlasov-Maxwell (RVM) system. At any given time t ∈ ]0 , ∞ [, the RVMsystem is characterized by the one-particle distribution function f = f ( t, x, p ) withposition x ∈ R and momentum p ∈ R . The self-induced electric and magneticfields are denoted by E = E ( t, x ) and B = B ( t, x ), respectively. Setting all physicalconstants to one, the model equations for a single particle species read(1.1) ∂f∂t + v · ∇ x f + ( E + v × B ) · ∇ p f = 0(1.2) ∂E∂t − ∇ × B = − πj (1.3) ∂B∂t + ∇ × E = 0(1.4) ∇ · E = 4 πρ , ∇ · B = 0,where v := p (cid:0) | p | (cid:1) − / denotes the relativistic velocity. The coupling of theVlasov (1.1) and Maxwell equations (1.2)-(1.4) is through the charge and currentdensities, which we denoted by ρ = ρ ( t, x ) and j = j ( t, x ) respectively. They aredefined by(1.5) ρ := R R f dp , j := R R vf dp .We define the Cauchy problem for the RVM system by (1.1)-(1.5) with initial data(1.6) f | t =0 = f , E | t =0 = E , B | t =0 = B ,satisfying (1.4) in the sense of distribution. It is not difficult to check that if (1.4)holds at t = 0, then it will do so for all time in which the solution exist. Thus, theequations (1.4) can be understood as a mere constraint on the initial data. Key words and phrases.
Vlasov-Maxwell, weak solutions, conservation of the total energy.
Now, define L kin ( R ) := (cid:26) g ∈ L ( R ) : g ≥ , Z Z q | p | g ( x, p ) dxdp < ∞ (cid:27) . For
T >
0, we say that ( f, E, B ) is a weak solution of the RVM system if(1.7) f ∈ L ∞ ([0 , T [; L kin ∩ L ∞ ( R )), E, B ∈ (cid:2) L ∞ ([0 , T [; L ( R )) (cid:3) and the equations (1.1)-(1.4) are satisfied in the sense of distributions. In particular,we say that the Vlasov equation (1.1) is satisfied in the sense of distributions if forall ϕ ∈ C ∞ ([0 , T ] × R ) with compact support in [0 , T [ × R Z T Z R × R f ( t, x, p ) [ ∂ t ϕ + v · ∇ x ϕ + K · ∇ p ϕ ] ( t, x, p ) dtdxdp = − Z R × R f ( x, p ) ϕ (0 , x, p )) dxdp. (1.8)We define analogous relations for the Maxwell equations (1.2)-(1.4) as well. Thevector field K := E + v × B in (1.8) denotes the Lorentz force acting on a referenceparticle of velocity v . Notice that it satisfies ∇ p · K ≡ t by its value at t = 0, namely E ( t ) := Z R × R q | p | f ( t, x, p ) dxdp + 18 π Z R | E ( t, x ) | + | B ( t, x ) | dx ≤ E (0) . (1.9)In the present note we show that if the electric and magnetic fields E and B arelocally of bounded variation and, for any λ >
0, the function(1.10) ρ λ ( t, x ) := Z R | p | λ f ( t, x, p ) dp is square integrable, then the relation (1.9) is in fact an equality for almost all0 ≤ t < T . Precisely, we prove the following result: Theorem 1.
Let λ > . Let f ∈ L kin ∩ L ∞ ( R ) , E , B ∈ (cid:2) L ( R ) (cid:3) anddenote by ( f, E, B ) a weak solution of the RVM system satisfying ( f, E, B ) | t =0 =( f , E , B ) . If E, B ∈ [ L loc (]0 , T [; BV loc ( R ))] and ρ λ as defined in (1.10) is in L ∞ loc (]0 , T [; L ( R )) , then the total energy defined by (1.9) satisfies E ( t ) = E (0) foralmost all ≤ t < T . The tools we use are basically those introduced by DiPerna and Lions in [4]to study renormalized solutions of transport equations. We shall also refer to [2],where applications to the Vlasov equation are given. We remark that the sameresult holds for the electromagnetic field in [ L loc (]0 , T [; W , loc ( R ))] since we havethe (strict) inclusion W , (Ω) ⊂ BV (Ω) for any open set Ω ⊆ R . For a detailedaccount on functions of bounded variation cf. [1]. We would like to include here thereference [5], where the uniqueness of weak solutions for the Vlasov-Poisson system NERGY CONSERVATION BY WEAK SOLUTIONS 3 has been obtained under the sole assumption that the spatial density is bounded.Similar results would be desirable for the more demanding Vlasov-Maxwell system.Formally, the law of the conservation of the total energy is derived as follows.Multiply the Maxwell equations (1.2) and (1.3) by E and B respectively and inte-grate on R to find that(1.11) 18 π ddt Z R | E | + | B | dx = − Z R j · Edx.
