Diffusion Limit for the Vlasov-Maxwell-Fokker-Planck System

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1 IAENG International Journal o Applie Mathematics, 40:3, IJAM_40_3_05 Diusion Limit or the Vlasov-Maxwell-Fokker-Planck System Najoua El Ghani Abstract In this paper we stuy the iusion limit o the Vlasov-Maxwell-Fokker-Planck System. Here, we generalize the one an one hal imensional case, obtaine in [5] to the case o several space imensions using global renormalize solutions. The limit equation consists in a Drit-Diusion equation where the rit velocity is eine by means o the Maxwell relation. Keywors: Vlasov-Maxwell-Fokker-Planck system, Drit-Diusion moel, moment metho, velocity averaging lemma, renormalize solution. Introuction We consier a plasma in which the ilute charge particles interact both through collisions an through the action o their sel-consistent electro-magnetic iel. The evolution o the plasma is governe by the ollowing equations t + v. x + ( E + δ v B ). v = θ L F P ( ) are respectively the charge an current ensities o. The imensionless parameter is proportional with the scale thermal mean ree path an also with the scale macroscopic velocity. We are intereste in the asymptotic regime 0 < <<. For simplicity, we take δ = θ =. I we neglect the magnetic iel we obtain the Vlasov- Poisson-Fokker-Planck system (VPFP or short) on where, t + v. x xφ. v = L F P ( ), L F P () = v.(v + v ), φ = p(x) (6) where p(x) is the ensity o a backgroun o positive charges which is assume to be ixe. The asymptotic behavior o this system when goes to 0 was stuie in [], [7] an [4]. It was shown that the limit (, φ) = lim(, φ ) solves the ollowing Drit-Diusion 0 limit { t + x.( x x φ) = 0, (7) x φ = p(x). := θ v.(v + v ), (t, x, v) ]0, T [, () t E curl x B = J, (t, x) ]0, T [, () δ t B + curl x E = 0, (t, x) ]0, T [, (3) iv x E =, (t, x) ]0, T [, (4) iv x B = 0, (t, x) ]0, T [, (5) where δ, θ an are imensionless parameters. The system (), (), (3), (4), (5) is reere to as the rescale Vlasov-Maxwell-Fokker-Planck (VMFP or short) system on where. Here (t, x, v) 0 is the istribution unction o particles, E, B stan or the electric an magnetic iels respectively. (t, x) = v, J (t, x) = v v. Département e Mathématiques, Faculté es Sciences e Tunis, TN-09 El Manar Tunis, Tunisia. Tel: Fax: Najoua.ElGhani@st.rnu.tn By taking as a small parameter the square o the ratio o the thermal mean ree path with respect to the Debye length an by assuming that the istance travele by the light uring the relaxation time is o the Debye length, Bostan-Gouon [5] has prove, in this case, the convergence o the ollowing VMFP system in one an one hal imensional case, i.e = (t, x, p, p ), E = (E (t, x), E (t, x), 0) an B = (0, 0, B (t, x)) or any (t, x, p, p ) [0, T ] 3. t + ( v (p). x + E + δ v (p)b ) p +( E δ v (p)b ) p = θ L( ) t E = := θ iv p( p + v(p) ), (8) v (p) (t, x, v)v + J(t, x), (9) (Avance online publication: 9 August 00)

