ANALYSIS OF THE TIME-DEPENDENT MAXWELL EQUATIONS 1. The Maxwell equations

uR+

j∈KS

cj(e⁻^j^√^t−1)Sj



+cjSj (85)

As the bracket belongs to _R⁻_⊂R[H₋²(!)]_⊂R[H₋^s(!)] (Proposition 4.8). Equation (85) coin-cides with the decomposition of u in the direct sum E. Then we use Sobolev injections and interpolation arguments to show that the R[H₋^s(!)] norm of the bracket is O(t^s=2−1); as cjSj

decays faster, (84) is proven.

Finally, consider u∈D_A# and write u=A⁻^#A^#u, i.e.

u=sin#

_+∞

t⁻^#(A+tI)⁻¹A^#udt

By (84), the norm of the integrand inE is O(t^s=2−1−^#)A^#uH, and the integral converges for s¡2#.

6. ANALYSIS OF THE TIME-DEPENDENT MAXWELL EQUATIONS 6.1. The Maxwell equations

Given T ¿0, let Q= _×]0; T[ and = _×]0; T[; let c and₀ be the speed of light and the dielectric permittivity; % and _J the source terms (charge and current densities). First, there are the evolution equations

@_E

@t ⁻c²curlB=−1

₀^J; @_B

@t +curlE= 0 in Q (86) Then, the constraint equations, viz. divergence and boundary conditions:

divE= %

₀; divB= 0 in Q (87) E×n= 0; B·n= 0 on (88) The charge conservation equation,

@t +divJ= 0 in Q (89) appears as a compatibility condition for the rst equation in (86), given the rst in (87). Last, initial conditions are provided to close the system of equations,

E(0) =_E₀; _B(0) =_B₀ in (90)

The existence and uniqueness of the electromagnetic eld can be proven under suitable as-sumptions on the data and the initial conditions, cf. [1, Theorem 6.1]. In the axisymmetric case, we have:

Proposition 6.1

If %, _J and (_E₀;_B₀) are axisymmetric, so is the solution to (86) – (90). And, provided that J∈C¹(0; T; L²()) and that %∈C⁰(0; T; L²()), there holds

E∈C⁰(0; T; _X^d)_∩_C¹(0; T; L²()); _B_∈_C⁰(0; T; _X⁰ⁿ)_∩_C¹(0; T; L²()) (91)

6.2. Reduction to two-dimensional problems and basic regularity results

We now examine the simplication of the time-dependent Maxwell problem (86) – (90) in-duced by the axial symmetry. Similarly to the static problems (see Section 4), there is a decoupling of meridian and azimuthal components: namely, the problem (86) – (90) is decou-pled into two sub-systems:

• the ‘rst system’ links the meridian electric eld and the azimuthal magnetic eld;

• the ‘second system’ links the azimuthal electric and meridian magnetic elds.

Like in Section 4, it is convenient to introduce the product by r of the ‘natural’ variables u=r_E; v=r_B; f= (r=₀)_J; g=r%=₀

The following forms for the two systems are obtained through simple calculations.

The rst system. The evolution and constraint equations are:

@um

@t ⁻c²curlv=₋fm in q (92)

@t +curl₋um= 0 in q (93)

divum=g in q (94)

um·]= 0 on a; um·= 0 on b (95) The compatibility condition between fm andg reads

divfm+ @g

@t = 0 in q (96)

As for the initial data, they are

um(0) =u_m0=rE_0m; v(0) =v₀=rB₀ in ! (97) they satisfy

divu_m0=g(0) in !; u_m0_·_]= 0 on a; u_m0_·= 0 on b (98)

The second system. The evolution and constraint equations read:

@t ⁻c²curl₋vm=₋f in q (99)

@vm

@t +curlu= 0 in q (100)

divvm= 0 in q (101)

u= 0and vm·]= 0 on (102)

There is no compatibility condition for this problem. The initial data are:

vm(0) =vm0=rB_0m; u(0) =u₀=rE₀ in ! (103) they satisfy

divv_m0= 0 in !; v_m0_·_]= 0 on ; u₀= 0 on (104) Basic regularity results. Combining Proposition 6.1 with the results of Sections 2.2 and 4, we obtain the following regularity result:

(um; v)∈C⁰(0; T;U^d×H₋₁¹ (!))∩C¹(0; T;L²₋₁(!)²×L²₋₁(!)) (105) (vm; u)_∈_C⁰(0; T;U⁰ⁿ×H^◦¹₋₁(!))_∩_C¹(0; T;L²₋₁(!)²_×L²₋₁(!)) (106) The rest of this section is devoted to the improvement of this result, as announced in Reference [12]. For the azimuthal components, there holds by interpolation:

