dorsal/arxiv
View SchemaGlobal Existence for General Systems of Isentropic Gas Dynamics via a Weighted Pressure Perturbation Approach
| Authors | Kewang Chen |
|---|---|
| Categories | |
| ArXiv ID | 2601.08157vv2 |
| URL | https://arxiv.org/abs/2601.08157 |
| License | http://creativecommons.org/licenses/by/4.0/ |
Abstract
This paper establishes the global existence of bounded entropy solutions to the one-dimensional system of isentropic gas dynamics with general pressure laws, allowing for the presence of vacuum states. We introduce a novel regularization scheme based on a \textbf{weighted pressure perturbation} designed to enforce acoustic degeneracy at a cut-off density $\rho=\delta$. A distinguishing feature of our approach is the \textbf{strict preservation of the mass conservation law}, in contrast to previous regularization methods that require modifying the continuity equation. We prove that the perturbed pressure $\tilde{P}$ exactly inherits the convexity structure of the original equation of state (satisfying $2\tilde{P}'+\rho\tilde{P}'' = 2P'+\rho P''$), thereby ensuring the strict convexity of the mechanical entropy without auxiliary transformations. This structural preservation allows for the direct application of the Chueh-Conley-Smoller theory to establish uniform $L^\infty$ bounds via invariant regions. The convergence to a global weak solution in the vanishing regularization limit is finally proved using the method of compensated compactness and a singular analysis of the entropy equation near the vacuum.
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"abstract": "This paper establishes the global existence of bounded entropy solutions to the one-dimensional system of isentropic gas dynamics with general pressure laws, allowing for the presence of vacuum states. We introduce a novel regularization scheme based on a \\textbf{weighted pressure perturbation} designed to enforce acoustic degeneracy at a cut-off density $\\rho=\\delta$. A distinguishing feature of our approach is the \\textbf{strict preservation of the mass conservation law}, in contrast to previous regularization methods that require modifying the continuity equation. We prove that the perturbed pressure $\\tilde{P}$ exactly inherits the convexity structure of the original equation of state (satisfying $2\\tilde{P}\u0027+\\rho\\tilde{P}\u0027\u0027 = 2P\u0027+\\rho P\u0027\u0027$), thereby ensuring the strict convexity of the mechanical entropy without auxiliary transformations. This structural preservation allows for the direct application of the Chueh-Conley-Smoller theory to establish uniform $L^\\infty$ bounds via invariant regions. The convergence to a global weak solution in the vanishing regularization limit is finally proved using the method of compensated compactness and a singular analysis of the entropy equation near the vacuum.",
"arxiv_id": "2601.08157",
"authors": [
"Kewang Chen"
],
"categories": [
"math.AP",
"math-ph",
"math.MP"
],
"license": "http://creativecommons.org/licenses/by/4.0/",
"title": "Global Existence for General Systems of Isentropic Gas Dynamics via a Weighted Pressure Perturbation Approach",
"url": "https://arxiv.org/abs/2601.08157",
"version": "v2"
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