Low regularity well-posedness of the modified and the generalized Korteweg-de Vries equations
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Date
27/11/2021Author
Chapouto, Andreia
Metadata
Abstract
In this thesis, we study the well-posedness of the modified and generalized Korteweg-de
Vries equations on the one-dimensional torus. We first consider the complex-valued modified
Korteweg-de Vries equation (mKdV). We observe that the momentum, a formally conserved
quantity of the equation, plays a crucial role in the well-posedness theory. In particular, following the method by Guo-Oh (2018), we show the ill-posedness of the complex-valued mKdV, in
the sense of non-existence of solutions, when the momentum is infinite. This result motivates the
introduction of a novel renormalization of the equation, which we propose as the correct model
to study at low regularity. Moreover, we establish the global well-posedness of the renormalized equation in the Fourier-Lebesgue spaces following two approaches: the Fourier restriction
norm method and the recent method by Deng-Nahmod-Yue (2020). Lastly, by imposing a new
notion of finite momentum at low regularity, we show the existence of distributional solutions
to the original equation, with the nonlinearity interpreted in a limiting sense. Regarding the
generalized Korteweg-de Vries equations (gKdV), we present a joint work with N. Kishimoto
(RIMS, Kyoto University) on the well-posedness with Gibbs initial data. To bypass the analytical ill-posedness of gKdV in the Sobolev support of the Gibbs measure, we prove local
well-posedness in the Fourier-Lebesgue spaces. Key ingredients are novel bilinear and trilinear
Strichartz estimates adapted to the Fourier-Lebesgue setting. Finally, by applying Bourgain’s
invariant measure argument (1994), we construct almost sure global-in-time dynamics and show
the invariance of the Gibbs measure for gKdV.