Maxwell-Dirac: Null structure and almost optimal local - PowerPoint PPT Presentation

Maxwell-Dirac: Null structure and almost optimal local well-posedness Sigmund Selberg (joint work with P. D’Ancona and D. Foschi) Department of Mathematical Sciences Norwegian University of Science and Technology (NTNU) HYP 2008 University of Maryland, June 9–13 2008 S. Selberg (NTNU) Maxwell-Dirac null structure HYP08, June 9–13 2008 1 / 39

Outline Study Maxwell-Dirac system (MD). System of nonlinear wave equations. Describes: electron self-interacting with electromagnetic field. We would like to understand the nonlinear structure of MD. The structure cannot be seen in each component equation, only in system as whole. Structure is expressed in terms of trilinear and quadrilinear integral forms with special cancellation properties expressed in terms of the spatial frequencies. 3D case: Use structure to prove multilinear space-time Fourier restriction estimates at scale invariant regularity up to a logarithmic loss. As a consequence, we are able to prove local well-posedness almost down to the critical regularity in 3D. S. Selberg (NTNU) Maxwell-Dirac null structure HYP08, June 9–13 2008 2 / 39

Minkowski space-time R 1+3 = R t × R 3 x Coordinates/partials: t = x 0 ∂ 0 = ∂ t time x = ( x 1 , x 2 , x 3 ) ∂ j = ∂ x j ∇ = ( ∂ 1 , ∂ 2 , ∂ 3 ) space � − 1 � 0 0 0 Metric (raise/lower indices): ( g µν ) = 0 1 0 0 0 0 1 0 0 0 0 1 Summation convention applies for repeated upper/lower indices ( j , k , · · · = 1 , 2 , 3; µ, ν, · · · = 0 , 1 , 2 , 3) For example: � = ∂ µ ∂ µ = − ∂ 2 0 + ∂ 2 1 + ∂ 2 2 + ∂ 2 3 = − ∂ 2 t + ∆ x S. Selberg (NTNU) Maxwell-Dirac null structure HYP08, June 9–13 2008 3 / 39

Maxwell-Dirac (MD) Couple Maxwell’s equations and Dirac equation: � ∇ · E = ρ, ∇ · B = 0 , (Maxwell) ∇ × E + ∂ t B = 0 , ∇ × B − ∂ t E = J , ( α µ D µ + m β ) ψ = 0 . (Dirac) m ≥ 0 constant; α µ and β are 4 × 4 Dirac matrices . Unknowns: E , B : R 1+3 → R 3 electric and magnetic fields ψ : R 1+3 → C 4 Dirac four-spinor Represent EM field by real four-potential A µ , µ = 0 , 1 , 2 , 3: B = ∇ × A , E = ∇ A 0 − ∂ t A ( A = ( A 1 , A 2 , A 3 )) . S. Selberg (NTNU) Maxwell-Dirac null structure HYP08, June 9–13 2008 4 / 39

Maxwell-Dirac (MD) We have � ∇ · E = ρ, ∇ · B = 0 , (Maxwell) ∇ × E + ∂ t B = 0 , ∇ × B − ∂ t E = J , ( α µ D µ + m β ) ψ = 0 , (Dirac) B = ∇ × A , E = ∇ A 0 − ∂ t A (Potentials) Complete the coupling: J µ = � α µ ψ, ψ � C 4 Dirac four-current ρ = J 0 = | ψ | 2 charge density J = ( J 1 , J 2 , J 3 ) three-current density = 1 D µ = D ( A ) i ∂ µ − A µ gauge covariant derivative µ S. Selberg (NTNU) Maxwell-Dirac null structure HYP08, June 9–13 2008 5 / 39

Gauge invariance Result is the following nonlinear system: � A µ − ∂ µ ( ∂ ν A ν ) = −� α µ ψ, ψ � C 4 , (Maxwell) ( − i α µ ∂ µ + m β ) ψ = A µ α µ ψ, (Dirac) Invariant under the gauge transformation → ψ ′ = e i χ ψ, → A ′ ψ − A µ − µ = A µ + ∂ µ χ, (GT) for any χ : R 1+3 → R (the gauge function ). Observables E , B , ρ, J not affected, so solutions related by GT are physically undistinguishable; considered equivalent . Pick representative whose potential A µ simplifies the analysis. Natural: impose Lorenz gauge condition ∂ µ A µ = 0 ( ⇐ ⇒ ∂ t A 0 = ∇ · A ) (LG) S. Selberg (NTNU) Maxwell-Dirac null structure HYP08, June 9–13 2008 6 / 39