(u; v)_∈_C^0;¹⁻(0; T;H^◦₋₁(!)_×H₋₁(!)) for 0¡¡1 (107) We now focus on the meridian components and their splitting, valid at any time, into a regular and a singular part, the latter being chosen within the explicit singular spaces of Section 4.4:

um(t) =uR(t) +

i∈KS^e

i(t)S^d_i (108)

vm(t) =vR(t) +

j∈KS^b

j(t)S_j⁰ⁿ (109)

By the continuity of projections, there holds: (uR;vR)∈C⁰(0; T;U^dR×U⁰ⁿR ) and, for (i; j)∈K_S^e

×K_S^b, (i; j)∈C⁰(0; T;R×R).

6.3. Regularity in time of the singular coecients and global space – time regularity of the elds

The rst system. We consider the Hodge decomposition, valid at any time:

um(t) =curlW(t)₋rgradV(t) =hodge(V(t); W(t)) (110) If we assume: _∀t, V(t)_∈^d+ and W(t)_∈ⁿ⁻, the results of Subsection 4.2 allow us to state that they are uniquely dened and satisfy

(V; W)_∈_C⁰(0; T; ^d+_×ⁿ⁻)_∩_C¹(0; T;V^d+×Vⁿ⁻) (111) Combining (110) with (94) and (95) shows that, for all t, V(t) is the variational solution to

−⁺V(t) =%(t) 0

in !; V(t) = 0 on b; @V(t)

@ = 0 on a (112)

Obviously, the time regularity of % and V are related:

Proposition 6.2

If %_∈_C^m(0; T; [V^d+]) resp. W^{s; p}(0; T; [V^d+]) or _C^0;(0; T;L²₁(!)), then V_∈_C^m(0; T;V^d+) resp. W^{s; p}(0; T;V^d+) or _C^0;(0; T; ^d+).

Let us now look for the equation satised by W(t). Plugging (110) into (92) and (93) yields

@t ⁻⁻W= 0 (113)

curl @W

@t ⁻c²v

=₋fm+ @

@t(rgradV) (114)

The right-hand side of (114) is divergence free, thanks to (96) and (112). Hence, there exists a function such that curl=₋fm+@t(rgradV). But the left-hand side of (114), by (111) and (105), is at any time in L²₋₁(!)². Hence, it has a unique potential in H₋₁¹ (!), and we can choose =@tW ₋c²v. Combining this equation with (113) yields

@²W

@t² ⁻c²⁻W=@

@t ^def= (115)

Proposition 6.3

Assumejm∈W^1;¹(0; T; L²()), hence fm∈W^1;¹(0; T;L²₋₁(!)²). Then there exists a strong so-lutionW(t)_∈_C¹(0; T;Vⁿ⁻)_∩_C⁰(0; T; ⁿ⁻) to the evolution equation (115), supplemented with the boundary and initial conditions





W= 0 on a; @W= 0 on b

W(0) =W₀; W(0) =W₁ in !

(116) where the initial conditions satisfy

W₀_∈ⁿ⁻; W₁_∈H₋₁¹ (!); curlW₀₋rgradV(0) =u_m0; W₁=c²v₀+(0) (117)

Proof

jm∈W^1;¹(0; T; L²()) implies @t%_∈W^1;¹(0; T; H⁻¹()) by (89) and@tV_∈W^1;1(0; T;V^d+) by Proposition 6.2. Hencecurl=−fm+@t(rgradV)∈W^1;¹(0; T;L²₋₁(!)²),∈W^1;¹(0; T;H₋¹₁(!)), and nally ∈L¹(0; T;H₋¹₁(!)). Hence the existence of W by Theorem 5.1. Now (117) is clear, since and @t(rgradV), as W^1;¹ functions of time, are continuous up to t= 0, like um

and v.

Conversely, checking thatV andW dened, respectively, by (112) and (115) – (116) satisfy (110), provided (117) holds, is straightforward. Thus, we can apply the results of Section 5.

