Polynomial decay rate for the Maxwell s equations with Ohm s law

Transcription

1 Polynomial decay rate for the Maxwell s equations with Ohm s law Kim Dang PHUNG Sichuan University. November 2010, Paris, IHP

2 Maxwell s equation with Ohm s law Let be a smooth bounded domain in R 3. 8 " t E curlh + (x) E = 0 >< t H + curle = 0 div ( >: o H) = 0 E j@ = H j@ = 0 2 L 1 () and 0 take " o = o = 1

3 Energy E (t) = 1 2 Z je (x; t)j 2 + jh (x; t)j 2 dx E (t 2 ) E (t 1 ) = Z t2 t 1 Z (x) je (x; t)j 2 dxdt 0 E 1 (t) = 1 2 Z j@ t E (x; t)j 2 + j@ t H (x; t)j 2 dx

4 Di culties Free divergence is not preserved by the 2 t E + curl curl E t E = 0 curl curl E = E+r div E and div E 6= 0 on (0; +1) but div E = 0 on f (x) = 0g from now div E (; t = 0) = 0 on f (x) = 0g

5 Remedies "scalar potential, vector potential and Coulomb gauge". Suppose is simply connected has only one connected component. E = t A H = curl A and 8 < t 2 A + curl curl A t rp + E div A = 0 A j@ = 0 kek 2 L 2 () 3 = krpk2 L 2 () 3 + k@ tak 2 L 2 () 3 k@ t rpk L 2 () 3 kek L 2 () 3

6 Results (x) constant > 0 8x 2! (x) = 0 8x 2 n! =) treatment of the divergence part. KNOWN RESULTS lim t!+1 E (t) = 0 " GCC " = no trapped ray =) E (t) ce t E (0) NEW RESULT! = small neighborhood outside parallel trapped ray parallel trapped rays =) E (t) C t (E (0) + E 1 (0))

7 Interpolation observation inequality Polynomial decay =) 8h > 0 Z ( C E (0) 0 h ) 1= Z jej 2 dxdt + h (E (0) + E 1 (0)) Now, (= also true

8 Interpolation observation inequality for the Wave equation 8 < 2 t u u = 0 u j@ = 0 u j@ = 0 1 c ku (; t)k2 L 2 () E (u; 0) = t) u (; 0)k 2 L 2 () 4! = small neighborhood outside parallel trapped ray For parallel trapped ray, 8h > 0 Z ( C E (u; 0) 0 h ) 1= Z! t u)j 2 dxdt + h (E (u; 0) + E (@ t u; 0))

9 New operator h 2 (0; 1], L 1. Add new variable s. i@s + t 2 F = 0 F (x; t; 0) = " u (x; t) " Multiply by u over (x; t; s) 2 [ parts, T; T ] [0; L] and integrate by term at s = 0! closed to u (x; t) = term at s = L + h term at t = T + h term \ (outside )! term on! + h term \ )

10 New operator h 2 (0; 1], L 1. Add new variable s. i@s + t 2 F = 0 F (x; t; 0) = " u (x; t) " Multiply by u over (x; t; s) 2 [ parts, T; T ] [0; L] and integrate by term at s = 0! closed to u (x; t) = term at s = L + h term at t = T + h term \ (outside )! term on! + h term \ )

11 Fourier inversion formula f (x; t) = 1 (2) 4 Fourier Integral Operator Fourier transform Z R 4 e i(x+t) b f (; ) dd F (x; t; s) = 1 (2) 4 Z R 4 e i(jj 2 2 )hs a (x x o 2hs; t + 2hs; s) e i(x+t) b f (; ) dd with a (x; t; s) smooth and localized around (x; t) = (0; 0).

12 When s = 0, Properties F (x; t; 0) = a (x x o ; t; 0) f (x; t) If i@ s + 2 t a = 0, i@s + t 2 F = 0 Now, it remains to take a good a (x; t; s) solution of i@s + h a = 0 to 2 t the term at s = L the term at t = T the term \ )

13 Construction of a = a (x; t; s) 2 C h jxj 2 is + 1 e a (x; t; s) = (is + 1) 3=2 1 0 e C B R 5 ; C 1 t ihs + 1 p ihs + 1 C A Then i@s + t 2 a (x; t; s) = 0 ja (x; t; s)j = e jxj 2 =4 h s ps 3= t 2 =4 e (hs) q (hs) =2

14 At the end h 2 (0; 1], L 1. Add new variable s.construction of a = a (x; t; s) 2 C 1 R 5 ; C. i@s + t 2 F = 0 F (x; t; 0) = " u (x; t) " Multiply by u over (x; t; s) 2 [ parts, T; T ] [0; L] and integrate by term at s = 0! closed to u (x; t) = term at s = L! small for large L 1 + h term at t = T! small for large T 1 + h term \ (outside )! term on! + h term \ )! by microlocal analysis

15 At the end h 2 (0; 1], L 1. Add new variable s.construction of a = a (x; t; s) 2 C 1 R 5 ; C. i@s + t 2 F = 0 F (x; t; 0) = " u (x; t) " Multiply by u over (x; t; s) 2 [ parts, T; T ] [0; L] and integrate by term at s = 0! closed to u (x; t) = term at s = L! small for large L 1 + h term at t = T! small for large T 1 + h term \ (outside )! term on! + h term \ )! by microlocal analysis

16 Localization in Fourier variables Partition for 3 2 ( o3 1; o3 + 1), o3 2 (2Z + 1) and jj <, F (x; t; s) = 1 (2) 4 ZR 2 Z o3 +1 o3 1 Z jj< e i(jj2 2 )hs a (x x o 2hs; t + 2hs; s) e i(x+t) c'u (; ) dd with a (x; t; s) smooth and localized around (x; t) = (0; 0), ' (x; t) smooth.

17 References F. Cardoso and G. Vodev, On the stabilization of the wave equation by the boundary, Serdica Math. J. 28 (2002), N. Burq and M. Hitrik, Energy decay for damped wave equations on partially rectangular domains, Math. Res. Lett. 14 (2007), H. Nishiyama, Polynomial decay rate for damped wave equations on partially rectangular domains, Math. Res. Lett. 16 (2009), K.-D. Phung, Polynomial decay rate for the dissipative wave equation, J. Di erential Equations 240 (2007), K.-D. Phung, Boundary stabilization for the wave equation in a bounded cylindrical domain, Discrete Contin. Dyn. Syst. 20 (2008), K. D. Phung, Contrôle et stabilisation d ondes électromagnétiques, ESAIM Control Optim. Calc. Var. 5 (2000),

18 THANK YOU!