Stability of generic inflow wave patterns for the one-dimensional Navier–Stokes–Fourier system
For the one-dimensional Navier–Stokes–Fourier system on the half-line with inflow boundary data in the subsonic region, consider the composite wave consisting of a degenerate boundary layer, a rarefaction wave, a viscous contact wave, and a viscous shock. If the initial perturbation and the strengths of the boundary layer, viscous contact wave, and viscous shock are sufficiently small, while the rarefaction wave may have arbitrarily large strength, prove that the corresponding solution is nonlinearly stable and converges as to this composite wave, up to a time-dependent spatial shift of the viscous shock.
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Progress summary
A new paper claims to settle the stability question, including cases with a very large rarefaction wave, but the claim has not been independently verified.
The problem asks for nonlinear stability of a generic composite wave in the one-dimensional half-line inflow problem for the Navier–Stokes–Fourier system. Earlier work established important special cases, but not the full stated pattern.
Known results
- Matsumura and Nishihara, 1985: viscous-shock stability for isentropic compressible Navier–Stokes equations.
- Earlier work by Huang, Li, and Shi: stability of certain full-system inflow boundary layers when both end states are subsonic.
- 2009: stability of a boundary-layer, contact-wave, and rarefaction composite, allowing arbitrarily large rarefaction amplitude but requiring small other amplitudes.
- 2010: stability of a four-wave inflow superposition under suitably small wave strengths and initial perturbation.
August 2026 claimed resolution
A newly reported arXiv preprint, Long-time dynamics toward a generic composite wave for the inflow problem of the Navier--Stokes--Fourier system, claims to resolve the stated inflow stability problem even for large-amplitude rarefaction. No independent verification, gap analysis, or retraction was found.
arxiv.org · aimsciences.org · actams.apm.ac.cn · deepmind.google · preprints.org · academia.edu · quantamagazine.org · quantamagazine.org · arxiv.org · ar5iv.labs.arxiv.org · ar5iv.labs.arxiv.org · ar5iv.labs.arxiv.org · mathstodon.xyz · mathstodon.xyz · quantamagazine.org · quantamagazine.org · mathstodon.xyz · quantamagazine.org · quantamagazine.org · quantamagazine.org · quantamagazine.org · quantamagazine.org
Current status (as of August 2026): A preprint claims the stated stability result, including large rarefaction, but the claim remains unverified; the result is therefore not settled independently.
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Solutions 0
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