2 IAENG International Journal o Applie Mathematics, 40:3, IJAM_40_3_05 t E + x B = v (p) v, (0) δ t B + x E = 0, () x E = (t, x) D(t, x), () where D, J : [0, T ] are the charge an current ensities o a backgroun particle istribution o an opposite sign, satisying D 0 an the continuity equation t D + x J = 0. The ollowing limit system was obtaine { θ t E + E xe = x D + θj, x E = D (3) In this paper, we generalize the stuy one in [5]. Inee, i we want to work with global solutions we have only to eal with solutions to the VMFP system which are eine in the renormalize sense. Notice that in () the main iiculty relies on the non linear term E. v which is the same orer o magnitue in the iusion Fokker-Planck term. The system is complete by prescribing the initial conitions or the istribution unction an the electro-magnetic iel (E, B ) (t = 0, x, v) = 0 (x, v), (4) E (t = 0, x) = E 0(x), B (t = 0, x) = B 0(x), (5) satisying iv x E0(x) = 0 (x, v)v, iv x B0(x) = 0. (6) Now, note that the Fokker-Planck operator can be written as the ollowing L F P () = iv v (e v v (e v )), (7) an thereore () becomes t + v. x + (v B ). v = iv v(e v E v (e v E )) (8) We can expect rom (8) that when 0, the istribution converge to (t, x, v) (t, x)m(v), (9) where M is the normalize Maxwellian with zero mean velocity: M(v) = (π) e v. (0) Let us eine the Hilbert space L M ( ) as L M ( ) = { L ( )/ v M < + }, equippe with the inner prouct <, g > = v L g M ( ) M. The operator L F P acting on L M ( ) is unboune, with omain D(L F P ) = { L M ( )/ v (M v ( M )) L M ( )}, an it satisies the ollowing proposition Proposition. [4] The operator L F P is continuous on L (v) an satisies. L F P is sel ajoint on L M ( ).. Ker(L F P ) = M. 3. (L F P ) = {g L M ( )/ g(v)v = 0}, 4.For all g (L F P ), there exists D(L F P ) such that L F P () = g. This solution is unique uner the solvability conition (v)v = 0. We enote that = L F P (g). 5. L F P satisies the ollowing H-theorem or all L (v) D(L F P ), L F P () ln( M )v = v ln( M ) v. The VMFP system is motivate to the plasma physics, as or instance in the theory o semiconuctors, the evolution o laser-prouce plasmas or the escription o tokamaks. The iusion limit has been analyze in the kinetic theory o semiconuctors in [0] an [9] an we can see also [4] or the VPFP system. Beore giving the main result, let us assume that the sequence o initial ata satisies: A : 0 0, 0 (+ x + v + ln( 0 ) ) vx < C, or some constant C > 0 inepenent o. A : E0 an B0 are uniormly boune in L ( ). Here, our main result to state as the ollowing (see section. or the einition o renormalize solution) Theorem. Assume that the assumptions A an A are satisie. Let (, E, B ) be a renormalize solution o the VMFP system ()-(5). Then, M(v) in L (0, T ; L ( )), E E in L (0, T ; L ( )) weakly. B B in L (0, T ; L ( )) weakly. In particular, converges weakly in L (0, T ; L ( )) towars an (, E, B) is a weak solution o the Drit- (Avance online publication: 9 August 00)

3 IAENG International Journal o Applie Mathematics, 40:3, IJAM_40_3_05 Diusion-Maxwell system t + x.j = 0, (t, x) ]0, T [, t E curl x B = J, (t, x) ]0, T [, t B + curl x E = 0, (t, x) ]0, T [, iv x E =, (t, x) ]0, T [, iv x B = 0, (t, x) ]0, T [, (t = 0, x) = 0 (x) = 0(x, v)v, where 0 is the weak limit o 0 an J := x + E. () The proo o this Theorem is as the ollowing. In section, we recall the existence o renormalize solutions o the VMFP system. Then, in section 3, we establish some a priori uniorm estimates. In section 4, we prove the compactness o using an averaging lemma. This result will be essential in orer to pass to the limit in the equation which will be one in section 5. In the last section we prove the regularity estimates o (, E) which will en the proo o Theorem... Existence o renormalize solutions Let us now present the einition o solutions we are going to eal with. Deinition. We say that (, E, B ) is a renormalize solution to the VMFP system ()-(5) i it satisies. β C ( + ), β(t) C min( t, ), tβ (t) C an tβ (t) C, β( ) is a weak solution o t β( ) + v. x β( ) + v.