The Chern-Simons-Schrödinger equation

The Chern-Simons-Schrödinger equation Low regularity local wellposedness Baoping Liu, Paul Smith, Daniel Tataru University of California, Berkeley July 16, 2012 Paul Smith (UC Berkeley) Chern-Simons-Schrödinger UNC, July 2012 1 / 32

Chern-Simons-Schrödinger We study the Chern-Simons-Schrödinger equation D t φ = id l D l φ + ig φ 2 φ. The covariant derivatives D l act on functions v according to D l v = l v + ia l v, where the A l are real-valued connection coefficients. Paul Smith (UC Berkeley) Chern-Simons-Schrödinger UNC, July 2012 2 / 32

Full Chern-Simons-Schrödinger system In addition to the evolution equation D t φ = id l D l φ + ig φ 2 φ, we impose the curvature constraints t A 1 1 A t = Im( φd 2 φ) t A 2 2 A t = Im( φd 1 φ) 1 A 2 2 A 1 = 1 2 φ 2. Paul Smith (UC Berkeley) Chern-Simons-Schrödinger UNC, July 2012 3 / 32

Conserved quantities and scaling The Chern-Simons-Schrödinger (CSS) system has conserved charge M(φ) := φ 2 dx and energy E(φ) := D x φ 2 g 2 φ 4 dx. The scaling symmetry preserves charge. φ(t, x) λφ(λ 2 t, λx), λ > 0, Paul Smith (UC Berkeley) Chern-Simons-Schrödinger UNC, July 2012 4 / 32

Choosing a gauge The Chern-Simons-Schrödinger system exhibits gauge freedom in that it is invariant with respect to the transformations for real-valued functions θ(t, x). φ e iθ φ, A α A α + α θ, Therefore in order for the system to be well-posed, a gauge must be selected. This is achieved by imposing an extra condition on the connection coefficients. Paul Smith (UC Berkeley) Chern-Simons-Schrödinger UNC, July 2012 5 / 32

CSS evolution in expanded form The function φ evolves according to the nonlinear Schrödinger equation (i t + )φ = 2iA l l φ i( l A l )φ + (A t + A 2 x)φ g φ 2 φ, where A x, A t satisfy the constraints t A 1 1 A t = Im( φd 2 φ) t A 2 2 A t = Im( φd 1 φ) 1 A 2 2 A 1 = g 2 φ 2. Paul Smith (UC Berkeley) Chern-Simons-Schrödinger UNC, July 2012 6 / 32

History Introduction of the model: Jackiw-Pi, 1991, 1992 Ezawa-Hotta-Iwazaki, 1991 Local wellposedness in H 2 : Bergè-de Bouard-Saut, 1995 Paul Smith (UC Berkeley) Chern-Simons-Schrödinger UNC, July 2012 7 / 32

Main result Theorem For initial data φ 0 H s (R 2 ), s > 0, there is a positive time T, depending only upon φ 0 H s, such that the Chern-Simons-Schrödinger system with appropriate gauge and initialization has a unique solution φ(t, x) C([0, T ], H s (R 2 )). In addition, φ 0 φ is Lipschitz continuous from H s (R 2 ) to C([0, T ], H s (R 2 )). Paul Smith (UC Berkeley) Chern-Simons-Schrödinger UNC, July 2012 8 / 32

First try: the Coulomb gauge The Coulomb gauge condition is A x = 0. It leads to A t = 1 Im( φd 2 φ) 2 Im( φd 1 φ) and { A 1 = Im( φd 2 φ) + 1 2 2 φ 2 A 2 = Im( φd 1 φ) 1 2 1 φ 2. Paul Smith (UC Berkeley) Chern-Simons-Schrödinger UNC, July 2012 9 / 32

First try: the Coulomb gauge From A t = 1 Im( φd 2 φ) 2 Im( φd 1 φ) we conclude A t 1 Q 12 ( φ, φ), where Q 12 (, ) denotes the null form defined by Q 12 (f, g) = 1 f 2 g 2 f 1 g. This has bad high high low frequency interactions. Paul Smith (UC Berkeley) Chern-Simons-Schrödinger UNC, July 2012 10 / 32

Second try: the caloric gauge The appeal: Introduced to study wave maps. Works well for Schrödinger maps. Geometric, nonlinear, and behaves similarly to the Coulomb gauge, but better handles high high low frequency interactions. Paul Smith (UC Berkeley) Chern-Simons-Schrödinger UNC, July 2012 11 / 32

Second try: the caloric gauge The problem: How does one define the caloric gauge in our setting? Paul Smith (UC Berkeley) Chern-Simons-Schrödinger UNC, July 2012 12 / 32

