On the Cauchy problem for acoustic waves in hereditary fluids: decay properties and inviscid limits

Abstract

This manuscript considers the viscous/inviscid Moore-Gibson-Thompson (MGT) equations with memory of type I in the whole space $\mathbb{R}^n$. For one thing, associating with a new condition on initial data, we derive the optimal $L^2$ estimates and the optimal leading term of the acoustic velocity potential for large time, in which we analyze different contributions from viscous, thermally relaxing, as well as hereditary fluids on large time asymptotic behavior for the acoustic waves models. For another, via the multi-scale analysis and energy methods in the Fourier space, we demonstrate the $L^{\infty}$ inviscid limits in the sense of the diffusivity of sound tending to zero, which match our WKB expansion of the solution. Finally, we give a further application of our results on large time behavior for the nonlinear Jordan-MGT equation in viscous hereditary fluids.