Convergence Rate for Compressible Euler Equations with Damping and Vacuum

Abstract

We study the asymptotic behavior of L ∞ weak-entropy solutions to the compressible Euler equations with damping and vacuum. Previous works on this topic are mainly concerned with the case away from the vacuum and small initial data. In the present paper, we prove that the entropy-weak solution strongly converges to the similarity solution of the porous media equations in L p (R) (2 p < ∞) with decay rates. The initial data can contain vacuum and can be arbitrary large. A new approach is introduced to control the singularity near vacuum for the desired estimates.

Initial boundary value problem for compressible Euler equations with damping

“…We will see that an easy lemma plays an important role. It should be pointed out that the key approach in [13][14][15][16][17][18][19][20][21] is to compare the solution of (1.2)-(1.3) with the similarity solution of (1.4) via energy estimates. Unfortunately, the exponential decay rate cannot be achieved by this approach, due to the boundary effects.…”

“…In this direction, the readers are referred to [31], [13], [14] and [12] for existence of small smooth solutions; to [27], [4], and [6] for solutions in BV ; to [7] and [20] for L ∞ solutions. For large time behavior of solutions, we refer [13], [14], [33], [34] and [45] fore small smooth solutions; and we refer [19], [21], [39] and [47] for weak solutions. For initial boundary value problems, see [16], [28] and [35] for small smooth solutions.…”

Asymptotic behavior and strong convergence for hyperbolic systems of conservation laws with damping

“…Similarly, Huang and Pan in [4] proved this kind of stronger convergence in connection with the result of [8],…”

“…The idea in this paper is to obtain results similar to [15] and [4] for the general 2x2 strictly hyperbolic systems in 1 -D by coupling the convergence obtained in [11] by means of compensated compactness arguments with energy estimates. Although in [11] other cases are also treated, in this paper these limitations are essential because of the difficulties concerning the properties of the similarity solution for the limiting equations.…”

Large time behavior of Euler-Poisson system for semiconductor

“…Following the ideas introduced in [5] and [6], we will now explore the entropy dissipation. In view of the definition of η * and q * in (19), we substitute them into the entropy inequality (7) to obtain…”