Slow Modulation & Large-Time Dynamics Near Periodic Waves

Slow Modulation & Large-Time Dynamics Near Periodic Waves Miguel Rodrigues IRMAR Université Rennes 1 France SIAG-APDE Prize Lecture Jointly with Mathew Johnson (Kansas), Pascal Noble (INSA Toulouse), Kevin Zumbrun (Indiana). Formerly Lyon/Indiana. Part of a larger project also including Blake Barker (Brown). L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 0 / 23

Outline. 1 Motivation Model result Some surface waves 2 Structure of the spectrum 3 Dynamical stability 4 Averaged dynamics 5 Conclusion L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 0 / 23

Ordinary differential equations. F : R n R n smooth. U : R + R n solving Steady solution, 0 = F(U). U t = F(U) and U(0) = U 0. L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 1 / 23

Ordinary differential equations. F : R n R n smooth. U : R + R n solving Steady solution, 0 = F(U). Stability δ > 0, ε > 0, U t = F(U) and U(0) = U 0. d(u 0, U) < ε (! U and ( t 0, d(u(t), U) < δ)). Asymptotic stability Stability + ε 0 > 0, d(u 0, U) < ε 0 ( )! U and d(u(t), U) t 0. L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 1 / 23

From the spectrum to the nonlinear dynamics. Spectral stability σ(df(u)) {λ Reλ < 0}. Theorem Spectral stability implies asymptotic stability. Decay is exponentially fast. L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 2 / 23

From the spectrum to the nonlinear dynamics. Spectral stability σ(df(u)) {λ Reλ < 0}. Theorem Spectral stability implies asymptotic stability. Decay is exponentially fast. Goal : PDE version for periodic plane waves. U : R + R R d solving U t = F(U, U x, ) and U(0, ) = U 0 ( ). Adapted notion of nonlinear stability for periodic waves. Adapted notion of spectral stability, depending on the PDE type. L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 2 / 23

Roll-waves in thin films. In a channel. Courtesy of Neil Balmforth (British Columbia). The Saint-Venant system (SV) h t + (hu) x = 0, (hu) t + ) (hu 2 + h2 2F 2 x = h u u + (h u x ) x. F > 2, primary instability. The Korteweg-de Vries/Kuramoto-Sivashinsky equation (KdV-KS) U T + ( 1 2 U2) X + U XXX + δ(u XX + U XXXX ) = 0. Near threshold, δ F 2. L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 3 / 23

Outline. 1 Motivation 2 Structure of the spectrum Floquet-Bloch theory Diffusive spectral stability 3 Dynamical stability 4 Averaged dynamics 5 Conclusion L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 3 / 23

About a wave. t > 0 time, x R space, U(x, t) R d unknown. f smooth. (P) U t + (f(u)) x = U xx. Traveling wave. U(t, x) = U(x ct) moving with speed c. Profile U solving c U x + (f(u)) x = U xx. L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 4 / 23

About a wave. t > 0 time, x R space, U(x, t) R d unknown. f smooth. (P) U t + (f(u)) x = U xx. Traveling wave. U(t, x) = U(x ct) moving with speed c. Profile U solving c U x + (f(u)) x = U xx. Generator in the frame of the wave. L := 2 x + c x x df(u) L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 4 / 23

Integral transform. Bloch-wave decomposition g(x) = π π e iξx ǧ(ξ, x) dξ, where ξ is a Floquet exponent ǧ(ξ, x + 1) = ǧ(ξ, x), e iξ(x+1) ǧ(ξ, x + 1) = e iξ e iξx ǧ(ξ, x). L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 5 / 23

Integral transform. Bloch-wave decomposition g(x) = π π e iξx ǧ(ξ, x) dξ, where ξ is a Floquet exponent ǧ(ξ, x + 1) = ǧ(ξ, x), e iξ(x+1) ǧ(ξ, x + 1) = e iξ e iξx ǧ(ξ, x). From the Fourier decomposition g(x) = R e iξx ĝ(ξ)dξ. Floquet-Bloch transform ǧ(ξ, x) := e i 2jπx ĝ(ξ + 2jπ) = e i ξ (x+k) g(x + k). j Z k Z L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 5 / 23

First observations. h(x) = π π e iξx ȟ(ξ, x) dξ. Observations (g h)ˇ(ξ, x) = g(x) ȟ(ξ, x), g of period 1, ( x h)ˇ(ξ, ) = ( x + iξ) ȟ(ξ, ), ȟ (ξ, x) = ĥ(ξ), h low-frequency. L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 6 / 23

