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

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