Polynomial decay rate for the Maxwell s equations with Ohm s law Kim Dang PHUNG Sichuan University. November 2010, Paris, IHP
Maxwell s equation with Ohm s law Let be a smooth bounded domain in R 3. 8 " o @ t E curlh + (x) E = 0 >< o @ t H + curle = 0 div ( >: o H) = 0 E j@ = H j@ = 0 2 L 1 () and 0 take " o = o = 1
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
Di culties Free divergence is not preserved by the system. @ 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
Remedies "scalar potential, vector potential and Coulomb gauge". Suppose is simply connected and @ has only one connected component. E = rp @ 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
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))
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
Interpolation observation inequality for the Wave equation 8 < : @ 2 t u u = 0 u j@ = 0 or @ u j@ = 0 1 c ku (; t)k2 L 2 () E (u; 0) = k(r; @ 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! j(u; @ t u)j 2 dxdt + h (E (u; 0) + E (@ t u; 0))
New operator h 2 (0; 1], L 1. Add new variable s. i@s + h @ 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 at @ \ (outside parallel @ )! term on! + h term at @ \ (parallel @ )
New operator h 2 (0; 1], L 1. Add new variable s. i@s + h @ 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 at @ \ (outside parallel @ )! term on! + h term at @ \ (parallel @ )
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).
When s = 0, Properties F (x; t; 0) = a (x x o ; t; 0) f (x; t) If i@ s + h @ 2 t a = 0, i@s + h @ t 2 F = 0 Now, it remains to take a good a (x; t; s) solution of i@s + h a = 0 to treat @ 2 t the term at s = L the term at t = T the term at @ \ (parallel @ )
Construction of a = a (x; t; s) 2 C 1 0 1 4h jxj 2 is + 1 e a (x; t; s) = B @ (is + 1) 3=2 1 0 e C B A @ R 5 ; C 1 t 2 1 4 ihs + 1 p ihs + 1 C A Then i@s + h @ t 2 a (x; t; s) = 0 ja (x; t; s)j = e jxj 2 =4 h s 2 + 1 ps 3=2 2 + 1 t 2 =4 e (hs) 2 + 1 q (hs) 2 + 1 1=2
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 + h @ 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 at @ \ (outside parallel @ )! term on! + h term at @ \ (parallel @ )! by microlocal analysis
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 + h @ 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 at @ \ (outside parallel @ )! term on! + h term at @ \ (parallel @ )! by microlocal analysis
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.
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