First observations. h(x) = π π e iξx ȟ(ξ, x) dξ. Observations (g h)ˇ(ξ, x) = g(x) ȟ(ξ, x), g of period 1, ( x h)ˇ(ξ, ) = ( x + iξ) ȟ(ξ, ), ȟ (ξ, x) = ĥ(ξ), h low-frequency. Remark : If g is of period 1 and h is low-frequency (g h)ˇ(ξ, x) = g(x) ĥ(ξ). L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 6 / 23

Bloch symbols. U(t, x) = U(k(x ct)), U of period 1. Generator L := k 2 x 2 + k c x k x df(u) with Bloch symbols L ξ := k 2 ( x + iξ) 2 + k ( x + iξ)(c df(u)), ξ [ π, π]. Bloch diagonalization (L g)(x) = π π e iξx (L ξ ǧ(ξ, ))(x) dξ. Each L ξ acts on functions of period 1. L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 7 / 23

Diffusive spectral stability. Spectral decomposition σ(l) = ξ [ π,π] σ per (L ξ ). 0.1 0.08 0.06 0.04 0.02 0 0.02 0.04 0.06 0.08 T= 5.8 c= 0.56669987 0.1 0.1 0.05 0 0.05 0.1 Spectum of a stable wave of (SV). Barker-Johnson-Noble-LMR- Zumbrun, Proc. Port-d Albret 2010. (D1) Critical spectrum reduced to {0}. σ(l) {λ Reλ < 0} {0}. (D2) Diffusion. θ > 0, ξ [ π, π], σ per (L ξ ) {λ Reλ θ ξ 2 }. (D3) λ = 0 of multiplicity d + 1 for L 0 (minimal dimension). (H) Distinct group velocities (strict hyperbolicity). L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 8 / 23

Outline. 1 Motivation 2 Structure of the spectrum 3 Dynamical stability Space-modulated stability The parabolic case 4 Averaged dynamics 5 Conclusion L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 8 / 23

Direct simulation : space-time diagramm. About a stable wave of (KdV-KS). Barker-Johnson-Noble-LMR-Zumbrun, Phys. D 2013. Peaks. Troughs. Theory. 25 20 Time 15 10 5 140 160 180 x 200 L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 9 / 23

Space-modulation of distances. Allow for resynchronization by comparing functions with δ H (u, v) = inf u Ψ v H + x (Ψ Id R ) H. Ψ one-to-one U reference wave. U(t, ) U L p (R) 1 t σ t p while, in a diffusive context, δ L p (R)( U(t, ), U ) t σ t 1 2 (1 1/p). L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 10 / 23

Nonlinear diffusive stability. Theorem (Johnson-Noble-LMR-Zumbrun, Inventiones Math. 2014.) A diffusively spectrally stable periodic wave of (P) is δ L 1 (R) H K (R) to δ H K (R) asymptotically stable when K 3, with algebraic decay rates. L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 11 / 23

Elements of proof : introducing the phase. Seek for (V, ψ) with (V, x ψ) small and such that V(t, ) = U(t, ) (Id R ψ(t, )) U. L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 12 / 23

Elements of proof : introducing the phase. Seek for (V, ψ) with (V, x ψ) small and such that V(t, ) = U(t, ) (Id R ψ(t, )) U. Equation ( t L) (V(t) + U x ψ(t)) = N (t). L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 12 / 23

Elements of proof : introducing the phase. Seek for (V, ψ) with (V, x ψ) small and such that V(t, ) = U(t, ) (Id R ψ(t, )) U. Equation ( t L) (V(t) + U x ψ(t)) = N (t). Integral formulation V(t) + U x ψ(t) = S(t) (V(0) + U x ψ 0 ) + with S(t) := e t L. t 0 S(t s)n (s)ds, L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 12 / 23

Isolating the phase. Separation S(t) = U x s φ (t) + S(t) t ψ(t) = s φ (t) (V 0 + U x ψ 0 ) + s φ (t s)n (s)ds 0 t V(t) = S(t) (V 0 + U x ψ 0 ) + 0 S(t s)n (s)ds L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 13 / 23

Isolating the phase. Separation S(t) = U x s φ (t) + S(t) t ψ(t) = s φ (t) (V 0 + U x ψ 0 ) + 0 [ χ(t) s φ (t) (V 0 + U x ψ 0 ) ψ 0 + t 0 s φ (t s)n (s)ds ] s φ (t s)n (s)ds t V(t) = S(t) (V 0 + U x ψ 0 ) + + χ(t) U x [s φ (t) (V 0 + U x ψ 0 ) ψ 0 + with χ cutting off large times. 0 t 0 S(t s)n (s)ds ] s φ (t s)n (s)ds L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 13 / 23

Critical evolution. Expand with 1 ikξ d+1 α=1 e λα(ξ)t ψ α (ξ, ) φ α (ξ, ) ψ α (ξ, ) = U x ( ) β (α) (ξ) + O(ξ) L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 14 / 23