Multiply the Vlasov equation by q | p | and integrate on R × R to get(1.12) ddt Z R × R q | p | f dxdp = Z R j · Edx.
Then the sum of (1.11) and (1.12) provide the desired result.As for weak solutions, we shall follow the same scheme. We find relations anal-ogous to (1.11) and (1.12) in sections 2 and 3 respectively. The difficulty is toovercome the lack of regularity and the need of justifying the operations taken forgranted when the solutions are smooth.2.
Energy balance for the Maxwell equation
Here we show that if the current j is square integrable for almost all time, thenthe weak solution of the RVM system satisfies the energy balance associated to theMaxwell equations, i.e., the relation (1.11). This result is reminiscent of the dualitytheorem for transport equations given by DiPerna and Lions in [4]. Lemma 1.
Let ( f, E, B ) be a weak solution of the RVM system with initial data ( f , E , B ) . If j as defined in (1.5) is in [ L ∞ (]0 , T [; L ( R ))] , then (2.1) 18 π ( k E ( t ) k L x + k B ( t ) k L x ) + Z t Z R j · Edsdx = 18 π ( k E k L x + k B k L x ) for almost all t ∈ [0 , T [ .Proof. Let ǫ > κ ∈ C ∞ ( R ), κ even, be a standard mollifier. Define theregularization kernel κ ǫ := ǫ κ ( xǫ ). Since mollification and distributional differenti-ation commute, i.e., ( ∂ x u ) ∗ κ ǫ = ∂ x ( u ∗ κ ǫ ), we can convolute (1.2) and (1.3) with κ ǫ to obtain ∂E ǫ ∂t − ∇ × B ǫ = − πj ǫ (2.2) ∂B ǫ ∂t + ∇ × E ǫ = 0 , (2.3)where j ǫ := j ∗ κ ǫ and ( E ǫ , B ǫ ) := ( E, B ) ∗ κ ǫ .Consider the family of smooth cut-off functions φ R := φ ( · R ), R ≥ φ ∈ C ∞ ( R ), φ ≥ φ ≡ B ⊂ supp φ ⊂ B . The smoothness ofthe fields B ǫ and E ǫ with respect to x imply via (2.2) and (2.3) that ∂ t E ǫ , ∂ t B ǫ ∈ L loc (cid:2) (]0 , T [ × R ) (cid:3) . Thus, E ǫ , B ǫ ∈ h W , loc (]0 , T [ × R ) i and we can apply the chainrule in Sobolev spaces, i.e., for almost all t ∈ ]0 , T [12 ∂∂t (cid:16) | E ǫ ( t ) | + | B ǫ ( t ) | (cid:17) = E ǫ · ∂E ǫ ∂t + B ǫ · ∂B ǫ ∂t . REINEL SOSPEDRA-ALFONSO
Therefore, we can multiply (2.2) and (2.3) by E ǫ φ R and B ǫ φ R respectively, sumthe resultant equations and integrate by parts to find that18 π Z R (cid:16) | E ǫ ( t ) | + | B ǫ ( t ) | (cid:17) φ R − π Z R (cid:16) | E ǫ (0) | + | B ǫ (0) | (cid:17) φ R = 14 π Z t Z R ( E ǫ × B ǫ ) · ∇ φ R − Z t Z R j ǫ · E ǫ φ R . (2.4)Let ǫ →
0. The terms on the left side converge as a consequence of the theoremof smooth approximations [6, Theorem 3, p.196]. Also, the same theorem and theassumption made on the current j easily implies that R j ǫ · E ǫ → R j · E for almostall s ∈ [0 , T [. Thus, we may invoke the Lebesgue dominated convergence theoremand the convergence of the second term in the right side follows as well. Clearly,the same reasoning applies to the remaining term. Then, for almost all t ∈ [0 , T [18 π Z R (cid:16) | E ( t ) | + | B ( t ) | (cid:17) φ R − π Z R (cid:16) | E (0) | + | B (0) | (cid:17) φ R = 14 π Z t Z R ( E × B ) · ∇ φ R − Z t Z R j · Eφ R . (2.5)Finally, since for some constant C T that does not depend on R (cid:12)(cid:12)(cid:12)(cid:12)Z t Z R ( E × B ) · ∇ φ R (cid:12)(cid:12)(cid:12)(cid:12) ≤ C T R k E k L ∞ , t,x k B k L ∞ , t,x , it is easy to check that (2.1) follows from (2.5) by letting R → ∞ . The proof of thelemma is complete. (cid:3) Energy Balance for the Vlasov Equation
In this section we deduce the duality formula [4] resulting from the Vlasov equa-tion (1.1) and (the identity) K ·∇ p q | p | ≡ v · E , which gives the energy balanceassociated to the Vlasov equation. Since we now face a nonlinear term in (1.1), weneed to first prove the following lemma, a particular case of Lemma 3.5 in [2]. Lemma 2.
