Lp convergence rates of planar waves for multi-dimensional Euler equations with damping

In this paper, the Lp convergence rates of planar diffusion waves for multi-dimensional Euler equations with damping are considered. The analysis relies on a newly introduced frequency decomposition and Green function based energy method. It is a combination of the Lp estimate on the low frequency component by using an approximate Green function and L2 estimate on the high frequency component through the energy method. By noticing that the low frequency component in the approximate Green function has the algebraic decay which governs the large time behavior, while the high frequency component has the exponential decay but with singularity, their combination leads to a global algebraic decay estimate. To use the decay property only of the low frequency component in the approximate Green function avoids the singularity in the high frequency component so that it simplifies and improves the previous works on this system. This new approach of the combination of the Green function and energy method through the frequency decomposition can also be applied to the hyperbolic–parabolic systems satisfying the Kawashima condition, and also the systems whose derivatives of the coefficients have suitable time decay properties.