Abstract
In this paper, we establish a priori estimates for three-dimensional compressible Euler equations with the moving physical vacuum boundary, the $\gamma$-gas law equation of state for $\gamma=2$ and the general initial density $\rho_0 \in H^5$. Because of the degeneracy of the initial density, we investigate the estimates of the horizontal spatial and time derivatives and then obtain the estimates of the normal or full derivatives through the elliptic-type estimates. We derive a mixed space-time interpolation inequality which plays a vital role in our energy estimates and obtain some extra estimates for the space-time derivatives of the velocity in $L^3$.
Highlights
In recent years, the motion of physical vacuum in compressible fluids has been received much attention due to its great physical importance and mathematical challenges
It is the physical vacuum that makes the study of free boundary problems of compressible fluids challenging and very interesting, because standard methods of symmetric hyperbolic systems can not be applied directly
We introduce the following rth order energy function r
Summary
The motion of physical vacuum in compressible fluids has been received much attention due to its great physical importance and mathematical challenges (cf. [14,15,16,17,18, 21]). It is the physical vacuum that makes the study of free boundary problems of compressible fluids challenging and very interesting, because standard methods of symmetric hyperbolic systems (cf [11]) can not be applied directly. Weighted estimates show that this wave equation loses derivatives with respect to the uniformly hyperbolic non-degenerate case of a compressible liquid, wherein the density takes the value of a strictly positive constant on the moving boundary [3]. Since η3 = x3 = 0 on the fixed boundary Γ0, according to (1.9), the components a31 = a32 = 0 on Γ0, and v3 = 0 on Γ0 due to v · (0, 0, −1) = 0 where (0, 0, −1) is the outward unit normal vector to Γ0, the Lagrangian version of (1.4) can be written in the fixed reference domain Ω as ft + f Aji vi,j = 0 f vti
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