Global Small Solutions of the 3D Kerr-Debye Model Mohamed Kanso - - PowerPoint PPT Presentation

Models Properties and known results Main result Sketch of the proof Conclusion and perspectives

Global Small Solutions of the 3D Kerr-Debye Model

Mohamed Kanso

Institut de Mathématiques de Bordeaux, UMR CNRS 5251

Models Properties and known results Main result Sketch of the proof Conclusion and perspectives

Global Small Solutions of the 3D Kerr-Debye Model

• Models
• Properties and known results
• Main result
• Sketch of the proof

Models Properties and known results Main result Sketch of the proof Conclusion and perspectives

Physical context

The propagation of electromagnetic waves in a homogeneous isotropic nonlinear material (cristal) is described by Maxwell’s equations ∂tD − curl H = ∂tB + curl E = divD = divB = 0 E : electric field H : magnetic field D : electric displacement B : magnetic induction

Models Properties and known results Main result Sketch of the proof Conclusion and perspectives

Physical context

Maxwell’s equations: ∂tD − curl H = ∂tB + curl E = divD = divB = 0 Kerr medium ⇒ constitutive relations :

Models Properties and known results Main result Sketch of the proof Conclusion and perspectives

Physical context

Maxwell’s equations: ∂tD − curl H = ∂tB + curl E = divD = divB = 0 Kerr medium ⇒ constitutive relations : Kerr Model: instantaneous response B = H and D = (1 + |E|2)E

Models Properties and known results Main result Sketch of the proof Conclusion and perspectives

Physical context

Maxwell’s equations: ∂tD − curl H = ∂tB + curl E = divD = divB = 0 Kerr medium ⇒ constitutive relations : Kerr Model: instantaneous response B = H and D = (1 + |E|2)E Kerr-Debye Model: finite response time B = H and D = (1 + χ)E with ∂tχ + 1 τ χ = 1 τ |E|2 τ: relaxation parameter Y.- R. Shen, The Principles of Nonlinear Optics, Wiley Interscience, 1994. R.W. Ziolkowski. The incorporation of microscopic material models into FDTD approach for ultrafast optical pulses simulations, IEEE Transactions on Antennas and Propagation 45(3):375-391, 1997.

Models Properties and known results Main result Sketch of the proof Conclusion and perspectives

Properties

Kerr-Debye is a relaxation model of Kerr in the sense of Chen-Levermore-Liu (CPAM 1994 Equilibrium manifold: V = n (D, H, χ), χ = |E|2 = (1 + χ)−2|D|2o Reduced system: Kerr is the reduced system of Kerr-Debye on the equilibrium manifold

Models Properties and known results Main result Sketch of the proof Conclusion and perspectives

Properties

Kerr-Debye is a relaxation model of Kerr in the sense of Chen-Levermore-Liu (CPAM 1994 Equilibrium manifold: V = n (D, H, χ), χ = |E|2 = (1 + χ)−2|D|2o Reduced system: Kerr is the reduced system of Kerr-Debye on the equilibrium manifold Entropy relations (electromagnetic energy): strictly convex PK (D, H) = 1 2 (|E|2 + |H|2 + 3 2 |E|4) PKD(D, H, χ) = 1 2(1 + χ)−1|D|2 + 1 2 |H|2 + 1 4 χ2 On the equilibrium manifold, PKD(D, H, χ(D)) = PK (D, H) Kerr: hyperbolic symmetrizable system of conservation laws. Kerr-Debye: hyperbolic symmetrizable, partially dissipative system

Models Properties and known results Main result Sketch of the proof Conclusion and perspectives

Known results

Local existence of smooth solutions

• Kato (1975) or Majda (1984): Cauchy problem
• Picard-Zajaczkowski (1995): Initial-Boundary Value Problem (IBVP)

Models Properties and known results Main result Sketch of the proof Conclusion and perspectives

Known results

Local existence of smooth solutions

• Kato (1975) or Majda (1984): Cauchy problem
• Picard-Zajaczkowski (1995): Initial-Boundary Value Problem (IBVP)

Convergence of Kerr-Debye smooth solutions towards Kerr smooth solutions when τ → 0.

