Low-frequency stability analysis of periodic traveling-wave - - PowerPoint PPT Presentation

Low-frequency stability analysis of periodic traveling-wave solutions of viscous conservation laws in several dimensions

Myunghyun Oh Department of Mathematics University of Kansas

Stability Analysis, HYP2006 – p.1/16

Outline of the talk

• Introduction

Stability Analysis, HYP2006 – p.2/16

Outline of the talk

• Introduction
• Basic Ideas

Stability Analysis, HYP2006 – p.2/16

Outline of the talk

• Introduction
• Basic Ideas
• Known Results

Stability Analysis, HYP2006 – p.2/16

Outline of the talk

• Introduction
• Basic Ideas
• Known Results
• Spectral Stability

Stability Analysis, HYP2006 – p.2/16

Outline of the talk

• Introduction
• Basic Ideas
• Known Results
• Spectral Stability
• More...

Stability Analysis, HYP2006 – p.2/16

Introduction We investigate stability of periodic traveling waves: specifically, the spectrum of the linearized operator about the wave.

Stability Analysis, HYP2006 – p.3/16

Introduction We investigate stability of periodic traveling waves: specifically, the spectrum of the linearized operator about the wave. ut + f(u)x = (B(u)ux)x, where x ∈ R and u, f ∈ Rn and a periodic traveling-wave solution u(x, t) = ¯ u(x − st) of period X.

Stability Analysis, HYP2006 – p.3/16

Introduction We investigate stability of periodic traveling waves: specifically, the spectrum of the linearized operator about the wave. ut + f(u)x = (B(u)ux)x, where x ∈ R and u, f ∈ Rn and a periodic traveling-wave solution u(x, t) = ¯ u(x − st) of period X. Our main result generalizes the works of Oh-Zumbrun, Serre about stability of periodic traveling waves of systems of vis- cous conservation laws from the one-dimensional to the multi- dimensional setting.

Stability Analysis, HYP2006 – p.3/16

Evans function

Choose bases [u1(λ), . . . , uk(λ)] and [uk+1(λ), . . . , un(λ)] of Eu

0 (λ) and Es 0(λ), respectively.

Stability Analysis, HYP2006 – p.4/16

Evans function

Choose bases [u1(λ), . . . , uk(λ)] and [uk+1(λ), . . . , un(λ)] of Eu

0 (λ) and Es 0(λ), respectively.

The Evans function is defined by D(λ) = det[u1(λ), . . . , un(λ)].

Stability Analysis, HYP2006 – p.4/16

Evans function

Choose bases [u1(λ), . . . , uk(λ)] and [uk+1(λ), . . . , un(λ)] of Eu

0 (λ) and Es 0(λ), respectively.

The Evans function is defined by D(λ) = det[u1(λ), . . . , un(λ)].

• D(λ) ∈ R whenever λ ∈ R
• D(λ) = 0 if, and only if, λ is an eigenvalue of the ODE.
• The order of λ as a zero of D(λ) is equal to the algebraic

multiplicity of λ as an eigenvalue of the ODE.

Stability Analysis, HYP2006 – p.4/16

Floquet theory

The spectral analysis of differential operators with periodic coefficients.

Stability Analysis, HYP2006 – p.5/16

Floquet theory

The spectral analysis of differential operators with periodic coefficients. The eigenvalue equation Lw = λw has a non-trivial bounded solution.

Stability Analysis, HYP2006 – p.5/16

Floquet theory

The spectral analysis of differential operators with periodic coefficients. The eigenvalue equation Lw = λw has a non-trivial bounded solution. The eigenvalue equation Lw = λw has a non-trivial solution with w(x + kX) = eikθw(x), k ∈ Z.

Stability Analysis, HYP2006 – p.5/16

Floquet theory

The spectral analysis of differential operators with periodic coefficients. The eigenvalue equation Lw = λw has a non-trivial bounded solution. The eigenvalue equation Lw = λw has a non-trivial solution with w(x + kX) = eikθw(x), k ∈ Z. λ ∈ if, and only if, det(M(X; λ) − eiθI2n) = 0 for some real number θ.

Stability Analysis, HYP2006 – p.5/16

Floquet theory

The spectral analysis of differential operators with periodic coefficients. The eigenvalue equation Lw = λw has a non-trivial bounded solution. The eigenvalue equation Lw = λw has a non-trivial solution with w(x + kX) = eikθw(x), k ∈ Z. λ ∈ if, and only if, det(M(X; λ) − eiθI2n) = 0 for some real number θ.

