Well-posedness of Boussinesq Equation with Linear Restoring Force

Abstract

We establish a local well-posedness theory for the $n-$dimensional Boussinesq equation with a linear restoring force in a modified Sobolev spaces $H_ω^s$ and Bourgain-type spaces $X_ω^{s,b}$, which are adapted to its modified dispersion relation. The restoring force introduces a new phase function whose behavior in the low-frequencies is different from ones of the classical Boussinesq equation, that in turn, affects the associated energy functional. We prove bilinear and trilinear estimates for quadratic and cubic nonlinearities, respectively, which in turn imply local well-posedness under suitable conditions on $s$ and $ω$. To our knowledge, this is the first well-posedness result for this model.