Initial data System becomes � A µ = −� α µ ψ, ψ � C 4 , (Maxwell) ( − i α µ ∂ µ + m β ) ψ = A µ α µ ψ, (Dirac) ∂ µ A µ = 0 . (LG) Initial data: ψ | t =0 = ψ 0 , E | t =0 = E 0 , B | t =0 = B 0 . Maxwell imposes constraints ∇ · E 0 = | ψ 0 | 2 , ∇ · B 0 = 0 . Scale invariant data regularity (3D): ( E 0 , B 0 ) ∈ ˙ ψ 0 ∈ L 2 ( R 3 ; C 4 ) , H − 1 / 2 ( R 3 ; R 3 ) . S. Selberg (NTNU) Maxwell-Dirac null structure HYP08, June 9–13 2008 7 / 39

Construction of Lorenz data The data for the four-potential, A µ | t =0 = a µ , ∂ t A µ | t =0 = ˙ a µ , must be constructed from the observables ( E 0 , B 0 ). Set a 0 = ˙ a 0 = 0 . Then a = ( a 1 , a 2 , a 3 ) and ˙ a = (˙ a 1 , ˙ a 2 , ˙ a 3 ) determined by ∇ · a = 0 , ∇ × a = B 0 , a = − E 0 ˙ . � �� � � �� � by LG condition by Maxwell LG condition automatically satisfied in the evolution starting from Lorenz data. LG equation can be discarded from the system. Next step: Solve wave equation for A µ and plug into Dirac equation. Result is a single nonlinear Dirac equation. S. Selberg (NTNU) Maxwell-Dirac null structure HYP08, June 9–13 2008 8 / 39

Reduction to nonlinear Dirac equation Recall Duhamel’s formula for � u = F , ( u , ∂ t u ) | t =0 = ( u 0 , u 1 ): � t u ( t ) = cos( t |∇| ) u 0 + sin( t |∇| ) sin(( t − s ) |∇| ) u 1 + F ( s ) ds |∇| |∇| 0 � �� � � �� � u hom . 1 � F Thus, A µ = A hom . − 1 � � α µ ψ, ψ � C 4 . µ Result: MD in LG becomes a single nonlinear Dirac equation ( − i α µ ∂ µ + m β ) ψ = A hom . α µ ψ − N ( ψ, ψ, ψ ) , (MDL) µ where � 1 � α µ · N ( · , · , · ) = � � α µ · , ·� C 4 S. Selberg (NTNU) Maxwell-Dirac null structure HYP08, June 9–13 2008 9 / 39

Local well-posedness (almost optimal) Theorem Let s > 0 . Assume given initial data ψ 0 ∈ H s ( R 3 ; C 4 ) , E 0 , B 0 ∈ H s − 1 / 2 ( R 3 ; R 3 ) , satisfying the Maxwell constraints. Prepare Lorenz data: ∇ · a = 0 , ∇ × a = B 0 , a = − E 0 , a 0 = ˙ a 0 = 0 , ˙ and use these to define A hom . . Then ∃ T > 0 , depending continuously on µ the data norm, and there exists ψ ∈ C ([ − T , T ]; H s ( R 3 ; C 4 )) which solves the MDL equation on ( − T , T ) × R 3 : ( − i α µ ∂ µ + m β ) ψ = A hom . α µ ψ − N ( ψ, ψ, ψ ) , ψ | t =0 = ψ 0 . µ S. Selberg (NTNU) Maxwell-Dirac null structure HYP08, June 9–13 2008 10 / 39

