On the occasion of our cooperation partner Kevin Zumbrun receiving a Humbold Research Award, he will be in Konstanz and and hold a series of 3 lectures on
"Stability and behavior of periodic solutions in the presence of conservation laws",
July 17, 16:30-18:00, and July 18, 9:00-10:00 and 10:30-11:30.
Everybody who is interested is cordially invited to attend. In case you need a hotel room reservation for Thursday evening, please write soon to Heinreich Freistühler.
Part 1: Formal modulation expansion and nonlinear stability and behavior.
Topics:
A. Connection through modulation to hyperbolic conservation laws/shock theory: contrast formal modulation expansion (system) with conservation laws to those without (scalar, Burgers).
B. Bloch-Floquet framework, Serre’s lemma and appearance for conser-vation laws of Jordan blocks in neutral modes.
C. Key insight: “Rebalancing” (suggested by formal modulation structure) and analytic unfolding of Jordan block. With Haussdorf-Young and Prüss theorem, gives linear Lq → Lp estimates.
D. Nonlinear stability: multiple characteristic speeds prevents classical renormalization methods; instead borrow from conservation laws/shock stability
(i) dynamical removal of phase shift (similar to Beyn “freezing” technique for traveling fronts and pulses) and key nonlinear cancellation estimate, and
(ii) nonlinear damping estimate regaining apparently lost derivatives.
CONCLUSION: nonlinear stability follows from (an appropriate natural generalization of) the classical “diffusive sta-
bility condition” of Schneider in the case without conservation laws.
E. Asymptotic behavior (stated only, with directions to Inventiones reference where it is proved): the principal be-
havior of a perturbed solution is given by formal modulation system- compare to very early paper formally relating Kuramoto-Sivashinsky behavior to mass-spring system between periodic cells, yielding viscoelastic wave equation as asymptotic behavior, of which this is a far-reaching generalization.
Part 2: Amplitude equations and stability of small-amplitude waves.
Topics:
A. Initiation of shallow-water waves: Kuramoto-Sivashinski-KdV and the viscous Saint Venant equations; far from Turing scenario of classical case, this is degenerate Bogdanov-Takens type bifurcation from the constant homogeneous solution.
B. Stability diagram for small-amplitude waves: stability of Cnoidal KdV waves and its strange mirror history across
the iron curtain; Bar-Nepomnyasky analysis, and computer assisted proof of Barker (discus-sion only, with directions to the various references).
C. Turing bifurcation with conservation laws: 3x3 example of Barker-Jung-Z and generalization of Wheeler-Z.
D. Amplitude equa-tions: classical complex Ginzburg Landau (cGL) vs. models of Matthews-Cox (MC) and Häcker-Schneider-Zimmerman (HSZ), the latter of novel coupled cGL/*singular* convection-diffusion type resistant to usual spectral analysis.
E. Eckhaus-type stability conditions for HSZ: 2-parameter spectral perturbation problem and reappearance of the key “rebalancing” trick from lecture 1 and the study of nonlinear stability.
CONCLUSION: simple stability conditions like those for the classical case of cGL, yielding again complete stability dia-
gram for bifurcating waves. Applications to B´enard-Marangoni and shallow-water flow and modern biomorphology models (e.g., for vasculogenesis).
Part 3: Stability of large-amplitude waves: numerical Evans studies and rigorous theory for Saint Venant.
Topics:
A. Numerical evaluation of Gardner’s periodic Evans function, and “shock-type” remedy for pitfalls in the large-period limit.
B. Viscous examples: Kuramoto-Sivashinsky, viscous Saint Venant and anomalous power-law behavior in the large Froude-number limit.
C. Inviscid Saint Venant, or “Dressler wave”, analysis explaining limiting power-law behavior.
D. Doubly periodic Herringbone patterns for Saint-Venant equations in multi-d (Yang-Z, Tice et al; description only).
E. Subsequent/current developments: multi-d modulation (bounded time) using Kreiss symmetrizer techniques (Metivier-Z); extension to multi-d (Melinand-Rodrigues and Rodrigues-Wheeler) of the Kuramoto-Sivashinsky “interacting cells” picture of asymptotic behavior to doubly periodic cells; 1D damping estimates for Dressler waves of Saint Venant (Rodrigues-Z).
F. OPEN PROBLEMS: nonlinear stability of Dressler waves (!); bounded-time modulation in the presence of conservation laws; systematic study of Turing/Eckhaus conditions in biomorphology.