• Hanouzet-Huynh (2000): Cauchy problem using the results of Yong (1999).
• Carbou-Hanouzet (2009): Initial-Boundary Value Problem

Models Properties and known results Main result Sketch of the proof Conclusion and perspectives

Known results

Local existence of smooth solutions

• Kato (1975) or Majda (1984): Cauchy problem
• Picard-Zajaczkowski (1995): Initial-Boundary Value Problem (IBVP)

Convergence of Kerr-Debye smooth solutions towards Kerr smooth solutions when τ → 0.

• Hanouzet-Huynh (2000): Cauchy problem using the results of Yong (1999).
• Carbou-Hanouzet (2009): Initial-Boundary Value Problem

Global existence of smooth solutions ?

• global existence without smallness condition holds for the Cauchy problem as well

as for the impedance IBVP for Kerr-Debye model in the 1D and 2D Tranverse Electric cases (Carbou-Hanouzet 2009).

• apparition of shocks for the Kerr model in both 1D and 2D-TE cases.

Models Properties and known results Main result Sketch of the proof Conclusion and perspectives

Global existence in 3D

For Kerr Model

Global small solution for the Cauchy problem: based on a decay estimate for the linear wave equation

• R. Racke, Lectures on nonlinear evolution equations. Initial value problems. Aspects
• f Mathematics, E19. Friedr. Vieweg & Sohn, Braunschweig, 1992

Models Properties and known results Main result Sketch of the proof Conclusion and perspectives

Global existence in 3D

For Kerr Model

Global small solution for the Cauchy problem: based on a decay estimate for the linear wave equation

• R. Racke, Lectures on nonlinear evolution equations. Initial value problems. Aspects
• f Mathematics, E19. Friedr. Vieweg & Sohn, Braunschweig, 1992

For Kerr-Debye Model

The Shizuta-Kawashima [SK] condition does not hold: the linearized system around the null constant equilibrium writes: ∂t „ E H « + „ −curl curl « „ E H « = 0, ∂tχ = −χ, i.e the dissipative variable χ and the variable (E, H) are completely uncoupled.

• All known results around [SK] condition do not applied
• Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with

application to the discrete Boltzmann equation. Hokkaido Math.J. 14 (1985)

Models Properties and known results Main result Sketch of the proof Conclusion and perspectives

Global existence in 3D

Kerr-Debye Model

8 > > > > > > > > > > > < > > > > > > > > > > > : ∂tD − curl H = 0, ∂tH + curl E = 0, ∂tχ = |E|2 − χ, D = (1 + χ)E, div D = div H = 0. The dispersion of the Maxwell equations in the 3-D case + The partial dissipative character of the Kerr-Debye model = ⇒ Global small solution:

Models Properties and known results Main result Sketch of the proof Conclusion and perspectives

Global existence in 3D

Kerr-Debye Model

8 > > > > > > > > > > > < > > > > > > > > > > > : ∂tD − curl H = 0, ∂tH + curl E = 0, ∂tχ = |E|2 − χ, D = (1 + χ)E, div D = div H = 0. The dispersion of the Maxwell equations in the 3-D case + The partial dissipative character of the Kerr-Debye model = ⇒ Global small solution:

Theorem 3.1

There exist an integer s ≥ 7 and a δ > 0 such that the following holds: if the initial data V 0 = (E0, H0, χ0) satisfies V 0s,2 + V 0s, 6

5 < δ, with χ0 ≥ 0 and div H0 = div [(1 + χ0)E0] = 0,

then there exists a unique solution V for the Cauchy problem of the KD model, with: V = (E, H, χ) ∈ C0` [0, ∞), W s,2´ ∩ C1` [0, ∞), W s−1,2´ .