We set

V(t) =VR(t) +

i∈KS^e

_i^d(t)_{^d_iS_i^d+_} (118)

W(t) =WR(t) +

i∈KS^e

_iⁿ(t)_{ⁿ_iS_iⁿ⁻_} (119) As stated in Subsection 4.4, we can choose the constants _i^d and _iⁿ so as to have

−rgrad{^d_iS_i^d+}+curl{ⁿ_iS_iⁿ⁻}= 2S^d_i + 2w_i⁺; w_i⁺∈U^dR

−rgrad_{^d_iS_i^d+_{} −}curl_{ⁿ_iS_iⁿ⁻_}= 2w_i⁻_∈U^dR

Combining (110) with (118) and (119), we nd um(t) =curlWR(t)₋rgradVR(t) +

i∈K_S^e

{(_i^d(t)₋_iⁿ(t))w_i⁻+ (_i^d(t) +_iⁿ(t))w_i⁺_}

i∈KS^e

(_i^d(t) +_iⁿ(t))S^d_i Comparing this equation to (108), we infer

uR(t) =curlWR(t)−rgradVR(t) +

i∈K_S^e

{(_i^d(t)−_iⁿ(t))w_i⁻+i(t)w_i⁺} (120)

i(t) =_i^d(t) +_iⁿ(t) _∀i_∈_K_S^e (121)

Theorem 6.4

Assume that the sources enjoy the following space – time regularity:

%_∈_C^0;¹⁻^M^e⁻(0; T;L²₁(!)) _∀¿0 and fm∈W^1;¹(0; T;L²₋₁(!)²) Then the following regularity results hold:

i∈C^0;¹⁻^Mê⁻(0; T;_R) _∀¿0 (122) um∈C^0;¹⁻^Mê⁻(0; T;H₋₁^mê⁻(!)²) ∀; ¿0 (123)

Proof

The projection onto closed subspaces, such as _R^d+, span S_iⁿ⁻, etc., is smooth. Hence, the assumed regularity of %, together with Proposition 6.2, yields

VR∈C^0;1⁻^M^e⁻(0; T; _R^d+); _i^d_∈_C^0;¹⁻^M^e⁻(0; T;_R) and Theorem 5.8 implies

WR∈C^0;¹⁻^Mê⁻(0; T;R[H₋¹⁺^Mê⁺(!)]); _iⁿ_∈_C^0;1−^Mê⁻(0; T;_R) So, under the above hypotheses, i∈C^0;¹⁻^Mê⁻(0; T;_R).

Moreover, we know from Theorem 4.10 that for all t, um(t)_∈H₋₁^m^e⁻(!); and this space regularity is optimal. Since _R^d+_⊂H₊²(!)_⊂H₊¹⁺^m^e⁻(!), one has

−rgradVR∈C^0;1−^Mê⁻(0; T;R[H₁^mê⁻(!)²]) =_C^0;¹⁻^Mê⁻(0; T;H₋^mê₁⁻(!)²) by Proposition 2.1. The same proposition yields

WR∈C^0;¹⁻^Mê⁻(0; T;_R[H₋¹⁺^mê⁻(!)])_⊂_C^0;1−^Mê⁻(0; T;H₋¹⁺₁ ^mê⁻(!)) thus curlWR∈C^0;¹⁻^Mê⁻(0; T;H₋₁^mê⁻(!)²). Then (123) follows from (120).

Corollary 6.5

Under the hypotheses of the above theorem, there holds

E∈C^0;¹⁻^M^e⁻(0; T; H^m^e⁻()) (124) Proof

It stems from (123) that Em∈C^0;¹⁻^Mê⁻(0; T;H₁^mê⁻(!)²), where H₁^mê⁻(!)²= H₋^mê⁻(!)_×H₊^mê⁻(!) is the meridian component of H^mê⁻(). On the other hand, u sat-ises (107); taking =_mê₋ yields u∈C^0;¹⁻^mê⁺(0; T;H₋₁^mê⁻(!)) or E∈C^0;¹⁻^mê⁺(0; T;

H₋^m^e⁻(!)). The conclusion follows.

The second system. According to Section 4.3, we set at any time

vm(t) =curl’(t) (125)

with ’(t)∈^d⁻. It stems from the regularity (106) of vm, and Section 4.3 that

’∈C⁰(0; T; ^d⁻)∩C¹(0; T;V^d⁻) (126) Let us look for the equation satised by ’. It follows from (100) that curl(@t’+u) = 0.