Let κ ǫ and κ ǫ be two regularization kernels defined on R x and R p respectively. Let ( f, E, B ) be a weak solution of the RVM system. If E, B ∈ (cid:2) L (]0 , T [; BV loc ( R )) (cid:3) , then there exist two sequences ǫ n > , ǫ n > , ǫ n → , ǫ n → such that ∇ x · (cid:2) v ( κ ǫ n κ ǫ n ∗ f ) (cid:3) + ∇ p · (cid:2) K ( κ ǫ n κ ǫ n ∗ f ) (cid:3) − ( ∇ x · [ vf ]) ∗ κ ǫ n κ ǫ n − ( ∇ p · [ Kf ]) ∗ κ ǫ n κ ǫ n (3.1) converges to in L (]0 , T [; L loc ( R × R )) .Proof. First we omit the dependence in time and show the corresponding conver-gence on R × R . Then we study the convergence on time as well. NERGY CONSERVATION BY WEAK SOLUTIONS 5
Indeed, the compact support of the mollifiers and the divergence theorem allowus to rewrite (3.1) as I v ( x, p ) + I K ( x, p ):= Z Z [ f ( x, p ) − f ( x − y, p − q )][ v ( p ) − v ( p − q )] · ∇ y κ ǫ ( y ) κ ǫ ( q ) dydq + Z Z [ f ( x, p ) − f ( x − y, p − q )][ K ( x, p ) − K ( x − y, p − q )] · ∇ q κ ǫ ( q ) κ ǫ ( y ) dydq. In addition, since we have that K ( x, p ) − K ( x − y, p − q ) = E ( x ) − E ( x − y ) + v ( p ) × [ B ( x ) − B ( x − y )]+[ v ( p ) − v ( p − q )] × B ( x − y ) , (3.2)we may decompose the second integral by I K = I K,x + I K,p where I K,x involves the first two terms in the right side of (3.2) and I K,p involvesthe third term. Now, let
R > B R × B R =: Ω ⊂ R × R suchthat Ω + supp κ ǫ κ ǫ ⊂ B R +1 × B R +1 . In view of the assumptions of the lemma k E ( x ) − E ( x − y ) k L x ( B R ) ≤ k∇ x E k M ( B R +1 ) | y | , | y | < ǫ ,(similarly for B ), where k∇ x E k M ( B R +1 ) < ∞ denotes the norm of the measure ∇ x E (resp. ∇ x B ), which coincides with the variation of E (resp. B ) on the ball B R +1 . Hence, since the relativistic velocity v ∈ (cid:2) C ∞ b ( R ) (cid:3) satisfies | v | ≤
1, we findthat for some positive constant C R that depends on R (cid:13)(cid:13) I K,x (cid:13)(cid:13) L x,p (Ω) ≤ C R ǫ ǫ (cid:16) k∇ x E k M ( B R +1 ) + k∇ x B k M ( B R +1 ) (cid:17) × (cid:18)Z | ǫ ∇ q κ ǫ | (cid:19) sup | y |≤ ǫ , | q |≤ ǫ k f ( x, p ) − f ( x − y, p − q ) k L ∞ x,p (Ω) . (3.3)Similarly, we find the estimates (cid:13)(cid:13) I K,p (cid:13)(cid:13) L x,p (Ω) ≤ k B k L x k∇ p v k L p ( B R +1 ) × (cid:18)Z | ǫ ∇ q κ ǫ | (cid:19) sup | y |≤ ǫ , | q |≤ ǫ k f ( x, p ) − f ( x − y, p − q ) k L x,p (Ω) (3.4)and k I v k L x,p (Ω) ≤ C R ǫ ǫ (cid:18)Z | ǫ ∇ y κ ǫ | (cid:19) k∇ p v k L p ( B R +1 ) × sup | y |≤ ǫ , | q |≤ ǫ k f ( x, p ) − f ( x − y, p − q ) k L x,p (Ω) . (3.5)Now, we have (cid:0)R | ǫ ∇ κ ǫ | (cid:1) ≤ C , and we also have thatsup | y |≤ ǫ , | q |≤ ǫ k f ( x, p ) − f ( x − y, p − q ) k L x,p (Ω) → , as ǫ , ǫ → . Hence, we can choose two sequences ǫ n , ǫ n → ǫ n /ǫ n = 1 /n such that forsome n sufficiently large the right-hand sides of (3.4) and (3.5) are less than 1 /n .Therefore, since we also have f ∈ L ∞ (Ω), it follows that (3.3), (3.4) and (3.5) goto zero as n → ∞ , and so does (3.1) in L loc ( R × R ).Finally, we consider the