article

Title: Local well-posedness of a dispersive Navier-Stokes system

Authors: C. David Levermore and Weiran Sun

Issue: Volume 60 (2011), Issue 2, 517-576

Abstract:

$We establish local well-posedness and smoothing results for the Cauchy problem of a degenerate dispersive Navier-Stokes (DNS) system that arises from kinetic theory. Under assumptions that the initial data have enough regularity and satisfy asymptotic flatness and nontrapping conditions, we show there exists a unique smooth solution for a finite time. Due to degeneracies in both dissipation and dispersion for the system, different components of the solution gain different regularity. The full system is decomposed accordingly into its strictly dispersive part and non-dispersive part. We apply the strategy of Kenig, Ponce, and Vega [C.E. Kenig, G. Ponce, and L. Vega, \textit{The Cauchy problem for quasi-linear Schr\"odinger equations}, Invent. Math. \textbf{158} (2004), no. 2, 343--388] to treat the strict dispersive part of the DNS system. Couplings of these two parts are then analyzed using normal form reductions and regularization from dispersion and dissipation.$