This textbook introduces the well-posedness theory for initial-value problems of nonlinear,
dispersive partial differential equations, with special focus on two key models, the Korteweg …

This book is the second edition of the book Lectures on nonlinear evolution equations. Initial
value problems [150] from 1992. Additionally, it now includes a new Chapter 13 on initial …

Proving local well-posedness for quasi-linear problems in partial differential equations
presents a number of difficulties, some of which are universal and others of which are more …

We prove strong nonlinear illposedness results for the generalized SQG equation
$$\partial_t\theta+\nabla^\perp\Gamma [\theta]\cdot\nabla\theta= 0$$ in any sufficiently …

We provide a unified viewpoint on two illposedness mechanisms for dispersive equations in
one spatial dimension, namely degenerate dispersion and (the failure of) the Takeuchi …

In part I of this project we examined low-regularity local well-posedness for generic
quasilinear Schrödinger equations with small data. This improved, in the small data regime …

We study the well-posedness of the initial value problem for fully nonlinear evolution
equations, where f may depend on up to the first three spatial derivatives of u. We make …

We present a novel method for establishing large data local well-posedness in low regularity
Sobolev spaces for general quasilinear Schrödinger equations with non-degenerate and …