Critical evolution. Expand with to U x ( ) α=1 Decomposition 1 ikξ d+1 with s φ (t) = sα(t). φ d+1 α=1 e λα(ξ)t ψ α (ξ, ) φ α (ξ, ) ψ α (ξ, ) = U x ( ) β (α) (ξ) + O(ξ) d+1 1 ikξ eλα(ξ)t β (α) (ξ) φ α (ξ, ) + O(e θ ξ 2 t ). α=1 S(t) = U x s φ (t) + S(t) L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 14 / 23

Bounds. 1 Bloch analysis. Hausdorff-Young inequalities For 2 p, One estimate g L p (R) ǧ L p ([ π,π],l p ([0,1])), ǧ L p ([ π,π],l p ([0,1])) g L p (R). ψ 0 ( ) = ψ 0 ( ). For t 0, 2 p, x sα(t)(ψ φ 0 U x ) (1 + L p (R) t) 1 2 (1 1/p) x ψ 0 L 1 (R). 2 Energy estimates and resolvent estimates in a Hilbertian framework. Out of time? L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 15 / 23

Outline. 1 Motivation 2 Structure of the spectrum 3 Dynamical stability 4 Averaged dynamics Formal derivation Nonlinear validation 5 Conclusion L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 15 / 23

Parametrization. At fixed k wavenumber, U(t, x) = U(k(x ct)) with U of period 1 determined by averaged values M:= U. L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 16 / 23

Parametrization. At fixed k wavenumber, U(t, x) = U(k(x ct)) with U of period 1 determined by averaged values M:= U. Profile : U = U (M,k). Phase velocity and time frequency c = c(m, k), ω(m, k) = k c(m, k). L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 16 / 23

Two-scale ansatz. Two-timing method. Fast oscillation/slow variation. U(t, x) = ( U 0 + εu 1 + ε 2 ) ( U 2 }{{} εt, }{{} εx, (Ψ 0 + εψ 1 )(εt, εx) ) + o(ε 2 ) ε T X }{{} θ with U 0, U 1 and U 2 of period 1 in θ. L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 17 / 23

Slow modulation behavior. ( ) (U 0 + εu 1 )(T, X, θ) = U ((M 0,κ 0 )+ɛ(m 1,κ 1 ))(T,X ) θ + o(ε) with κ 0 + ɛκ 1 = (Ψ 0 + ɛψ 1 ) X local wave number, M 0 + ɛm 1 local averages. L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 18 / 23

Averaged dynamics : matching with slow evolution. Evolution of (M 0, κ 0 ), (M 1, κ 1 ) coincide with expansion of slow ansatz ( )( ) (M, κ)(x, t) = (M 0, κ 0 ) + ε(m 1, κ 1 ) }{{} εt, }{{} εx + o(ε 2 ) into T X (W ) { Mt + (F (M, κ)) x = (d 11 (M, κ)m x + d 12 (M, κ)κ x ) x, κ t (ω(m, κ)) x = (d 21 (M, κ)m x + d 22 (M, κ)κ x ) x. L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 19 / 23

Averaged dynamics : matching with slow evolution. Evolution of (M 0, κ 0 ), (M 1, κ 1 ) coincide with expansion of slow ansatz ( )( ) (M, κ)(x, t) = (M 0, κ 0 ) + ε(m 1, κ 1 ) }{{} εt, }{{} εx + o(ε 2 ) into T X (W ) { Mt + (F (M, κ)) x = (d 11 (M, κ)m x + d 12 (M, κ)κ x ) x, κ t (ω(m, κ)) x = (d 21 (M, κ)m x + d 22 (M, κ)κ x ) x. (W ) phase Ψ t ω(m, κ) = d 21 (M, κ)m x + d 22 (M, κ)κ x. L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 19 / 23

Averaged dynamics : matching with slow evolution. Evolution of (M 0, κ 0 ), (M 1, κ 1 ) coincide with expansion of slow ansatz ( )( ) (M, κ)(x, t) = (M 0, κ 0 ) + ε(m 1, κ 1 ) }{{} εt, }{{} εx + o(ε 2 ) into T X (W ) { Mt + (F (M, κ)) x = (d 11 (M, κ)m x + d 12 (M, κ)κ x ) x, κ t (ω(m, κ)) x = (d 21 (M, κ)m x + d 22 (M, κ)κ x ) x. (W ) phase Ψ t ω(m, κ) = d 21 (M, κ)m x + d 22 (M, κ)κ x. I shall hide a choice leading to a canonical artificial viscosity system. Out of time? L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 19 / 23

