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. u t + f ( u ) x = ( B ( u ) u x ) x , where x ∈ R and u, f ∈ R n 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. u t + f ( u ) x = ( B ( u ) u x ) x , where x ∈ R and u, f ∈ R n 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 [ u 1 ( λ ) , . . . , u k ( λ )] and [ u k +1 ( λ ) , . . . , u n ( λ )] of E u 0 ( λ ) and E s 0 ( λ ) , respectively. Stability Analysis, HYP2006 – p.4/16
Evans function Choose bases [ u 1 ( λ ) , . . . , u k ( λ )] and [ u k +1 ( λ ) , . . . , u n ( λ )] of E u 0 ( λ ) and E s 0 ( λ ) , respectively. The Evans function is defined by D ( λ ) = det[ u 1 ( λ ) , . . . , u n ( λ )] . Stability Analysis, HYP2006 – p.4/16
Evans function Choose bases [ u 1 ( λ ) , . . . , u k ( λ )] and [ u k +1 ( λ ) , . . . , u n ( λ )] of E u 0 ( λ ) and E s 0 ( λ ) , respectively. The Evans function is defined by D ( λ ) = det[ u 1 ( λ ) , . . . , u n ( 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 ) = e ikθ w ( 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 ) = e ikθ w ( x ) , k ∈ Z . λ ∈ � if, and only if, det( M ( X ; λ ) − e iθ I 2 n ) = 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 ) = e ikθ w ( x ) , k ∈ Z . λ ∈ � if, and only if, det( M ( X ; λ ) − e iθ I 2 n ) = 0 for some real number θ . � pt is empty. Stability Analysis, HYP2006 – p.5/16
Evans function for periodic waves Choose a basis W 1 ( · ; λ ) , . . . , W 2 n ( · ; λ ) of the kernel of L − λ . The Evans function is defined by W j ( X ; λ ) − e iθ W j (0; λ ) D ( λ, θ ) = det . j ( X ; λ ) − e iθ W ′ W ′ j (0; λ ) 1 ≤ j ≤ 2 n 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 � � f j ( u ) x j = ( B jk ( u ) u x k ) x j , u t + j j, k u ∈ U ( open ) ∈ R n , f j ∈ R n , B jk ∈ R n × n , x ∈ R d , d ≥ 2 , Stability Analysis, HYP2006 – p.8/16
Spectral Stability Consider a system of conservation laws � � f j ( u ) x j = ( B jk ( u ) u x k ) x j , u t + j j, k u ∈ U ( open ) ∈ R n , f j ∈ R n , B jk ∈ R n × n , x ∈ R d , 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 � � f j ( u ) x j = ( B jk ( u ) u x k ) x j , u t + j j, k u ∈ U ( open ) ∈ R n , f j ∈ R n , B jk ∈ R n × n , x ∈ R d , d ≥ 2 , and a periodic traveling-wave solution u = ¯ u ( x · ν − st ) , of period X , satisfying the traveling-wave ODE u ′ ) ′ = ( u )) ′ − s ¯ � � ν j ν k B jk (¯ ν j f j (¯ u ′ . ( u )¯ j j,k with ¯ u (0) = ¯ u ( X ) =: u 0 . Stability Analysis, HYP2006 – p.8/16
Example In the constant-coefficient case B 11 w ′′ − A 1 w ′ + i B j 1 ξ j w ′ + i � � B 1 k ξ k w ′ j � =1 k � =1 � � A j ξ j w − B jk ξ k ξ j w − λw = 0 , − i j � =1 j � =1 ,k � =1 Stability Analysis, HYP2006 – p.9/16
Example In the constant-coefficient case B 11 w ′′ − A 1 w ′ + i B j 1 ξ j w ′ + i � � B 1 k ξ k w ′ j � =1 k � =1 � � A j ξ j w − B jk ξ k ξ j w − λw = 0 , − i j � =1 j � =1 ,k � =1 an elementary computation yields ξ ) X − e iξ 1 X ) , l =1 ( e µ l ( λ, ˜ D ( λ, ξ ) = Π 2 n Stability Analysis, HYP2006 – p.9/16
Example In the constant-coefficient case B 11 w ′′ − A 1 w ′ + i B j 1 ξ j w ′ + i � � B 1 k ξ k w ′ j � =1 k � =1 � � A j ξ j w − B jk ξ k ξ j w − λw = 0 , − i j � =1 j � =1 ,k � =1 an elementary computation yields ξ ) X − e iξ 1 X ) , l =1 ( e µ l ( λ, ˜ D ( λ, ξ ) = Π 2 n where µ l are roots of ( µ 2 B 11 + µ ( − A 1 + i � � B j 1 ξ j + i B 1 k ξ k ) j � =1 k � =1 � � A j ξ j + B jk ξ k ξ j + λI )) ¯ − ( i w = 0 , j � =1 j � =1 ,k � =1 where w = e µx 1 ¯ w . Stability Analysis, HYP2006 – p.9/16
Remarks Setting µ = iξ 1 , det ( − B ξ − iA ξ − λI ) = 0 where A ξ = � j A j ξ j and B ξ = � j,k B jk ξ k ξ j . Stability Analysis, HYP2006 – p.10/16
Remarks Setting µ = iξ 1 , det ( − B ξ − iA ξ − λI ) = 0 where A ξ = � j A j ξ j and B ξ = � j,k B jk ξ k ξ j . λ j ( ξ ) = 0 − ia j ( ξ ) + O ( | ξ | ) , j = 1 , . . . , n, for the n roots bifurcating from λ (0) = 0 , where a j denote the eigenvalues of A ξ . Thus we obtain the necessary stability condition of hyperbolicity, σ ( A ξ ) real. Stability Analysis, HYP2006 – p.10/16
Profile equation Integrating the traveling-wave ODE, we reduce to a first-order profile equation u ′ = � � ν j ν k B jk (¯ ν j f j (¯ u )¯ u ) − s ¯ u − q. j j,k Stability Analysis, HYP2006 – p.11/16
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