GLOBAL ANALYSIS OF WAVELET METHODS FOR EULER’S EQUATION

GLOBAL ANALYSIS OF WAVELET METHODS FOR EULER’S EQUATION Wayne Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543 Email matwml@nus.edu.sg Tel (65) 874-2749

RIGID BODIES Euler’s equation for their inertial motion angular velocity in the body inertia operator (from mass distribution) Theoria et ad motus corporum solidorum seu rigodorum ex primiis nostrae cognitionis principiis stbilita onmes motus qui inhuiusmodi corpora cadere possunt accommodata, Memoirs de l'Acad'emie des Sciences Berlin, 1765.

IDEAL FLUIDS Euler’s equation for their inertial motion velocity in space pressure outward normal of domain Commentationes mechanicae ad theoriam corporum fluidorum pertinentes, M'emoirs de l'Acad'emie des Sciences Berlin, 1765.

GEODESICS Moreau observed that these classical equations describe geodesics, on the Lie groups that parameterize their configurations, with respect to the left, right invariant Riemannian metric determined by the inertia operator (determined from kinetic energy) on the associated Lie algebra Une method de cinematique fonctionnelle en hydrodynamique, C. R. Acad. Sci. Paris 249(1959), 2156-2158

EULER’S EQUATION ON LIE GROUPS Arnold derived Euler’s equation that describe geodesics on Lie groups with respect to left, right invariant Riemannian metrics Mathematical Methods of Classical Mechanics, Springer, New York, 1978

GLOBAL ANALYSIS based on this geometric formulation provides a powerful tool for studying fluid dynamics Arnold used it to explain sensitivity to initial conditions in terms of curvature Ebin, Marsden, and Shkoller used it to derive existence, uniqueness and regularity results for both Euler’s and Navier-Stokes equations These ideas are fundamental for the study of a large class of nonlinear partial differential equations and have developed into the extensive field of topological hydrodynamics

REPRESENTATIONS Lie group Lie algebra dual For define the adjoint and coadjoint representations

WEAK FORMULATION The inertia operator is self-adjoint and positive definite and defines a bilinear form Then iff in this case the energy is constant

LAGRANGIAN FORMULATION A trajectory is a geodesic if and only if the associated momentum satisfies where for a left, right invariant Riemannian metric is the angular velocity in the body, in space The momentum lies within a coadjoint orbit which has a sympletic structure and thus even dimension

CARTAN-KILLING OPERATOR Define the Cartan-Killing operator B is self-adjoint and satisfies B is nonsingular iff G is semisimple (Cartan) B is positive semidefinite iff G is compact (Weyl)

VORTICITY EQUATION Consider a trajectory and associated Then satisfies the vorticity equation iff in this case the enstrophy is constant

VORTICITY FORMULATION Define the Greens operator satisfies the vorticity equation If then satisfies Euler’s equation If is nonsingular then the converse holds then is a stationary point If iff

CANONICAL FOURIER BASIS There exist a basis for and such that and If then

CANONICAL REPRESENTATION Structure Constants defined by yield vorticity equation

SPARSER REPRESENTATIONS ??? Symmetric forms (each k) can be diagonalized Can they be simultaneously sparsified? Existence of higher order invariants suggests so For any representation are constant; furthermore, Ado’s theorem ensures the existence of faithful finite dimensional representations

IDEAL FLUID FLOW IN Poisson bracket (commutator) The weak form of Euler’s equation provides the Faedo-Galerkin approximation method

STREAM FUNCTION Then (orthogonal coordinates The Green’s operator has convolution kernel is constant along particles in the flow, therefore the moments are invariant

IDEAL FLUID FLOW IN Identified with ideal flows in that are periodic with respect to the subgroup with average value zero, for the spectral basis of the complexified stream function Lie algebra

FAIRLIER, FLETCHER AND ZACHOS for odd defined the map

ZEITLIN used the approximation to approximate flow on by flows on

WAVELET BASES Neither the canonical Fourier basis nor the canonical sparse matrix basis provides a sparse representation of Euler’s equation on SU(n) • Wavelet vorticity bases provide nearly sparse • representations for Euler’s equations because • Green’s operator is Calderon-Zygmund • (ii) Poisson bracket is exponentially localized Wavelet bases provide simple approximations for invariant moments and energy We are using wavelet bases to study Okubo-Weiss criteria for two-dimensional turbulence

GLOBAL ANALYSIS OF WAVELET METHODS FOR EULER’S EQUATION