Large time behavior and pointwise estimates for compressible Euler equations with damping

Abstract

The Cauchy problem of the compressible Euler equations with damping in multi-dimensions is considered when the initial perturbation in H ³-norm is small. First, by using two new energy functionals together with Green’s function and iteration method, we improve the L ²-decay rate in Tan and Wang (2013) and Tan and Wu (2012) when $\left\| {(\rho _0 - \bar \rho ,m)} \right\|_{\dot B_{1,\infty }^{ - s} \times \dot B_{1,\infty }^{ - s + 1} }$ with s ∈ [0, 2] is bounded. In particular, it holds that the density converges to its equilibrium state at the rate $(1 + t)^{ - \tfrac{3} {4} - \tfrac{s} {2}}$ in L ²-norm and the momentum decays at the rate $(1 + t)^{ - \tfrac{5} {4} - \tfrac{s} {2}}$ in L ²-norm. Moreover, under a weaker and more general condition on the initial data, we show that the density and the momentum have different pointwise estimates in dimension d with d ⩾ 3 on both space variable x and time variable t as $\left| {D_x^\alpha (\rho - \bar \rho )} \right| \leqslant C(1 + t)^{ - \tfrac{d} {2} - \tfrac{{\left| \alpha \right|}} {2}} (1 + \tfrac{{\left| x \right|^2 }} {{1 + t}})^{ - r}$ with $r > \tfrac{d} {2}$ and $\left| {D_x^\alpha m} \right| \leqslant C(1 + t)^{ - \tfrac{d} {2} - \tfrac{{\left| \alpha \right| + 1}} {2}} (1 + \tfrac{{\left| x \right|^2 }} {{1 + t}})^{ - \tfrac{d} {2}}$ by a more elaborate analysis on the Green’s function. These results improve those in Wang and Yang (2001), where the density and the velocity (the momentum) have the same pointwise estimates.