The heat gauge We adopt the gauge condition A x = A t. Used in conjunction with the constraints, this leads to ( t )A t = 1 Im( φd 2 φ) + 2 Im( φd 1 φ) and { ( t )A 1 = Im( φd 2 φ) 1 2 2 φ 2 ( t )A 2 = Im( φd 1 φ) + 1 2 1 φ 2. Paul Smith (UC Berkeley) Chern-Simons-Schrödinger UNC, July 2012 13 / 32

Comparison of Coulomb and heat gauges Coulomb gauge: A t 1 Q 12 ( φ, φ) Heat gauge: A t H 1 Q 12 ( φ, φ) We define H 1 as the Fourier multiplier operator H 1 f := 1 1 (2π) 3 iτ + ξ 2 ei(tτ+x ξ) f (τ, ξ)dτdξ. The heat gauge provides extra decay when φ and φ interact so as to yield a high-modulation output. Paul Smith (UC Berkeley) Chern-Simons-Schrödinger UNC, July 2012 14 / 32

Gauged Chern-Simons-Schrödinger system The CSS with the heat gauge condition imposed: D t φ = id l D l φ + ig φ 2 φ t A 1 1 A t = Im( φd 2 φ) t A 2 2 A t = Im( φd 1 φ) 1 A 2 2 A 1 = 1 2 φ 2 A t = A x. Paul Smith (UC Berkeley) Chern-Simons-Schrödinger UNC, July 2012 15 / 32

Initial conditions We study the CSS system with the heat gauge selected and with the following initial conditions: φ(0, x) = φ 0 (x) A t (0, x) = 0 A 1 (0, x) A 2 (0, x) = 1 2 1 2 φ 0 2 (x) = 1 2 1 1 φ 0 2 (x). Paul Smith (UC Berkeley) Chern-Simons-Schrödinger UNC, July 2012 16 / 32

The connection coefficients The full expansion for A t is A t = H 1 ((ImQ 12 ( φ, φ))) H 1 ( 1 (A 2 φ 2 )) + H 1 ( 2 (A 1 φ 2 )), and for A 1, A 2, A 1 = H 1 (Re( φ 2 φ) + Im( φ 2 φ)) H 1 (A 2 φ 2 ) + H 1 A 1 (0) A 2 = H 1 (Re( φ 1 φ) + Im( φ 1 φ)) + H 1 (A 1 φ 2 ) + H 1 A 2 (0). Paul Smith (UC Berkeley) Chern-Simons-Schrödinger UNC, July 2012 17 / 32

Null form structure Consider 2i(A 1 1 φ + A 2 2 φ). A 1 = H 1 (Re( φ 2 φ) + Im( φ 2 φ)) H 1 (A 2 φ 2 ) + H 1 A 1 (0) A 2 = +H 1 (Re( φ 1 φ) + Im( φ 1 φ)) + H 1 (A 1 φ 2 ) + H 1 A 2 (0). Paul Smith (UC Berkeley) Chern-Simons-Schrödinger UNC, July 2012 18 / 32

Null form structure Consider 2i(A 1 1 φ + A 2 2 φ). A 1 = H 1 (Re( φ 2 φ) + Im( φ 2 φ)) H 1 (A 2 φ 2 ) + H 1 A 1 (0) A 2 = +H 1 (Re( φ 1 φ) + Im( φ 1 φ)) + H 1 (A 1 φ 2 ) + H 1 A 2 (0). Q 12 (f, g) = 1 f 2 g 2 f 1 g. Paul Smith (UC Berkeley) Chern-Simons-Schrödinger UNC, July 2012 21 / 32

Using the gauge We heuristically have A x = H 1 ( φ x φ) + error and we want to write (i t + )φ = main terms + error. Spaces for the connection coefficient error terms: L 4 t,x, L2,6 e. Paul Smith (UC Berkeley) Chern-Simons-Schrödinger UNC, July 2012 22 / 32

The nonlinearity with respect to the heat gauge We pair each term of the nonlinearity with a function ψ. The goal is to control the space-time integral of such expressions. Consider A j ( j φ) ψ. The term expands as (H 1 Q 12 )( φ, φ, φ) ψ + I 1 + H 1 (A 1 φ 2 ) 2 φ ψ H 1 (A 2 φ 2 ) 1 φ ψ, where (H 1 Q 12 )( φ, φ, φ) = H 1 ( φ 1 φ) 2 φ H 1 ( φ 2 φ) 1 φ and I 1 comes from the initial data. Paul Smith (UC Berkeley) Chern-Simons-Schrödinger UNC, July 2012 23 / 32

