The Cauchy problem for the nonlinear viscous Boussinesq equation in the $L^q$ framework

TL;DR

This paper investigates the linear and nonlinear viscous Boussinesq equation in the whole space, establishing $L^q$ well-posedness, solution estimates, and asymptotic behaviors using Fourier analysis, WKB method, and harmonic analysis techniques.

Contribution

It provides new $L^q$ well-posedness results and detailed analysis of solution properties for the nonlinear viscous Boussinesq equation.

Findings

$L^m-L^q$ estimates for solutions

Asymptotic profiles with small viscosity

Well-posedness for small initial data

Abstract

In this paper, we study the viscous Boussinesq equation in the whole space $\mathbb{R}^n$, which describes the propagation of small amplitude and long waves on the surface of water with viscous effects. Concerning the linearized Cauchy problem, some qualitative properties of solutions including $L^m-L^q$ estimates with $1\leqslant m\leqslant q\leqslant \infty$, inviscid limits and asymptotic profiles of solution with respect to the small viscosity are investigated by means of the Fourier analysis and the WKB method. For another, by applying some fractional order interpolations in the harmonic analysis, we derive the $L^q$ well-posedness and estimates for small data solutions to the nonlinear viscous Boussinesq equation under some conditions for the parameter of nonlinearity.