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Differential geometry. Global analysis --- 51 --- Mathematics --- 51 Mathematics --- Differentiaalvergelijkingen [Hyperbolische ] --- Differential equations [Hyperbolic] --- Equations différentielles hyperboliques --- Fourier-Bros-Iagolnitzer transformations --- Microlocal analysis --- Opérateurs pseudo-différentiels --- Pseudo-differentiale operatoren --- Pseudodifferential operators --- Differential equations, Hyperbolic. --- Microlocal analysis. --- Fourier-Bros-Iagolnitzer transformations. --- Pseudodifferential operators. --- Équations différentielles hyperboliques. --- Analyse microlocale. --- Opérateurs pseudo-différentiels. --- Équations différentielles hyperboliques. --- Opérateurs pseudo-différentiels.

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Normal forms (Mathematics) --- Perturbation (Mathematics) --- Wave equation --- Numerical solutions.

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Differential equations, Partial. --- Pseudodifferential operators. --- Schrödinger equation. --- Équations aux dérivées partielles. --- Opérateurs pseudo-différentiels --- Schrödinger, Équation de --- Equations aux derivees partielles non lineaires --- Equations aux derivees partielles non lineaires

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The goal of this monograph is to prove that any solution of the Cauchy problem for the capillary-gravity water waves equations, in one space dimension, with periodic, even in space, small and smooth enough initial data, is almost globally defined in time on Sobolev spaces, provided the gravity-capillarity parameters are taken outside an exceptional subset of zero measure. In contrast to the many results known for these equations on the real line, with decaying Cauchy data, one cannot make use of dispersive properties of the linear flow. Instead, a normal forms-based procedure is used, eliminating those contributions to the Sobolev energy that are of lower degree of homogeneity in the solution. Since the water waves equations form a quasi-linear system, the usual normal forms approaches would face the well-known problem of losses of derivatives in the unbounded transformations. To overcome this, after a paralinearization of the capillary-gravity water waves equations, we perform several paradifferential reductions to obtain a diagonal system with constant coefficient symbols, up to smoothing remainders. Then we start with a normal form procedure where the small divisors are compensated by the previous paradifferential regularization. The reversible structure of the water waves equations, and the fact that we seek solutions even in space, guarantees a key cancellation which prevents the growth of the Sobolev norms of the solutions.

Cauchy problem. --- Sobolev spaces. --- Spaces, Sobolev --- Function spaces --- Differential equations, Partial --- Differential equations, partial. --- Fourier analysis. --- Differentiable dynamical systems. --- Functional analysis. --- Partial Differential Equations. --- Fourier Analysis. --- Dynamical Systems and Ergodic Theory. --- Functional Analysis. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Analysis, Fourier --- Mathematical analysis --- Partial differential equations --- Partial differential equations. --- Dynamics. --- Ergodic theory. --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics)

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The goal of this monograph is to prove that any solution of the Cauchy problem for the capillary-gravity water waves equations, in one space dimension, with periodic, even in space, small and smooth enough initial data, is almost globally defined in time on Sobolev spaces, provided the gravity-capillarity parameters are taken outside an exceptional subset of zero measure. In contrast to the many results known for these equations on the real line, with decaying Cauchy data, one cannot make use of dispersive properties of the linear flow. Instead, a normal forms-based procedure is used, eliminating those contributions to the Sobolev energy that are of lower degree of homogeneity in the solution. Since the water waves equations form a quasi-linear system, the usual normal forms approaches would face the well-known problem of losses of derivatives in the unbounded transformations. To overcome this, after a paralinearization of the capillary-gravity water waves equations, we perform several paradifferential reductions to obtain a diagonal system with constant coefficient symbols, up to smoothing remainders. Then we start with a normal form procedure where the small divisors are compensated by the previous paradifferential regularization. The reversible structure of the water waves equations, and the fact that we seek solutions even in space, guarantees a key cancellation which prevents the growth of the Sobolev norms of the solutions.

Differential geometry. Global analysis --- Functional analysis --- Harmonic analysis. Fourier analysis --- Ergodic theory. Information theory --- Partial differential equations --- Differential equations --- Fourieranalyse --- differentiaalvergelijkingen --- differentiaal geometrie --- functies (wiskunde) --- informatietheorie