Persistence of regularity for electromagnetic field Theorem Assume same hypotheses as in previous theorem, and let ψ be the solution of MDL on ( − T , T ) × R 3 . Define � � ρ = | ψ | 2 � α j ψ, ψ � C 4 J = j =1 , 2 , 3 , and solve Maxwell’s equations � ∇ · E = ρ, ∇ · B = 0 , ∇ × E + ∂ t B = 0 , ∇ × B − ∂ t E = J . Solution retains data regularity: ( E , B ) ∈ C ([ − T , T ]; H s − 1 / 2 ) S. Selberg (NTNU) Maxwell-Dirac null structure HYP08, June 9–13 2008 11 / 39

Some earlier existence results for MD Gross ’66: Local existence for smooth data Georgiev ’91: Global existence for small, smooth data Bournaveas ’96: Local well-posedness (LWP) for data s > 1 ψ 0 ∈ H s ( R 3 ; C 4 ) , E 0 , B 0 ∈ H s − 1 / 2 ( R 3 ; R 3 ) , 2 Masmoudi and Nakanishi ’03: LWP s = 1 2 (Coulomb gauge) Latter result analogous to Klainerman and Machedon’s result for Maxwell-Klein-Gordon (MKG) from 1993 (finite energy well-posedness). But for MD, energy is not positive definite. MKG: almost optimal LWP proved by Machedon and Sterbenz ’03 (Coulomb gauge) S. Selberg (NTNU) Maxwell-Dirac null structure HYP08, June 9–13 2008 12 / 39

Nonlinear structure of MD in Lorenz gauge Isolate most difficult part: Consider model � 1 � − i α µ ∂ µ ψ = −N ( ψ, ψ, ψ ) , α µ · N ( · , · , · ) = � � α µ · , ·� C 4 Diagonalize Dirac operator: − i α µ ∂ µ = − i ∂ t + − i α j ∂ j � �� � = |∇| Π + −|∇| Π − where the Dirac projections Π ± are multipliers � � I 4 × 4 + ξ j α j Π ( ξ ) = 1 Π ± f ( ξ ) = Π ( ± ξ ) � � ( ξ ∈ R 3 ) . f ( ξ ) , 2 | ξ | Split ψ = ψ + + ψ − where ψ ± = Π ± ψ . S. Selberg (NTNU) Maxwell-Dirac null structure HYP08, June 9–13 2008 13 / 39

Nonlinear structure of MD in Lorenz gauge Result is system ( − i ∂ t + |∇| ) ψ + = − Π + N ( ψ, ψ, ψ ) , ψ + | t =0 = Π + ψ 0 , ( − i ∂ t − |∇| ) ψ − = − Π − N ( ψ, ψ, ψ ) , ψ − | t =0 = Π − ψ 0 , � 1 � α µ · . where ψ = ψ + + ψ − and N ( · , · , · ) = � � α µ · , ·� C 4 Note that ψ ± = Π ± ψ , since Π + Π − = 0. Iterate in space X s , b with norm ± � � � � � � ξ � s � τ ± | ξ |� b � � u � X s , b ± = u ( τ, ξ ) � , L 2 τ,ξ u ( τ, ξ ) = F.t. of u ( t , x ), and � ξ � = (1 + | ξ | 2 ) 1 / 2 . where � For all ε > 0, s , 1 2 + ε → BC ( R ; H s ) . X ֒ ± S. Selberg (NTNU) Maxwell-Dirac null structure HYP08, June 9–13 2008 14 / 39

Linear estimate in X s , b ± Solution of linear IVP on S T = ( − T , T ) × R 3 , ( − i ∂ t ± |∇| ) u = F u | t =0 = u 0 , satisfies ( S T ) ≤ C ε � u 0 � H s + C ε T ε � F � � u � s , 1 s , − 1 2 + ε 2 +2 ε X X ( S T ) ± ± Apply to ( − i ∂ t ± 4 |∇| ) ψ ± 4 = − Π ± 4 N ( ψ, ψ, ψ ) � = − Π ± 4 N ( ψ ± 1 , ψ ± 2 , ψ ± 3 ) ± 1 , ± 2 , ± 3 � = − Π ± 4 N ( Π ± 1 ψ, Π ± 2 ψ, Π ± 3 ψ ) . ± 1 , ± 2 , ± 3 S. Selberg (NTNU) Maxwell-Dirac null structure HYP08, June 9–13 2008 15 / 39