((e + (v B ))β( )) = β ( ) L F P ( ) β( )(t = 0) = β(0 ). (). λ > 0, θ,λ = + λm satisies t θ,λ +v. x θ,λ + v.((e +(v B ))θ,λ ) = L F P ( ) θ,λ + λm θ,λ v.(e + (v B )) (3) emark. We point out here that the metho is base on a ouble renormalization. First, we write the equation satisie by β δ ( ) where β δ (s) = δ β(δs) or all s > 0 an ixe parameter δ > 0 an then weakly pass to the limit when goes to zero. then, we renormalize the resulting limit equation using the unction s + λm. Let us eine the ree energy unctional E (t) = + B ) + ( E ( v + ln( )) Proposition.3 The VMFP system ()-(5) has a renormalize solution in the sense o einition. which satisies in aition. The continuity equation. The entropy inequality E (t) + ln( t + x.j = 0, (4) M )vx E (0). (5) Proo. Ater integration o () with respect to v we euce that the charge an current ensities o the particles veriy the conservation law: t + iv x (J ) = 0. Now, multiplying () by v an integrating it with respect to (x, v), this implies that t v vx E.v vx = (v + v ).v vx (6) Multiplying (), (3) by E, resp B an integrating it with respect to x yiels + B )x = t ( E E.J x (7) by combining (6) an (7), we obtain t ( v vx + + B )x) ( E = (v + v ).v vx. (8) We multiply now () by (+ln( )) an ater integration with respect to (x, v), we get t ln( ) vx = (v + v ). v vx. (9) Combining (8) an (9), we euce that t E = (v + v ).(v + v ) vx = (v + v ) vx = (v + v ) vx = 4 v e v vx. ln( M )vx (Avance online publication: 9 August 00)

4 IAENG International Journal o Applie Mathematics, 40:3, IJAM_40_3_05 which is obtaine by using the logarithmic Sobolev inequality. 3. Uniorm estimates The goal o this section is to erive uniorm estimates neee or the proo o Theorem. one can establish or these renormalize solutions. We point out that we try here to generalize energy estimates obtaine in [5]. The ollowing Proposition states the usual bouns or the mass, energy an entropy. Proposition 3. Assume that assumptions A an A are satisie. Then, there exists a renormalize solution (, E, B ) o the VMFP system ()-(5)which satisies the conclusions o Proposition.. Besies, the ollowing quantities are boune or any t [0, T ], with bouns which are inepenent o an t: ( + x + v + ln( ) ) vx, + B ) x ( E an t 0 v e v vxs. Moreover, is weakly relatively compact in L ((0, T ) ). Proo. Integrating () with respect to (x, v) yiels the mass conservation t (t, x, v)vx = 0, which implies that (t, x, v)vx = 0 (x, v)vx. From Proposition. we euce that E (t) + 4 t 0 v e v E (0). (30) In orer to use the ollowing inequality with k = 4 which is base on classical arguments ue to Carleman ln( ) vx ln( )vx v +k ( x + ) vx + C k, (3) where the constant v C k = C exp( k ( x + )) vx with C = sup { y ln(y)}. Let us multiply () by x 0<y< an integrate with respect to (x, v). We euce that t x vx = implying that x + (v.x) vx x v + v vx x vx x 0 vx + t 