On the well-posedness of higher order viscous Burgersʹ equations

We consider higher order viscous Burgersʹ 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 = 2 k + 1 .