Since u∈V^d⁻, we infer u=₋@t’, and (99) becomes

@²’

@t² ⁻c²⁻’=f (127)

Given (126), Theorem 5.1 implies:

Proposition 6.6

If j∈L¹(0; T;H₋¹(!)) and j|= 0 — hence f∈L¹(0; T;H^◦₋¹₁(!)) —’ is the strong solution inC⁰(0; T; ^d⁻)∩C¹(0; T;V^d⁻) to the evolution equation (127), supplemented with the initial and boundary conditions

’= 0 on

’(0) =’0; ’(0) =’1 in ! (128) where the initial conditions satisfy

’0∈^d⁻; ’1∈V^d⁻; curl’0=vm0; ’1=−u0 (129) The proof is similar to Proposition 6.3, and simpler. Conversely, checking that ’ solution to (127) and (128) satises (125), provided (129) holds, is straightforward. To apply the results of Section 5, we set

’(t) =’R(t) +

j∈KS^b

j(t)S_j^d⁻ (130)

We recall that curlS_j^d⁻=S_j⁰ⁿ. Comparing (125) and (130), one sees that the singular coecients j(t) are indeed the same as in (109), and that vR(t) =curl’R(t).

Theorem 6.7

Assume that the current f belongs to W^1;¹(0; T;V^d⁻). Then the following regularity results holds:

j∈C^0;¹⁻^M^b⁻(0; T;_R) _∀¿0 (131) vm∈C^0;¹⁻^M^b⁻(0; T;H₋^m^b₁⁻(!)²) _∀; ¿0 (132)

Proof

By Theorem 5.8

’R∈C^0;¹⁻^M^b⁻(0; T;R[H₋¹⁺^M^b⁺(!)]); j∈C^0;1−^M^b⁻(0; T;R)

Moreover, we know from Theorem 4.10 that for all t, vm(t)_∈H₋₁^m^b⁻(!); again, this space regularity is optimal. But

’R∈C^0;¹⁻^M^b⁻(0; T;_R[H₋¹⁺^m^b⁻(!)])_⊂_C^0;¹⁻^M^b⁻(0; T;H₋¹⁺₁ ^m^b⁻(!))

by Proposition 2.1, hence, curl’R∈C^0;¹⁻^M^b⁻(0; T;H₋₁^m^b⁻(!)²). As j is an element of C^0;¹⁻^M^b⁻(0; T;R), this implies (132).

Corollary 6.8

Under the hypotheses of the above theorem, there holds

B∈C^0;¹⁻^M^b⁻(0; T; H^m^b⁻()) (133) Proof

Similar to Corollary 6.5.

REFERENCES

1. Assous F, Ciarlet Jr P, Labrunie S. Theoretical tools to solve the axisymmetric Maxwell equations.Mathematical Methods in the Applied Sciences2002;25:49 –78.

2. Bernardi C, Dauge M, Maday Y.Spectral Methods for Axisymmetric Domains, Series in Applied Mathematics, Gauthier-Villars, North Holland: Paris, Amsterdam, 1999.

3. Mercier B, Raugel G. Resolution d’un probleme aux limites dans un ouvert axisymetrique par elements nis en r,z et series de Fourier en.RAIRO Analyse Numerique. 1982;16:405 – 461.

4. Costabel M, Dauge M. Singularities of electromagnetic elds in polyhedral domains. Archive for Rational Mechanics and Analysis2000;151:221– 276.

5. Assous F, Ciarlet Jr P, Raviart PA, Sonnendrucker E. A characterization of the singular part of the solution to Maxwell’s equations in a polyhedral domain. Mathematical Methods in the Applied Sciences 1999; 22:

485 – 499.

6. Assous F, Ciarlet Jr P, Labrunie S. Resolution of axisymmetric Maxwell equations.Technical Reportno. 50, In-stitut Elie Cartan, Nancy, 2001. URL:http://www.iecn.u-nancy.fr/Preprint/publis/Textes/2001-50.ps. 7. Assous F, Ciarlet Jr P, Sonnendrucker E. Resolution of the Maxwell equations in a domain with reentrant

corners.Model. Math. Anal. Numer. 1998;32:359 – 389.

8. Grisvard P.Singularities in Boundary-Value Problems, RMA 22. Masson: Paris, 1992.

9. Abramowitz M, Stegun IA. Handbook of Mathematical Functions. Dover: New York, 1965.

10. Ciarlet Jr P, Zou J. Finite element convergence for the Darwin model to Maxwell’s equations. Model. Math.

Anal. Numer.1997;31:213 – 250.

11. Girault V, Raviart PA.Finite Element Method for Navier–Stokes Equations. Springer: Berlin, 1986.

12. Garcia E, Labrunie S. Regularite spatio-temporelle de la solution aux equations de Maxwell. Comptes Rendus des Seances de l’Academie des Sciences Serie I 2002;334:293 – 298.

Dans le document Solution of axisymmetric Maxwell equations (Page 29-36)