0 L ( ) + t 0 v Combining (30), (3) an (3) yiels + t 0 e v vxs. (3) v ( x + + ln( ) ) vx + ( E + B )x v e v vxs C 4 + E (0) + ( t + x ) 0 L ( ), which leas to the esire results. Lemma 3. The current ensity J is boune in L ((0, T ) ). Proo. The current ensity can be written as the ollowing J = (v + v ) v So, by using Cauchy-Schwartz we get the estimate. Corollary 3.3 Assume that assumption A is satisie. Then, v is boune in L ((0, T ) ). Proo. Notice that v = 4 v + v 4 v v. v. Hence, by integrating with respect to (t, x, v) we conclue by using Proposition 3.. Corollary 3.4 is weakly relatively compact in L ((0, T ) ). Proo. Consier the convex unction ϕ : [0, + [, (Avance online publication: 9 August 00)

5 IAENG International Journal o Applie Mathematics, 40:3, IJAM_40_3_05 ϕ(s) = s ln(s), s > 0, ϕ(0) = 0 an the probability measure on M(v)v. By applying Jensen inequality ln() = ϕ() = ϕ( M Mv) ϕ( M )Mv = ( v + ln())v + ln(π) v. Integrating with respect to x yiels ln()x ( v +ln())vx+ ln(π) vx. Thus, we have to use the inequality o Carlemann with C = that ln() ln() + x + Ce x 4, sup { y ln(y)} an thereore, we euce 0<y< x ln() x (ln() + x )x + C e 4 x. The proposition 3. lea to ( + ln() )x C T, which implies the L ((0, T ) ) weak compactness o the sequence. Let us eine r = M ( M). Using the logarithmic Sobolev inequality an Young inequality, we euce as in [4] that Proposition 3.5 r is such that r M is boune in L ((0, T ) ), r v M is boune in L ((0, T ) ) an r v M is boune in L ((0, T ) ). 4. Compactness o the ensity Proposition 4. The ensity is relatively compact in L ((0, T ) ) an there exists L ((0, T ) ) such that, up to extraction o a subsequence i necessary, in L an a.e. The proo o this proposition is one in two steps. We irst prove the compactness o with respect to the x variable an then show the compactness in time. In the sequel, we will use the ollowing β(s) = s + s, β δ(s) = β(δs), s > 0. δ We recall that or all ixe parameter δ > 0, we have. 0 β δ (s) min(s, δ ),. β δ (s) C δ min( s, ), 3. sβ δ (s) C δ, 4. sβ δ (s) C δ. We remark that i we want to prove the Proposition 4., we only nee to show or all (ixe) δ > 0, the compactness (β δ ( )). This is a consequence o the ollowing averaging lemma (see [0] or the proo). Lemma 4. [0] Assume that h is boune in L ((0, T ) ), that h 0 an h are boune in L ((0, T ) ), an that t h + v. x h = h 0 + v.h. (33) Then, or all ψ C0 ( ), (t, x + y, v) h (t, x, v))ψ(v) v (h L t,x 0 (34) when y 0 uniormly in where h (t, x, v) has been prolonge by 0 or x /. emark 4.3 This lemma only gives the compactness in the x variable o the averages in v o h (t, x, v). This is ue to the presence o an in ront o the time erivative in 33. Proo o Proposition 4.. Let δ be a (ixe) nonnegative parameter. Let us veriy that the rescale VMFP system (in the renormalize sense) satisies the assumptions o the previous lemma. Inee, β δ ( ) is a weak solution o where an t β δ ( ) + v. x β δ ( ) = h 0 + v.h. (35) h 0 = (v + v ) v β δ ( ) h = (v + v ) β δ( ) E β δ ( ) (v B)β δ ( ). We remark that we can rewrite h 0 an h as the ollowing h 0 = (v + v ) v β δ ( ) h = (v + v ) β δ( ) E β δ ( ) (Avance online publication: 9 August 00)