Asymptotic behavior, refined description. Theorem (Johnson-Noble-LMR-Zumbrun, Inventiones Math. 2014.) Let η > 0 and K 4. There exists ε > 0 and C > 0 such that if E 0 := U 0 (Id R ψ 0 ) U L 1 (R) H K (R) + x ψ 0 L 1 (R) H K (R) ε for some ψ 0, then, there exist (U, ψ) with initial data (U 0, ψ 0 ) and M such that, for t > 0 and 2 p, U(t, ψ(t, )) ( U k M+M(t, ), (1 ψx (t, )) ) ( ) L p (R) C E 0 ln(2 + t) (1 + t) 3 4 (M, k ψ x )(t, ) L p (R) C E 0 (1 + t) 1 2 (1 1/p). L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 20 / 23

Asymptotic behavior, validation of (W). Theorem (Johnson-Noble-LMR-Zumbrun, Inventiones Math. 2014.) Moreover, setting Ψ(t, ) = (Id R ψ(t, )) 1, κ = k x Ψ, M(t, ) = (M + M(t, )) Ψ(t, ), and letting (M W, κ W ) and Ψ W solve (W ) and (W ) phase with initial data κ W (0, ) = k x Ψ(0, ), Ψ W (0, ) = Ψ(0, ), M W (0, ) = M ( + U 0 U Ψ(0, ) ) 1 + x Ψ(0, ) 1 (U Ψ(0, ) M), L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 21 / 23

Asymptotic behavior, validation of (W). Theorem (Johnson-Noble-LMR-Zumbrun, Inventiones Math. 2014.) Moreover, setting Ψ(t, ) = (Id R ψ(t, )) 1, κ = k x Ψ, M(t, ) = (M + M(t, )) Ψ(t, ), and letting (M W, κ W ) and Ψ W solve (W ) and (W ) phase with initial data κ W (0, ) = k x Ψ(0, ), Ψ W (0, ) = Ψ(0, ), M W (0, ) = M ( + U 0 U Ψ(0, ) ) 1 + x Ψ(0, ) 1 (U Ψ(0, ) M), we have, for t 0, 2 p, (M, κ)(t, ) (M W, κ W )(t, ) L p (R) C E 0 (1 + t) 1 2 (1 1/p) 1 2 +η, Ψ(t, ) Ψ W (t, ) L p (R) C E 0 (1 + t) 1 2 (1 1/p)+η. L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 21 / 23

Corollaries when ψ 0 0. Question : Could we get usual asymptotic stability in some special cases? Roadmap : look at uncoupling of κ t (ω(m, κ)) x = (d 21 (M, κ)m x + d 22 (M, κ)κ x ) x. L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 22 / 23

Corollaries when ψ 0 0. Question : Could we get usual asymptotic stability in some special cases? Roadmap : look at uncoupling of κ t (ω(m, κ)) x = (d 21 (M, κ)m x + d 22 (M, κ)κ x ) x. Quadratic phase uncoupling d M ω(m, k) 0 and d 2 Mω(M, k) 0. Corollary (Johnson-Noble-LMR-Zumbrun, Inventiones Math. 2014.) Jointly with previous assumptions this implies asymptotic stability from L 1 (R) H K (R) to H K (R). with decay (1 + t) 1 2 (1 1/p) 1 2 +η in L p (R), 2 p. L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 22 / 23

Corollaries when ψ 0 0. Question : Could we get usual asymptotic stability in some special cases? Roadmap : look at uncoupling of κ t (ω(m, κ)) x = (d 21 (M, κ)m x + d 22 (M, κ)κ x ) x. Linear phase uncoupling d M ω(m, k) 0. Corollary (Johnson-Noble-LMR-Zumbrun, Inventiones Math. 2014.) Jointly with previous assumptions this implies asymptotic stability from L 1 (R;(1+ )) H K (R) to H K (R). with decay (1 + t) 1 2 (1 1/p) 1 4 +η in L p (R), 2 p. L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 22 / 23

Outline. 1 Motivation 2 Structure of the spectrum 3 Dynamical stability 4 Averaged dynamics 5 Conclusion L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 22 / 23

Open questions. Verification of spectral assumptions : case-by-case. Space-modulated instability. Genuinely multidimensional periodic waves. Dispersive nonlinear stability (not on a torus). Composed patterns. L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 23 / 23

Bonus 1 : diffusive instability. 80 Time 60 40 20 0 1400 1500 1600 1700 1800 1900 2000 x Failure of diffusivity in (KdV-KS). Barker-Johnson-Noble-LMR-Zumbrun, Phys. D 2013. L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 23 / 23

Bonus 2 : nonlinear dynamics of (KdV). LMR, in preparation. Right : graph of the full solution. Left : perturbation seen from above. L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 23 / 23

Bonus 3 : a dispersive shock in (KdV). A 2-rarefaction wave of the averaged system. LMR, in preparation. L.M. Rodrigues (Rennes 1) About periodic waves SIAM PD 15 23 / 23