The Pointwise Estimates of Solutions for Euler Equations with Damping in Multi-Dimensions

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The Optimal Decay Estimates on the Framework of Besov Spaces for Generally Dissipative Systems (2014, background): “It has been shown by [23] and [28] that the damping term could prevent the development singularities and the Cauchy problem (6.”
Existence and asymptotic behavior of C1 solutions to the multi-dimensional compressible Euler equations with damping (2007, background): “vacuum, we can cite [4, 3, 5, 6, 15, 13] and their references.”
Long Time Behavior of Solutions to the 3D Compressible Euler Equations with Damping (2003, result): “If the initial data are small in an appropriate norm, then damping can prevent the development of singularities and the Cauchy problem has a unique global smooth solution which decays in the maximum norm to the background state at a rate of t ð3=2Þ. Similar results have been previously obtained by W”
Global existence and asymptotic behavior of the solutions to the three-dimensional bipolar Euler-Poisson systems (2012, background): “1) seems like the Euler equations with damping (see [19]) in some sense.”
Global Classical Solutions for Partially Dissipative Hyperbolic System of Balance Laws (2012, background): “Furthermore, it is proved that the solution in [32] has the L∞ convergence rate (1 + t)−3/2(d = 3) to the constant background state and the optimal Lp(1 < p ≤ ∞) convergence rate (1+t)−d/2(1−1/p) in general several dimensions [34], respectively.”