6 IAENG International Journal o Applie Mathematics, 40:3, IJAM_40_3_05 (v B ) β δ( ). The sequence (β δ ( )) is boune in L L ((0, T ) ) an hence in L ((0, T ) ). Moreover, by applying Höler s inequality an using the uniorm boun o β δ ( ) in L (or ixe δ) an by using the Proposition 3. an Corollary 3.3 an 3.4, we obtain an E β δ ( ) L ((0,T ),L ( C sup E )) L t T (, ) (v B )β δ ( ) L ((0,T ),L ( )) C (sup B L t T ( + sup ) t T v vx), The assumptions o the above lemma are satisie an we get the compactness in x o β δ( )ψ(v) v or all ψ D( ), namely (34) hols with h replace by β δ ( ). Next, using that (β δ ( )) is boune in L (0, T ; L (( + v )xv)), we see that we can take ψ(v) to be constant equal to in (34) an hence we euce, ater also sening δ to 0 an using the equi-integrability o, that (t, x + y) (t, x) L t,x 0 when y 0 uniormly in. Finally, using that t = x.j is boune in L (0, T ; W, ( )), we euce that is relatively compact in L ((0, T ) ) which ens the proo o the proposition. emark 4.4 Notice that we are renormalizing the equation satisie by to get the compactness o an we nee the equation satisie by θ,λ to pass to the limit in J. 5. Passage to the limit We woul like to pass to the limit in the continuity equation t + x.j = 0. The question is to ientiy the limit o the current ensity. Corollary 5. Proo. We write M in L t,x,v an a.e. M = ( M) + ( )M. By using the previous section, there exists L ((0, T ) ) such that in L t,x an a.e. Hence, it remains to iscuss M. By the logarithmic Sobolev inequality, we obtain ln( M )vx γ v e v vx By Proposition 3., ater integration or some γ > 0. with respect to time, this quantity is ominate by C T. Eventually, we conclue by using the Csiszar-Kullback- Pinsker inequality ( M vx) µ or some µ > 0 which implies that ln( M )vx T 0 ( M) x v t T T ( ( 0 M v x) t) T µt ( 0 ln( M ) v x t) γµt C T 0 0 Let us enote by J the weak limit o J when goes to zero an r is the weak limit o r in L ((0, T ), M(v)txv). Proposition 5. J D rvm v, in L t,x. Proo. Using r, we can write = M + M r + r M. Then, we obtain J = r vm v + r vm v. Moreover, we know that in L t,x an a.e. Since ( a b) a b we have in L t,x an a.e. Thus, using Proposition 3.5 leas to J = r vm v + O( ) L t,x,v rvm v in L t,x. We enote by χ the unique solution in [(L F P ) D(L F P )] o L F P χ = vm (Avance online publication: 9 August 00)

7 IAENG International Journal o Applie Mathematics, 40:3, IJAM_40_3_05 Proposition 5.3 J D J := D.( x E ) where D is the iusion matrix eine by D = χ(v) v v. Proo. We can pass to the limit in (3) or λ > 0, up to extraction o a subsequence, we get v. x ( + λ)m + v.(e LF P (rm) ( + λ)m) = ( + λ)m + λmv.e ( + λ)m (36) where E is the L t,x weak limit o E an we have use that converges strongly to M in L t,x,v. Sening λ to 0, we iner that ( x E ).vm = L F P (rm). (37) Let us go back to the expression o the current ensity compute in Proposition 5.. Using (37), we get J = rvmv = rm L F P χ M v = L F P (rm)χ v M = [( x E ).vm]χ v M = [ χ v v].