Niveau: Supérieur, Doctorat, Bac+8
Standing Waves On An Infinitely Deep Perfect Fluid Under Gravity G. Iooss? P. I. Plotnikov† J. F. Toland ‡ Abstract The existence of two-dimensional standing waves on the surface of an infinitely deep perfect fluid under gravity is established. When formulated as a second order equation for a real-valued function w on the 2-torus and a positive parameter µ, the problem is fully nonlinear (the highest order x-derivative appears in the nonlinear term but not in the linearization at 0) and completely resonant (there are infinitely many linearly independent eigenmodes of the linearization at 0 for all rational values of the parameter µ). Moreover, for any prescribed order of accuracy there exists an explicit approximate solution of the nonlinear problem in the form of a trigonometric polynomial. Using a Nash-Moser method to seek solutions of the nonlinear problem as perturbations of the approximate solutions, the existence question can be reduced to one of estimating the inverses of linearized operators at non- zero points. After changing coordinates these operators become first order and non-local in space and second order in time. After further changes of variables the main parts become diagonal with constant coefficients and the remainder is regularizing, or quasi-one-dimensional in the sense of [22]. The operator can then be inverted for two reasons. First, the explicit formula for the approximate solution means that, restricted to the infinite-dimensional kernel of the linearization at zero, the inverse exists and can be estimated.
- small divisors
- standing waves
- nash-moser theorem
- prescribed order
- linear theory
- dimensional motion
- then linear eigenvalues
- face determined