• pt is empty.

Stability Analysis, HYP2006 – p.5/16

Evans function for periodic waves

Choose a basis W1(·; λ), . . . , W2n(·; λ) of the kernel of L − λ. The Evans function is defined by D(λ, θ) = det   Wj(X; λ) − eiθWj(0; λ) W ′

j(X; λ) − eiθW ′ j(0; λ)

 

1≤j≤2n

.

Stability Analysis, HYP2006 – p.6/16

Known Results

• Gardner proved a stability result for the large wavelength

periodic waves

Stability Analysis, HYP2006 – p.7/16

Known Results

• Gardner proved a stability result for the large wavelength

periodic waves

• Oh and Zumbrun established a general theory and method of

the stability analysis of periodic waves in the “quasi-Hamiltonian” case

Stability Analysis, HYP2006 – p.7/16

Known Results

• Gardner proved a stability result for the large wavelength

periodic waves

• Oh and Zumbrun established a general theory and method of

the stability analysis of periodic waves in the “quasi-Hamiltonian” case

• analytic instability

Stability Analysis, HYP2006 – p.7/16

Known Results

• Gardner proved a stability result for the large wavelength

periodic waves

• Oh and Zumbrun established a general theory and method of

the stability analysis of periodic waves in the “quasi-Hamiltonian” case

• analytic instability
• numerical instability

Stability Analysis, HYP2006 – p.7/16

Known Results

• Gardner proved a stability result for the large wavelength

periodic waves

• Oh and Zumbrun established a general theory and method of

the stability analysis of periodic waves in the “quasi-Hamiltonian” case

• analytic instability
• numerical instability
• point-wise bounds

Stability Analysis, HYP2006 – p.7/16

Known Results

• Gardner proved a stability result for the large wavelength

periodic waves

• Oh and Zumbrun established a general theory and method of

the stability analysis of periodic waves in the “quasi-Hamiltonian” case

• analytic instability
• numerical instability
• point-wise bounds
• Serre carried out the stability study in the general case

Stability Analysis, HYP2006 – p.7/16

Spectral Stability

Consider a system of conservation laws ut +

• j

f j(u)xj =

• j, k

(Bjk(u)uxk)xj, u ∈ U(open) ∈ Rn, f j ∈ Rn, Bjk ∈ Rn×n, x ∈ Rd, d ≥ 2,

Stability Analysis, HYP2006 – p.8/16

Spectral Stability

Consider a system of conservation laws ut +

• j

f j(u)xj =

• j, k

(Bjk(u)uxk)xj, u ∈ U(open) ∈ Rn, f j ∈ Rn, Bjk ∈ Rn×n, x ∈ Rd, d ≥ 2, and a periodic traveling-wave solution u = ¯ u(x · ν − st),

Stability Analysis, HYP2006 – p.8/16

Spectral Stability

Consider a system of conservation laws ut +

• j

f j(u)xj =

• j, k

(Bjk(u)uxk)xj, u ∈ U(open) ∈ Rn, f j ∈ Rn, Bjk ∈ Rn×n, x ∈ Rd, d ≥ 2, and a periodic traveling-wave solution u = ¯ u(x · ν − st),

• f period X, satisfying the traveling-wave ODE

(

• j,k

νjνkBjk(¯ u)¯ u′)′ = (

• j

νjf j(¯ u))′ − s¯ u′. with ¯ u(0) = ¯ u(X) =: u0.

Stability Analysis, HYP2006 – p.8/16

Example

In the constant-coefficient case B11w′′ − A1w′ + i

• j=1

Bj1ξjw′ + i

• k=1

B1kξkw′ − i

• j=1

Ajξjw −

• j=1,k=1

Bjkξkξjw − λw = 0,

Stability Analysis, HYP2006 – p.9/16

Example

In the constant-coefficient case B11w′′ − A1w′ + i

• j=1

Bj1ξjw′ + i

• k=1

B1kξkw′ − i

• j=1

Ajξjw −

• j=1,k=1

Bjkξkξjw − λw = 0, an elementary computation yields D(λ, ξ) = Π2n

l=1(eµl(λ,˜ ξ)X − eiξ1X),

Stability Analysis, HYP2006 – p.9/16

Example

In the constant-coefficient case B11w′′ − A1w′ + i

• j=1

Bj1ξjw′ + i

• k=1

B1kξkw′ − i

• j=1

Ajξjw −

• j=1,k=1

Bjkξkξjw − λw = 0, an elementary computation yields D(λ, ξ) = Π2n

l=1(eµl(λ,˜ ξ)X − eiξ1X),

where µl are roots of (µ2B11 + µ(−A1 + i

• j=1

Bj1ξj + i

• k=1

B1kξk) − (i

• j=1

Ajξj +

• j=1,k=1

Bjkξkξj + λI)) ¯ w = 0, where w = eµx1 ¯ w.