dependence in time. The difficulty here seems to arisebecause the sequences ǫ n and ǫ n may also depend on t . Otherwise we could just REINEL SOSPEDRA-ALFONSO invoke the Lebesgue dominated convergence theorem as (3.3) and (3.5) suggest. Inparticular, we must be careful with the estimate (3.5). Nevertheless, if we keeptrack of the time dependence along the calculation, we find that k I v k L , t,x,p ((0 ,T ) × Ω) ≤ C R ǫ ǫ sup | y |≤ ǫ , | q |≤ ǫ k f ( t, x, p ) − f ( t, x − y, p − q ) k L , t,x,p ((0 ,T ) × Ω) and we can reason as above. This concludes the proof of the lemma. (cid:3) We now turn to the energy balance (1.12) associated to the Vlasov equation.
Lemma 3.
Let λ > . In addition to the assumptions of Lemma 2, suppose that ρ λ as defined in (1.10) is in L ∞ (]0 , T [; L ( R )) . Then (3.6) Z R × R q | p | f ( t ) dxdp = Z R × R q | p | f (0) dxdp + Z t Z R E · jdsdx for almost all t ∈ [0 , T [ .Proof. ( f, E, B ) is a weak solution of the RVM system. Thus, as a straightforwardconsequence of Lemma 2, there are two sequences ǫ n > ǫ n > ǫ n → ǫ n → ∂ t f n + v · ∇ x f n + K · ∇ p f n = r n converges to 0 in L (]0 , T [; L loc ( R × R )) as n → ∞ , where f n := κ ǫ n κ ǫ n ∗ f and r n is defined by (3.1).Consider a family of smooth cut-off functions φ R = φ ( · R ), R ≥ φ ∈ C ∞ ( R ), φ ≥ φ ≡ B ⊂ supp φ ⊂ B . If we multiply (3.7) by q | p | φ R and integrate by parts, we find that Z R q | p | f n ( t ) φ R − Z R q | p | f n (0) φ R = Z t Z R E · vf n φ R + Z t Z R q | p | φ R r n + Z t Z R q | p | f n K · ∇ p φ R + Z t Z R q | p | f n v · ∇ x φ R . (3.8)Here we have used the identity K · ∇ p q | p | ≡ v · E .Let n → ∞ . In doing so, we notice that the second term in the right-hand sidevanishes as a consequence of Lemma 2. Also, the convergence of the two terms in theleft-hand side and the last term in the right-hand side follow by a straightforwardapplication of the theorem of smooth approximations. Thus, we are led to provethe convergence of the first and third terms in the right-hand side.Indeed, the reasoning done so far does not preclude us from writing φ R as theproduct of two suitable functions χ R = χ ( · R ) and ζ R = ζ ( · R ) where χ ∈ C ∞ ( R x )and ζ ∈ C ∞ ( R p ). Hence, (cid:12)(cid:12)(cid:12)(cid:12)Z R E · v ( f − f n ) φ R (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z R | E | χ R Z R ζ R | f − f n |≤ C R k E k L x k f − f n k L x,p , which converges to zero as n → ∞ . Then, a use of the Lebesgue dominated con-vergence theorem provide the convergence of the first term in the right-hand side. NERGY CONSERVATION BY WEAK SOLUTIONS 7