Models Properties and known results Main result Sketch of the proof Conclusion and perspectives

Sketch of the proof

Kerr-Debye Model

8 > > > > > > > > > > > < > > > > > > > > > > > : ∂tD − curl H = 0, ∂tH + curl E = 0, ∂tχ = |E|2 − χ, D = (1 + χ)E, div D = div H = 0. The dispersion of the Maxwell equations in the 3-D case + The partial dissipative character of the Kerr-Debye model = ⇒ Global small solution: The two principal steps of the proof, cf: Klainerman-Ponce (1983), O. Liess (1989)

• r R. Racke (1992):
• High energy estimate by using variational methods
• a weighted a priori estimate based on Lp − Lq decay estimates for the linear

wave equation

Models Properties and known results Main result Sketch of the proof Conclusion and perspectives

Sketch of the proof

Kerr-Debye Model

8 > > > > > > > > > > > < > > > > > > > > > > > : ∂tD − curl H = 0, ∂tH + curl E = 0, ∂tχ = |E|2 − χ, D = (1 + χ)E, div D = div H = 0. The dispersion of the Maxwell equations in the 3-D case + The partial dissipative character of the Kerr-Debye model = ⇒ Global small solution Main difficulty: degree of vanishing of the nonlinearity near zero is not great enough New idea: we split the model in two parts:

• Maxwell’s equations
• Ordinary differential equation satisfied by χ

Models Properties and known results Main result Sketch of the proof Conclusion and perspectives

High energy estimate

We denote by T ∗ the lifespan of the local solution We define Ms1(t) = max

0≤τ≤t(1 + τ)2/3(E, H)(τ)s1,6

Proposition 4.1

Let s, s1 ∈ N; 2 ≤ s1 ≤ s − 1. Then there exists a constant c = c(s, s1) such that if the initial data V 0 = (E0, H0, χ0) satisfies V 0s,2 ≤ 1/2, and χ0 ≥ 0, then V(t)s,2 ≤ cV 0s,2 ˆ (1 + M2

s1(t))exp{cM2 s1(t)}

˜ ∀ t ∈ [0, ¯ T], where ¯ T > 0 is defined by ¯ T := max ˘ T < T ⋆ such that VL∞(0,T;W s,2) ≤ 1 ¯ .

Models Properties and known results Main result Sketch of the proof Conclusion and perspectives

High energy estimate

Idea of the proof

• We use the variable (E, H, χ):

8 > > > < > > > : (1 + χ)∂tE + (∂tχ)E − curl H = 0 ∂tH + curl E = 0 ∂tχ = |E|2 − χ

• Classical variationnal estimates on the Maxwell part
• We solve the ODE to estimate χ:

χ(t) = χ0e−t + Z t e(s−t)|E(s)|2ds

• The introduction of the weight (1 + t)2/3 is necessary to control χ

Models Properties and known results Main result Sketch of the proof Conclusion and perspectives

Weighted a priori estimate

Proposition 4.2

Let s, s1 ∈ N; 3 ≤ s1 ≤ s − 4. Then for all M0 > 0 there exists a 0 < δ(M0) ≤ 1 such that if: V 0s,2 + V 0s1+3, 6

5 ≤ δ and χ0s, 3 2 ≤ δ,

with χ0 ≥ 0 and div H0 = div [(1 + χ0)E0] = 0, then Ms1(t) ≤ M0 ∀ t ∈ [0, ˜ T], where ˜ T := max ˘ T < T ⋆ such that VL∞(0,T;W s,2) ≤ δ(M0) ¯ .

Models Properties and known results Main result Sketch of the proof Conclusion and perspectives

Weighted a priori estimate

Idea of the proof

• We use the variable (D, H, χ):

8 < : ∂tD − curl H = 0 ∂tH + curl D = curl (χE)

Models Properties and known results Main result Sketch of the proof Conclusion and perspectives

Weighted a priori estimate

Idea of the proof

• We use the variable (D, H, χ):

8 < : ∂tD − curl H = 0 ∂tH + curl D = curl (χE)

• Representation of Duhamel : Λ is the linear Maxwell operator, U = (D, H) and

f(t) = t(0, curl (χE))) U(t) = etΛU0 +

t

Z e(t−τ)Λf(τ)dτ, 0 ≤ t ≤ ˜ T,

Models Properties and known results Main result Sketch of the proof Conclusion and perspectives

Weighted a priori estimate

Idea of the proof

• We use the variable (D, H, χ):

8 < : ∂tD − curl H = 0 ∂tH + curl D = curl (χE)