( x E ) 6. egularity o the ensity Now, we woul like to explain how we can rewrite the current J. Precisely, we shall prove that the limit L (0, T ; L ( )) an that L (0, T ; H ( )). Lemma 6. Let be a positive unction such that L (0, T ; L ( )), satisying then an x E = G L (0, T ; L ( )), iv x E =. E L (0, T ; L ( )), L (0, T ; L ( )), E L (0, T ; L ( )). L (0, T ; H ( )), (38) Proo. The irst an the thir equation o (38) imply that x L. Let β δ be an approximation o ientity, namely β δ (s) = δ β(δs) where β is a C unction satisying β(s) = s or s, 0 β (s) or all s an β(s) = or s 3. Hence, β δ (s) goes to s an β δ (s) goes to, when δ goes to 0. Now, we have x β δ ( ) = x β δ ( ). Hence, we can renormalize the irst equation appearing in (38), it gives x β δ ( ) E β δ( ) = Gβ δ( ). (39) Then, using that or ixe δ > 0, E β δ( ) 3 δ E L. We euce that x β δ ( ) L or ixe δ. Then, by taking the L norm o (39), we get x β δ ( ) L + 4 E β δ( ) L T 0 E xβ δ ( )β δ( ) xt G L. (40) Let β be given by β(s) = s 0 τβ (τ) τ an β δ (s) = β(δs). δ Hence β δ (s) goes to s when δ goes to 0. Computing the thir term in (40), we get E xβ δ ( )β δ( ) x = E β x δ ( )x = β δ ( )x. Hence, we euce that or all δ > 0, x β δ ( ) L + 4 E β δ( ) L T + 0 β δ ( )xt G L. Letting δ go to 0, we get x L + 4 E L + T 0 xt G L, which en the proo o the Lemma. Now, using the previous lemma, we can see easily that we can rewrite the current J = D[ x E] = D[ x E]. an we conclue by using that χ i = v i M an D = I. Now, using that E, B L (0, T ; L ( )) we see that t B = curl x E is boune in L (0, T ; H ( )) an t E = curl x B J is boune in L (0, T ; H ( )) + L ((0, T ) ). This ens the proo o the main Theorem.. (Avance online publication: 9 August 00)

8 IAENG International Journal o Applie Mathematics, 40:3, IJAM_40_3_05 Acknowlegment. The author woul like to thank her avisor Naer Masmoui or interesting iscussions an encouragements. eerences [] Ben Aballah N., Tayeb M. L., Asymptotique e iusion pour le systme e Boltzmann-Poisson uniimensionnel, C.. Aca. Sci. Paris Sr. I Math, N8, pp , 39/99. [] Ben Aballah N., Tayeb M. L., Diusion limit or the one imensional Boltzmann-Poisson system, Discete. Contin. Dyn. Syst. Ser. B, pp. 9-4, (004). [3] Ben Aballah N., Tayeb M. L., Diusion approximation an homogenization o the semiconuctor Boltzmann equation, Multiscale Moel. Simul, V4, N3, (005), pp [4] El Ghani N., Masmoui N., Diusion limit o a Vlasov-Poisson-Fokker-Planck system, Comm. Math. Sci, V8 (), (00), pp [5] Bostan M., Gouon T., Low iel regime or the relativistic Vlasov-Maxwell-Fokker-Planck system; the one an one hal imensional case, Kinetic relate moels, V, N (008), pp [6] Gouon T., Mellet A., Homogenization an iusion asymptotics o the linear Boltzmann, ESAIM Control Opti, Calc, Var, pp , 9 (003). [7] Gouon T., Hyroynamic limit or the Vlasov- Poisson-Fokker-Planck system: analysis o the twoimentional case, Math. Mo. Meth. Appl. Sci, V5, (005), pp [8] Gouon T., Poupau F., Approximation by homogenization an iusion o kinetic equations, Comm. PDE, 6 (00), pp [9] Masmoui N., Tayeb M. L., Homogenize Diusion or nonlinear semiconuctor Boltzmann system, J. Hyper. Di. Equ, V5, N (008), pp [0] Masmoui N., Tayeb M. L., Diusion limit o a semiconuctor Boltzmann-Poisson system, SIAM J. Math. Anal, V36, N6, (007), pp [] Poupau F., Soler J., Parabolic limit an stability o the Vlasov-Fokker-Planck system, Math. Moels Mrthos Appl. Sci, 0 (000), N7, [] P.-L. Lions,Mathematical topics in lui mechanics. Vol.. Compressible moels. Oxor Lecture Series in Mathematics an its Applications, 0. Oxor Science Publications. The Clarenon Press, Oxor University Press, New York, 998. xiv+348pp. (Avance online publication: 9 August 00)