Stability and asymptotic analysis for instationary gas transport via relative energy estimates. (English) Zbl 1521.35136
This paper first studies (an equivalent reformulation of) the one-dimensional compressible barotropic Euler equations with gravitation and friction on a bounded interval, which models the gas flows in a pipe. Under the assumptions of non-degenerate pipe cross-section, no-vacuum, and no-concentration, the high-friction (i.e., parabolic) limit and the stability results with respect to coefficients and initial data are established.
The key ingredient of the proof is the estimates for the relative energy/entropy functional of an abstract evolutionary system (Equations (11) and (12)), which encompasses the above PDE model for pipe flow. It, in particular, only requires minimal regularity of pipe flow solutions.
Notably, such analyses can be naturally extended to study the gas flows in networks of pipes. See Section 5 of the paper.
The key ingredient of the proof is the estimates for the relative energy/entropy functional of an abstract evolutionary system (Equations (11) and (12)), which encompasses the above PDE model for pipe flow. It, in particular, only requires minimal regularity of pipe flow solutions.
Notably, such analyses can be naturally extended to study the gas flows in networks of pipes. See Section 5 of the paper.
Reviewer: Siran Li (Shanghai)
MSC:
| 35Q31 | Euler equations |
| 35Q49 | Transport equations |
| 76N15 | Gas dynamics (general theory) |
| 35B25 | Singular perturbations in context of PDEs |
| 35B35 | Stability in context of PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35L60 | First-order nonlinear hyperbolic equations |
| 35L65 | Hyperbolic conservation laws |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
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