The nonlinearity with respect to the heat gauge We cannot quite show that H 1 (A 1 φ 2 ) 2 φ ψ, H 1 (A 2 φ 2 ) 1 φ ψ are error terms. By expanding A 1, A 2 once more, we obtain H 1 ( ψ x φ)h 1 ( φ x φ) φ 2 + H 1 ( ψ x φ)h 1 (A x φ 2 ) φ 2 + H 1 ( ψ x φ)(h 1 A x (0)) φ 2. The last two terms we can treat as error terms, while the first one we cannot. Paul Smith (UC Berkeley) Chern-Simons-Schrödinger UNC, July 2012 24 / 32

The main + error decomposition We also perform a similar analysis with ( j A j )φ ψ = A t φ ψ and A 2 xφ ψ. The main terms we obtain are the fourth-order terms (H 1 Q 12 )( φ, φ, φ) ψ, H 1 (Q 12 ( φ, φ))φ ψ, and the sixth-order terms H 1 ( φ x φ)h 1 ( ψ x φ) φ 2, H 1 ( φ x φ)h 1 ( φ x φ)φ ψ. Paul Smith (UC Berkeley) Chern-Simons-Schrödinger UNC, July 2012 25 / 32

An improved bilinear Strichartz estimate Let u(x, t) = e it u 0 (x), v(x, t) = e it v 0 (x), u 0, v 0 L 2 (R 2 ), with respective frequency supports Ω 1, Ω 2 satisfying dist(ω 1, Ω 2 ) > 0. Then u v L 2 t,x sup ξ,τ ξ=ξ 1 ξ 2 χ Ω (ξ 1, ξ 2 )dh 1 (ξ 1, ξ 2 ) τ= ξ 1 2 ξ 2 2 dist(ω 1, Ω 2 ) 1/2 u 0 L 2 v 0 L 2, where dh 1 denotes 1-dimensional Hausdorff measure (on R 4 ) and χ Ω (ξ 1, ξ 2 ) the characteristic function of Ω := Ω 1 Ω 2. Paul Smith (UC Berkeley) Chern-Simons-Schrödinger UNC, July 2012 26 / 32

Bilinear Strichartz with modulation restriction Let µ, ν, λ be dyadic frequencies satisfying µ λ and ν µλ. Let φ λ, ψ λ be free waves with frequency support contained in a λ-annulus. Then P µ Q ν ( φ λ ψ λ ) L 2 ν1/2 (µλ) 1/2 φ λ L 2 x ψ λ L 2 x. Paul Smith (UC Berkeley) Chern-Simons-Schrödinger UNC, July 2012 27 / 32

L 2 -bilinear estimate with fixed null form size Let u(x, t) = e it u 0 (x), v(x, s) = e is v 0 (x), u 0, v 0 L 2 (R 2 ), with respective frequency supports Ω 1, Ω 2. Assume that for all ξ 1 Ω 1, ξ 2 Ω 2, we have ξ 1 ξ 2 β. Then u v L 2 s,t,x β 1/2 u 0 L 2 v 0 L 2. Paul Smith (UC Berkeley) Chern-Simons-Schrödinger UNC, July 2012 28 / 32

Function spaces Define U p atoms a : R H via K a = χ [tk 1,t k )φ k 1, k=1 {φ k } K 1 k=0 H, K 1 k=0 φ k p H = 1. Define the atomic space Up (R, H) as the set of all functions u : R H admitting a representation u = λ j a j for U p -atoms a j, {λ j } l 1 j=1 and endow it with the norm u U p := inf λ j : u = j=1 λ j a j, λ j C, a j a U p -atom. j=1 Paul Smith (UC Berkeley) Chern-Simons-Schrödinger UNC, July 2012 29 / 32

Function spaces Define V p (R, H) as the space of all functions v : R H such that is finite. v V p := sup {t k } K k=0 Z ( K v(t k ) v(t k 1 ) p H k=1 Likewise, let V p rc(r, C) denote the closed subspace of all right-continuous functions v : R C such that lim t v(t) = 0. [Koch-Tataru, 07; Hadac-Herr-Koch, 09; Herr-Tataru-Tzvetkov, 10] ) 1 p Paul Smith (UC Berkeley) Chern-Simons-Schrödinger UNC, July 2012 30 / 32

Function spaces For U 2, V 2 spaces, we have the embeddings Ẋ s, 1 2,1 U 2 V 2 Ẋ s, 1 2,. From these inclusions we conclude that the U 2 and V 2 norms are equivalent when restricted in modulation to a single dyadic scale. Paul Smith (UC Berkeley) Chern-Simons-Schrödinger UNC, July 2012 31 / 32

Extending estimates to U 2, V 2 functions The atomic structure of the U 2 space allows us to easily extend estimates for free waves to estimates for U 2 functions. An interpolation property allows us to extend these same estimates to V 2 functions, but with a log loss. Because we work in H s, s > 0, we can handle (many) of these log losses in a straightforward way. Paul Smith (UC Berkeley) Chern-Simons-Schrödinger UNC, July 2012 32 / 32