Tesi di Laurea Magistrale
Bilinear Estimates in Bourgain Spaces with
Applications to the periodic KdV and Ostrovsky
equation
Candidato: Relatore:
Lars Eric Hientzsch Prof. Vladimir Georgiev
Abstract
The present work treats the Cauchy Problem for the nonlinear periodic (in space variable) Korteweg–de Vries (KdV) equation and the periodic Os-trovsky equation for initial data in Sobolev spaces with low regularity. These equations are model equations for the propagation of nonlinear dispersive waves and arise e.g. in the theory of water waves. Our concern is to study the well-posedness results for these problems. This means, we have to show that for given initial data there exists a unique solution whose initial regularity persists and that depends continuously on the initial data.
For this purpose, we apply the classical Fourier restriction method intro-duced by Bourgain. This approach reduces the well-posedness problem to multilinear estimates in suitable function spaces, the so called Xs,b spaces,
also known as Bourgain or Fourier restriction spaces.
Well-posedness for KdV
First we study the Cauchy Problem for the periodic KdV equation on the domain T = R/2πZ,
∂tu + ∂x3u + u∂xu = 0. (1)
We are concerned with the following problems:
1. local well-posedness for (1) in the L2-based Sobolev spaces Hs with s ≥ −12,
2. global well-posedness for (1) at the same regularity.
The key point to solving the first problem is to use a bilinear estimate of the type k∂x(f g)kXs,b−1 τ =k3 ≤ ckf kXs,b τ =k3 kgkXs,b τ =k3 , (2)
for f, g ∈ Xs,b, s ≥ −12 and appropriate c > 0 and b ∈ R. This estimate can be interpreted as smoothing effect. This approach is of particular interest when dealing with large initial data, provided we can improve (2) replacing c by cTδ where [0, T ] is the solution time existence interval and δ > 0. In
2
the case of periodic initial data, there is no longer a gain of a factor T in the bilinear estimates that forces us to use a rescaling argument in the large data theory. Since the rescaling dilates the circle, this changes the period of the initial data and causes a technical issue.
Regarding the second problem, J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao showed sharp global well-posedness in Hs(T) for s ≥ −12. To that end, they introduced the ‘I -method’ that allows them to construct almost conserved quantities that are sufficient to iterate the local result to a global one.
Local Well-posedness for Ostrovsky
Finally, we address the local well-posedness of the Cauchy problem for the periodic Ostrovsky equation
∂tu − β∂x3u − γ∂x−1u + uxu = 0. (3)
This equation differs from the KdV equation in a nonlocal term and was derived as model for weakly nonlinear long waves in a rotating frame of reference. The parameter γ measures the effect of rotation corresponding to the Coriolis force, whereas the parameter β determines the type of dis-persion. For γ = 0 the equation reduces to the KdV equation (1) and we consider it as perturbation of (1). Thus, it seems natural to ask for analo-gous properties of the solution for the Cauchy Problem associated to (3). In the nonperiodic setting, local and global well-posedness have been discussed in various papers.
Our main result is a bilinear estimate of the type (2) in the Xs,b spaces defined according to the symbol of the Ostrovsky equation. We obtain that the periodic Cauchy Problem for (3) is locally well-posed in Hs for s ≥ −1 2
if the Hs norm of the initial data is sufficiently small and if the initial data has 0-mean.
Dealing with the periodic case, there is a technical issue similar to the KdV case. Furthermore, the lack of a rescaling property for (3) is an essential obstacle in the large data theory. A similar result including the existence of a factor Tδ for small δ > 0 in the bilinear estimate has been announced, where some technical details in the proof are unclear (at least to the author of this master’s thesis).
We conclude by discussing the difficulties occuring if one wants to extend our local result to a global result. It is necessary to overcome the difficulties arising from the lack of rescaling for (3) before proceeding to implement the ‘I -method’ for the periodic Ostrovsky equation and address the global Cauchy Problem with initial data in Hs for s < 0.
