Low regularity well-posedness of the modified and the generalized Korteweg-de Vries equations

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.