Stability of multi-dimensional patterns

1. Introduction
1.1 Definition of nonlinear (Lyapunov) stability
1.2 Duhamel-based approach towards nonlinear stability
Outline main steps on the basis of the nonlinear stability argument for fixed points in autonomous ODEs.
1.3 Spectral, linear and nonlinear stability
1.4 Further reading: other approaches
Main goal is to outline the general structure of a Duhamel-based nonlinear stability argument and thereby motivate the upcoming material. That is, first control the spectrum of the linearization about the pattern to obtain suitable estimates on the linearized solution operator (i.e. the semigroup generated by the linearization), then combine these linear estimates with bounds on the nonlinearity to close a nonlinear iteration argument.

Reference:
-- Kapitula, Todd; Promislow, Keith. Spectral and dynamical stability of nonlinear waves. Applied Mathematical Sciences, 185. Springer, New York, 2013.

Exercise:
-- Failure of Duhamel-based approach: nonlinear stability of fixed points in Hamiltonian ODEs.
Goal is to raise awareness of the fact that the Duhamel-based method is typically not applicable in conservative, non-dissipative settings (such as Hamiltonian systems). Yet, there is another variational approach available (due to Grillakis, Shatah and Strauss) which extends the method for fixed points in Hamiltonian ODEs to Hamiltonian PDEs with symmetries and has been employed to establish nonlinear stability of periodic patterns against co-periodic perturbations.

References:
-- M. Grillakis, J. Shatah, and W. Strauss. Stability theory of solitary waves in the presence of symmetry, I. J. Funct. Anal., 74:160–197, 1987.
-- M. Grillakis, J. Shatah, and W. Strauss. Stability theory of solitary waves in the presence of symmetry, II. J. Funct. Anal., 94:308–348, 1990.
-- Benzoni-Gavage, S.; Mietka, C.; Rodrigues, L. M. Co-periodic stability of periodic waves in some Hamiltonian PDEs. Nonlinearity 29 (2016), no. 11, 3241--3308

2. Spectral stability
2.1 Exponential dichotomies
2.2 Floquet theory
2.3 Evans function for periodic wave trains
2.4 Floquet-Bloch transform
2.5 Extension to multiple spatial dimensions
2.6 Further reading: spectra associated with nonperiodic waves (such as pulses and fronts)
Main goal is to characterize the spectrum of the linearization about a periodic pattern both against co-periodic and against localized perturbations.

References:
-- B. Sandstede. Stability of travelling waves. In: Handbook of Dynamical Systems II (B Fiedler, ed.). North-Holland (2002) 983-1055.
-- R.A. Gardner. On the structure of the spectra of periodic travelling waves. J. Math. Pures Appl. 72 (1993)
415–439.
-- B. Scarpellini. L2-perturbations of periodic equilibria of reaction-diffusion systems. Nonlinear Diff. Eqns. Appl. 1 (1994) 281–311

Exercises:
-- Spectrum of the Laplacian on a bounded domain with Dirichlet or periodic BCs and on an infinite domain (in one and multiple spatial dimensions)
Goal: raise awareness that spectrum depends on choice of underlying space of perturbations
-- Spectral instability of wave trains in scalar reaction-diffusion systems with Sturm-Liouville theory (if time permits)
-- Spectral (in)stability of plane waves in real Ginzburg-Landau equation (accessible since, due to gauge symmetry, reduction to constant-coefficient operator possible). Find Eckhaus spectral stability criterion. Show transverse (in)stability of associated planar stripe patterns.

3. Methods to obtain spectral stability
3.1 Center manifold reduction for Turing patterns
The spectral stability of the (bifurcating) equilibria in the reduced equation on the center manifold is inherited by the associated patterns in the original system (at least against co-periodic perturbations). Additional perturbative arguments are necessary to extend to spectral stability against localized perturbations.
3.2 Lyapunov-Schmidt reduction for Turing patterns
3.3 Extension to multiple spatial dimensions
Note that the material discussed in sections 3.1-3.3 builds upon the associated material on the existence of the (bifurcating) Turing patterns (and their expansions) discussed in the sections 1-2 of Gabriela’s part.
3.4 Further reading: approaches for far-from-equilibrium patterns
Spectral stability of far-from-equilibrium patterns can for instance be obtained by bifurcating from a nearby pulse solution or taking advantage of the slow-fast structure of the system.

