On the well-posedness of higher order viscous Burgers' equations

We consider higher order viscous Burgers' equations with generalized nonlinearity and study the associated initial value problems for given data in the $L^2$-based Sobolev spaces. We introduce appropriate time weighted spaces to derive multilinear estimates and use them in the contraction mapping principle argument to prove local well-posedness for data with Sobolev regularity below $L^2$. We also prove ill-posedness for this type of models and show that the local well-posedness results are sharp in some particular cases viz., when the orders of dissipation $p$, and nonlinearity $k+1$, satisfy a relation $p=2k+1$.Comment: Withdrawn due to technical reaso

On KP-II type equations on cylinders

In this article we study the generalized dispersion version of the Kadomtsev-Petviashvili II equation, on \T \times \R and \T \times \R^2. We start by proving bilinear Strichartz type estimates, dependent only on the dimension of the domain but not on the dispersion. Their analogues in terms of Bourgain spaces are then used as the main tool for the proof of bilinear estimates of the nonlinear terms of the equation and consequently of local well-posedness for the Cauchy problem.Comment: 32 page