• Representation of Duhamel : Λ is the linear Maxwell operator, U = (D, H) and

f(t) = t(0, curl (χE))) U(t) = etΛU0 +

t

Z e(t−τ)Λf(τ)dτ, 0 ≤ t ≤ ˜ T,

• Semi group estimate: for 1 < p ≤ 2 ≤ q < ∞, 1/p + 1/q = 1 and

Nq > 3(1 − 2/q), etΛU0q ≤ c(1 + t)−(1− 2

q )U0Nq,p

Models Properties and known results Main result Sketch of the proof Conclusion and perspectives

Weighted a priori estimate

Idea of the proof

• We use the variable (D, H, χ):

8 < : ∂tD − curl H = 0 ∂tH + curl D = curl (χE)

• Representation of Duhamel : Λ is the linear Maxwell operator, U = (D, H) and

f(t) = t(0, curl (χE))) U(t) = etΛU0 +

t

Z e(t−τ)Λf(τ)dτ, 0 ≤ t ≤ ˜ T,

• Semi group estimate: for 1 < p ≤ 2 ≤ q < ∞, 1/p + 1/q = 1 and

Nq > 3(1 − 2/q), etΛU0q ≤ c(1 + t)−(1− 2

q )U0Nq,p

• 1. L2 − L2 estimate from classical result on Maxwell equations
• 2. L∞ − W 3,1 estimate by the Kirchoff representation formula on linear wave equations

Models Properties and known results Main result Sketch of the proof Conclusion and perspectives

Weighted a priori estimate

Idea of the proof

• We use the variable (D, H, χ):

8 < : ∂tD − curl H = 0 ∂tH + curl D = curl (χE)

• Representation of Duhamel : Λ is the linear Maxwell operator, U = (D, H) and

f(t) = t(0, curl (χE))) U(t) = etΛU0 +

t

Z e(t−τ)Λf(τ)dτ, 0 ≤ t ≤ ˜ T,

• Semi group estimate: for 1 < p ≤ 2 ≤ q < ∞, 1/p + 1/q = 1 and

Nq > 3(1 − 2/q), etΛU0q ≤ c(1 + t)−(1− 2

q )U0Nq,p

• 1. L2 − L2 estimate from classical result on Maxwell equations
• 2. L∞ − W 3,1 estimate by the Kirchoff representation formula on linear wave equations
• Estimate of the nonlinear term f

f(t)s1+3,6/5 ≤ c ˆ χ0s,3/2e−tMs1(t) + ` e−t + (1 + t)−4/3M2

s1(t)

´ U(t)s,2 ˜

Models Properties and known results Main result Sketch of the proof Conclusion and perspectives

Weighted a priori estimate

Idea of the proof We obtain Ms1(t) ≤ cδ ` 1 + Ms1(t) + (1 + M2

s1(t))2exp{cM2 s1(t)}

´ , 0 ≤ t ≤ ˜ T. We introduce x = Ms1(t) We study the function ϕ(x) := cδ(1 + x + (1 + x2)2ecx2) − x, using ϕ(Ms1(t)) ≥ 0, 0 ≤ t ≤ ˜ T.

Models Properties and known results Main result Sketch of the proof Conclusion and perspectives

A priori bound

Proposition 4.3

There exist c > 0, an integer s ≥ 7 and a δ > 0 sufficiently small such that if V 0s,2+V 0s, 6

5 ≤ δ/2 , χ0s, 3 2 ≤ δ/2, with χ0 ≥ 0 and div H0 = div [(1+χ0)E0] = 0,

then V = (E, H, χ) satisfies V(t)s,2 ≤ cV 0s,2 ˆ (1 + M2

0)exp{cM2 0}

˜ ∀ t ∈ [0, ˜ T], where ˜ T = max ˘ T < T ⋆ such that VL∞(0,T;W s,2) ≤ δ ¯ . . Asymptotic behavior V(t)∞ + V(t)6 = O(t−2/3), V(t)s,2 = O(1) as t → ∞.

Models Properties and known results Main result Sketch of the proof Conclusion and perspectives

Conclusion and perspectives

Global small solution in 3D

For the Cauchy problem:

• Kerr : Racke 1992
• Kerr-Debye

For the IBVP:

• Kerr with Dirichlet condition (Recently)

Global solution in 3D without smallness condition ? Global solution in 2D Transverse magnetic case ?

Models Properties and known results Main result Sketch of the proof Conclusion and perspectives

Thank you