Stability Analysis, HYP2006 – p.9/16

Remarks

Setting µ = iξ1 , det(−Bξ − iAξ − λI) = 0 where Aξ =

j Ajξj and Bξ = j,k Bjkξkξj.

Stability Analysis, HYP2006 – p.10/16

Remarks

Setting µ = iξ1 , det(−Bξ − iAξ − λI) = 0 where Aξ =

j Ajξj and Bξ = j,k Bjkξkξj.

λj(ξ) = 0 − iaj(ξ) + O(|ξ|), j = 1, . . . , n, for the n roots bifurcating from λ(0) = 0, where aj denote the eigenvalues of Aξ. Thus we obtain the necessary stability condition of hyperbolicity, σ(Aξ) real.

Stability Analysis, HYP2006 – p.10/16

Profile equation

Integrating the traveling-wave ODE, we reduce to a first-order profile equation

• j,k

νjνkBjk(¯ u)¯ u′ =

• j

νjf j(¯ u) − s¯ u − q.

Stability Analysis, HYP2006 – p.11/16

Profile equation

Integrating the traveling-wave ODE, we reduce to a first-order profile equation

• j,k

νjνkBjk(¯ u)¯ u′ =

• j

νjf j(¯ u) − s¯ u − q. Given (a, s, ν, q) ∈ U × R × Sd−1 × Rn, the profile equation admits a unique local solution u(y; a, s, ν, q) such that u(0; a, s, ν, q) = a.

Stability Analysis, HYP2006 – p.11/16

Profile equation

Integrating the traveling-wave ODE, we reduce to a first-order profile equation

• j,k

νjνkBjk(¯ u)¯ u′ =

• j

νjf j(¯ u) − s¯ u − q. Given (a, s, ν, q) ∈ U × R × Sd−1 × Rn, the profile equation admits a unique local solution u(y; a, s, ν, q) such that u(0; a, s, ν, q) = a. Denote X the period, ω := 1/X the frequency and M and F j the averages over the period: M := 1 X X u(y)dy, F j := 1 X X

• f j(u) −

d

• k=1

Bjk(u)ωνk∂yu

• dy

when u is a periodic solution of the profile equation.

Stability Analysis, HYP2006 – p.11/16

Class functions and assumptions

X = X( ˙ u), ω = Ω( ˙ u), s = S( ˙ u), ν = N( ˙ u), q = Q( ˙ u), M = M( ˙ u), F j = F j( ˙ u), where ˙ u is the equivalence class of translates of different periodic functions.

Stability Analysis, HYP2006 – p.12/16

Class functions and assumptions

X = X( ˙ u), ω = Ω( ˙ u), s = S( ˙ u), ν = N( ˙ u), q = Q( ˙ u), M = M( ˙ u), F j = F j( ˙ u), where ˙ u is the equivalence class of translates of different periodic functions. We assume: (H0) f j, Bjk ∈ C2. (H1) Re σ(

jk νjνkBjk) ≥ θ > 0.

(H2) The map H : R × U × R × Sd−1 × Rn → Rn taking (X; a, s, ν, q) → u(X; a, s, ν, q) − a is a submersion at point ( ¯ X; ¯ a, 0, e1, ¯ q).

Stability Analysis, HYP2006 – p.12/16

Homogenized equation and the family of eigenvalue equations

∂tM( ˙ u) +

• j

∂xj(F j( ˙ u)) = 0, ∂t(ΩN( ˙ u)) + ∇x(ΩS( ˙ u)) = with ˙ u in the vicinity of ¯ u.