Since we can do similarly with the remaining term, we find that as n → ∞ , (3.8)converges to Z R q | p | f ( t ) φ R = Z R q | p | f (0) φ R + Z t Z R E · vf φ R + Z t Z R q | p | f K · ∇ p φ R + Z t Z R q | p | f v · ∇ x φ R . Finally, we let R → ∞ and show that the above equality converges to (3.6). Theconvergences of the term in the left and the first term in the right-hand side arestraightforward, since for t = 0 and for almost all t > f ( t ) ∈ L kin ( R ). Also,since (cid:12)(cid:12)(cid:12)(cid:12)Z t Z R q | p | f v · ∇ x φ R (cid:12)(cid:12)(cid:12)(cid:12) ≤ CR Z t Z R q | p | f ≤ C T R , the last term converges to zero as R → ∞ . In order to obtain the convergence ofthe second term in the right, we first notice that for any λ > ρ ( t, x ) = Z | p |≤ f ( t, x, p ) dp + Z | p | > f ( t, x, p ) dp ≤ p π/ k f ( t, x ) k L p + ρ λ ( t, x ) . Then, the hypothesis made in the lemma implies that(3.9) k ρ ( t ) k L x ≤ p π/ k f ( t ) k L x,p + k ρ λ ( t ) k L x < ∞ . As a result, and since | v | ≤
1, we can easily verify that E · vf ∈ L (]0 , T [ × R ), sothe Lebesgue theorem provides the expected convergence. Hence, we are only leftto show that the third term in the right-hand side converges to zero. To this end,we first produce(3.10) (cid:12)(cid:12)(cid:12)(cid:12)Z t Z R q | p | f K · ∇ p φ R (cid:12)(cid:12)(cid:12)(cid:12) ≤ CR Z t Z R ( | E | + | B | ) Z R ≤| p |≤ R | p | f. To estimate the above inequality we observe that1 R Z R ≤| p |≤ R | p | f ≤ (cid:26) − λ ρ λ /R λ , < λ < ρ λ /R λ , ≤ λ . Thus, for any λ >
0, there exists a constant
C > R such that theright-hand side of (3.10) is less or equal than CR λ Z t (cid:16) k E ( s ) k L x + k B ( s ) k L x (cid:17) k ρ λ ( s ) k L x ds ≤ C T R λ . Therefore, (3.10) converges to zero as R → ∞ and the proof of the lemma iscomplete. (cid:3) Proof of Theorem 1
Proof.
Since | v | ≤
1, we have | j | ≤ ρ . Then, in view of (3.9), we can combineLemmas 1 and 3 to produce the equality for almost all 0 ≤ t < T claimed for(1.9). (cid:3) Acknowledgement.
I am grateful to Prof. R. Illner for useful discussions andinsightful comments concerning the subject matter of this paper.
REINEL SOSPEDRA-ALFONSO
References [1] Ambrosio, L., Fusco, N. and Pallara, D.,
Functions of bounded variation and free disconti-nuity , Oxford University Press, Inc. NY, 2000.[2] Bouchut, F.,
Renormalized solutions to the Vlasov equation with coefficients of boundedvariation , Arch. Rational Mech. Anal., 157:75-90, 2001.[3] DiPerna, R.J. and Lions, P.L.,
Global weak solutions of the Vlasov-Maxwell systems , Com-mun. Pure Appl. Math., 42(6):729-757, 1989.[4] DiPerna, R.J. and Lions, P.L.,
Ordinary differential equations, transport theory and Sobolevspaces , Invent. Math., 98:511-547, 1989.[5] Loeper, G.,
Uniqueness of the solution to the Vlasov-Poisson system with bounded density ,J. Math. Pures Appl., 86:68-79, 2006.[6] McOwen, R.,
Partial Differential Equations. Methods and Applications , Prentice Hall Inc.,Simon-Schuster/A Viacom Company, New Jersey-U.S.A., 1996.[7] Rein, G.,
Global weak solutions to the relativistic Vlasov-Maxwell system revisited , Comm.Math. Sci., 2(2):145-158, 2004.
Department of Mathematics and StatisticsUniversity of Victoria, PO BOX 3045 STN CSC, Victoria BC V8W 3P4
E-mail address , R. Sospedra-Alfonso:, R. Sospedra-Alfonso: