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Mathematics > Analysis of PDEs

arXiv:1807.07386 (math)
[Submitted on 19 Jul 2018 (v1), last revised 20 Jul 2018 (this version, v2)]

Title:Global Instability of the Multi-dimensional Plane Shocks for the isothermal flow

Authors:Ning-An Lai, Wei Xiang, Yi Zhou
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Abstract:In this paper, we are concerned with the long time behavior of the piecewise smooth solutions to the generalized Riemann problem governed by the compressible isothermal Euler equations in two and three dimensions. Non-existence result is established for the fan-shaped wave structure solution, including two shocks and one contact discontinuity and which is a perturbation of plane waves. Therefore, unlike the one-dimensional case, the multi-dimensional plane shocks are not stable globally. What is more, the sharp lifespan estimate is established which is the same as the lifespan estimate for the nonlinear wave equations in both two and three space dimensions.
Comments: 26 pages
Subjects: Analysis of PDEs (math.AP)
Cite as: arXiv:1807.07386 [math.AP]
  (or arXiv:1807.07386v2 [math.AP] for this version)
  https://doi.org/10.48550/arXiv.1807.07386

Submission history

From: Ning-An Lai [view email]
[v1] Thu, 19 Jul 2018 13:19:50 UTC (17 KB)
[v2] Fri, 20 Jul 2018 03:19:16 UTC (17 KB)
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