Buch, Englisch, Band 24, 269 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 429 g
Buch, Englisch, Band 24, 269 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 429 g
Reihe: Lecture Notes of the Unione Matematica Italiana
ISBN: 978-3-319-99485-7
Verlag: Springer International Publishing
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.
Zielgruppe
Research
Autoren/Hrsg.
Fachgebiete
- Mathematik | Informatik Mathematik Geometrie Dynamische Systeme
- Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
- Mathematik | Informatik Mathematik Mathematische Analysis Funktionalanalysis
- Mathematik | Informatik Mathematik Mathematische Analysis Harmonische Analysis, Fourier-Mathematik
Weitere Infos & Material
Introduction.- MainResult. - Paradifferential Calculus. - Complex Formulation of the Equation and Diagonalization of the Matrix Symbol. - Reduction to a Constant Coefficients Operator and Proof of the Main Theorem. - The Dirichlet–Neumann Paradifferential Problem. - Dirichlet–Neumann Operator and the Good Unknown. - Proof of Some Auxiliary Results.