Stability Analysis, HYP2006 – p.13/16

Homogenized equation and the family of eigenvalue equations

∂tM( ˙ u) +

• j

∂xj(F j( ˙ u)) = 0, ∂t(ΩN( ˙ u)) + ∇x(ΩS( ˙ u)) = with ˙ u in the vicinity of ¯ u. Without loss of generality taking S(¯ u) = 0, N(¯ u) = e1, ¯ u = ¯ u(x1) represents a stationary solution. Linearizing about ¯ u, taking Fourier transform in ˜ x = (x2, · · · , xd), and taking Laplace transform in t, we obtain 0 = (L˜

ξ − λ)w

= (B11w′)′ − (A1w)′ + i

• j=1

Bj1ξjw′ + i(

• k=1

B1kξkw)′ − i

• j=1

Ajξjw −

• j=1,k=1

Bjkξkξjw − λw.

Stability Analysis, HYP2006 – p.13/16

Main Result

ˆ ∆(ξ, λ) := det  λ∂(M, ΩN) ∂ ˙ u (˙ ¯ u) +

• j

iξj ∂(F j, SΩej) ∂ ˙ u (˙ ¯ u)   = 0.

Stability Analysis, HYP2006 – p.14/16

Main Result

ˆ ∆(ξ, λ) := det  λ∂(M, ΩN) ∂ ˙ u (˙ ¯ u) +

• j

iξj ∂(F j, SΩej) ∂ ˙ u (˙ ¯ u)   = 0. D(ξ, λ) = ∆1(ξ, λ) + O(|ξ, λ|n+2), where ∆1 is a homogeneous degree n + 1 polynomial.

Stability Analysis, HYP2006 – p.14/16

Main Result

ˆ ∆(ξ, λ) := det  λ∂(M, ΩN) ∂ ˙ u (˙ ¯ u) +

• j

iξj ∂(F j, SΩej) ∂ ˙ u (˙ ¯ u)   = 0. D(ξ, λ) = ∆1(ξ, λ) + O(|ξ, λ|n+2), where ∆1 is a homogeneous degree n + 1 polynomial. Relating these two expansions, D(ξ, λ) = Γ0λ1−d ˆ ∆(ξ, λ) + O(|ξ, λ|n+2), Γ0 = 0 constant, for |ξ, λ| sufficiently small.

Stability Analysis, HYP2006 – p.14/16

Corollary

With the nondegeneracy condition det ∂(M, ΩN) ∂ ˙ u (˙ ¯ u)

• = 0,

for λ, ξ sufficiently small,

Stability Analysis, HYP2006 – p.15/16

Corollary

With the nondegeneracy condition det ∂(M, ΩN) ∂ ˙ u (˙ ¯ u)

• = 0,

for λ, ξ sufficiently small, the zero-set of D(·, ·), corresponding to spectra of L, consists of n + 1 characteristic surfaces: λj(ξ) = −iaj(ξ) + O(ξ), j = 1, . . . , n + 1,

Stability Analysis, HYP2006 – p.15/16

Corollary

With the nondegeneracy condition det ∂(M, ΩN) ∂ ˙ u (˙ ¯ u)

• = 0,

for λ, ξ sufficiently small, the zero-set of D(·, ·), corresponding to spectra of L, consists of n + 1 characteristic surfaces: λj(ξ) = −iaj(ξ) + O(ξ), j = 1, . . . , n + 1, where aj(ξ) denote the eigenvalues of A :=

• j

ξj ∂(F j, SΩej) ∂(M, ΩN) .

Stability Analysis, HYP2006 – p.15/16

Asymptotic and Nonlinear Stability

If A has distinct real eigenvalues, i.e. strict hyperbolicity of the system, the strong spectral stability implies

Stability Analysis, HYP2006 – p.16/16

Asymptotic and Nonlinear Stability

If A has distinct real eigenvalues, i.e. strict hyperbolicity of the system, the strong spectral stability implies linearized L1 → Lp asymptotic stability for all p ≥ 2 and all dimensions d ≥ 1 with rate of decay v(t)Lp ≤ Ct− d

2 (1−1/p)v0L1,

for all t > 0,

Stability Analysis, HYP2006 – p.16/16

Asymptotic and Nonlinear Stability

If A has distinct real eigenvalues, i.e. strict hyperbolicity of the system, the strong spectral stability implies linearized L1 → Lp asymptotic stability for all p ≥ 2 and all dimensions d ≥ 1 with rate of decay v(t)Lp ≤ Ct− d

2 (1−1/p)v0L1,

for all t > 0, and nonlinear stability for all p ≥ 2 and all dimensions d ≥ 2 with respect to Hölder continuous initial perturbations that are sufficiently small in L1 ∩ L∞.

Stability Analysis, HYP2006 – p.16/16