Asymptotic behaviors for Blackstock's model of thermoviscous flow

Abstract

We study a fundamental model in nonlinear acoustics, precisely, the general Blackstock's model (that is, without Becker's assumption) in the whole space R n . This model describes nonlinear acoustics in perfect gases under the irrotational flow. By means of the Fourier analysis we will derive L 2 estimates for the solution of the linear homogeneous problem and its derivatives. Then, we will apply these estimates to study three different topics: the optimality of the decay estimates in the case n 5 and the optimal growth rate for the L 2 -norm of the solution for n = 3, 4; the singular limit problem in determining the first-and second-order profiles for the solution of the linear Blackstock's model with respect to the small thermal diffusivity; the proof of the existence of global (in time) small data Sobolev solutions with suitable regularity for a nonlinear Blackstock's model.