Short Wave-Long Wave Interactions for Compressible Navier-Stokes Equations

We consider a model describing the short wave vs. long wave interactions where the short waves are governed by a nonlinear Schrodinger equation, the long waves are governed by the Navier–Stokes equations for a compressible isentropic fluid flow, and these two modes are coupled nonlinearly by interaction terms. Our model is motivated by the general approach introduced by Benney in [Stud. Appl. Math., 56 (1977), pp. 81–94] and follows the reasoning adopted for the several models proposed and analyzed by Dias, Figueira, and Frid in [Arch. Ration. Mech. Anal., 196 (2010), pp. 981–1010]. For the model proposed here we address the problem of the global existence and uniqueness of the solution in the one-dimensional case. We also discuss the problem of the convergence of the sequence of solutions when the viscosity and the interaction coefficient vanish, applying in this part a recent compactness framework developed by Chen and Perepelitsa [Comm. Pure Appl. Math., 63 (2010), pp. 1469–1504].

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