References for 3.1:
-- Haragus, Mariana, and Gérard Iooss. Local bifurcations, center manifolds, and normal forms in infinite-dimensional dynamical systems. Vol. 3. London: Springer, 2011.
-- Delcey, Lucie; Haragus, Mariana. Periodic waves of the Lugiato-Lefever equation at the onset of Turing instability. Philos. Trans. Roy. Soc. A 376 (2018), no. 2117, 20170188, 21 pp.

References for 3.2:
-- Mielke, Alexander. Instability and stability of rolls in the Swift-Hohenberg equation. Comm. Math. Phys. 189 (1997), no. 3, 829--853.
-- Mielke, Alexander. "A new approach to sideband-instabilities using the principle of reduced instability." Nonlinear dynamics and pattern formation in the natural environment. Routledge, 2022. 206-222.
-- Sukhtayev, Alim, et al. "Diffusive stability of spatially periodic solutions of the Brusselator model." Communications in Mathematical Physics 358 (2018): 1-43.

References for 3.3:
-- Doelman, Arjen; Sandstede, Björn; Scheel, Arnd; Schneider, Guido. Propagation of hexagonal patterns near onset. European J. Appl. Math. 14 (2003), no. 1, 85—110
-- Schneider, Guido. Nonlinear stability of Taylor vortices in infinite cylinders. Arch. Rational Mech. Anal. 144 (1998), no. 2, 121—200
-- Bridges, Thomas J.; Mielke, Alexander. Instability of spatially-periodic states for a family of semilinear PDE's on an infinite strip. Math. Nachr. 179 (1996), 5--25.
-- Mielke, A. Mathematical analysis of sideband instabilities with application to Rayleigh-Bénard convection. J. Nonlinear Sci. 7 (1997), no. 1, 57--99.

References for 3.4:
-- B. Sandstede and A. Scheel. On the stability of periodic travelling waves with large spatial period. J. Differential Equations, 172(1):134–188, 2001
-- R. Gardner, Spectral analysis of long wavelength periodic waves and applications, J. Reine Angew. Math. 491 (1997), 149-181
-- B. de Rijk, A. Doelman, J.D.M. Rademacher. Spectra and stability of spatially periodic pulse patterns: Evans function factorization via Riccati transformation, SIAM J. Math. Anal. 48-1 (2016), pp. 61-121
-- B. de Rijk. Spectra and stability of spatially periodic pulse patterns II: the critical spectral curve. SIAM J. Math. Anal., 50(2):1958–2019, 2018

Exercise:
-- Spectral stability of periodic Turing patterns in the Brusselator model both against co-periodic and (if time permits) against localized perturbations
The existence (and associated expansions) of the Turing patterns is done in the exercise class of Gabriela


4. Linear stability
4.1 A short introduction to semigroup theory
4.2 Floquet-Bloch representation of semigroup (skip if short in time)
4.3 Inverse Laplace representation of semigroup and Green’s functions (skip if short in time)
4.4 Semigroup decomposition
4.5 High-frequency estimates and the Gearhart–Prüss theorem (skip if short in time)
The main goal is to explain how spectral bounds can be converted into estimates on the linearized solution operator (i.e. the semigroup generated by the linearization). If there is a lack of time one could restrict to co-periodic perturbations and the desired decomposition of the semigroup only requires defining a spectral projection onto the translational eigenmode. Moreover, one could restrict to parabolic problems, so that the linearization is sectorial and obeys a spectral mapping property (no need for Gearhart-Prüss).

References:

-- Kapitula, Todd; Promislow, Keith. Spectral and dynamical stability of nonlinear waves. Applied Mathematical Sciences, 185. Springer, New York, 2013.

-- Engel, Klaus-Jochen; Nagel, Rainer. One-parameter semigroups for linear evolution equations. Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000.

-- Margaret Beck's lecture notes on "Linear stability theory" @ the workshop "The stability of coherent structures and patterns" in June 2012.

-- M. A. Johnson and K. Zumbrun. Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction-diffusion equations. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 28(4):471–483, 2011.

-- S. Jung. Pointwise asymptotic behavior of modulated periodic reaction-diffusion waves. J. Differential Equations, 253(6):1807–1861, 2012

Exercises:

-- Example showing that the growth bound and the spectral bound of a C0-semigroup are in general not the same