Long Time Behavior of Solutions to the 3D Compressible Euler Equations with Damping

The strong relaxation limit of the multidimensional isothermal Euler equations

“…We refer the reader to [8] for the existence of global smooth solutions in the isentropic case. We also refer the reader to [6] for the derivation of the porous media equation as the limit of the isentropic Euler equations in one space dimension.…”

The strong relaxation limit of the multidimensional Euler equations

“…An analogous global existence result in a bounded domain was proved in [15]. However, it is not obvious that energy estimates in [15,17] can be made uniform with respect to the relaxation time and our first main goal here is to obtain such a uniform existence result. Let us observe that the global existence of smooth solutions also follows from the abstract result of [21] but it is not more clear that this result can be made independent of the relaxation time ε.…”

“…In one space dimension, the derivation of the porous media equation as the asymptotic behavior of the isentropic Euler equations was proved in [13]. In three space dimensions with a fixed relaxation time ε, the global existence of smooth solutions to the isentropic Euler equations (1) with the γ-law was proved in [17] by using an equivalent reformulation of the problem to obtain effective energy estimates. An analogous global existence result in a bounded domain was proved in [15].…”

“…The definition of the energy functional is the same as in [2], but it is different from the one used in [17], where the sound speed was chosen as a new dependent variable in order to symmetrize the system (while here we use the entropy variables). Remark that since s > n/2 + 1, the Sobolev imbedding theorem yields…”

A Note on the Relaxation-Time Limit of the Isothermal Euler Equations

“…If fixed τ > 0, there are some efforts on the global existence of smooth solutions to the system (1.1)-(1.2) for the isentropic gas or the general hyperbolic system, the interested readers can refer to [4][5][6][7]. Now, we state main results as follows.…”