In this paper, we study two problems appearing in two-dimensional fluid mechanics in a constant gravity field \({g}=-g{e}_{\tilde x}\). These two problems—the Rayleigh convection problem and the ablation front problem—generalize the Rayleigh–Taylor model in compressible flows. The analysis of their stability relies on semiclassical techniques for the linearized system around a reference solution. We consider normal modes which are approximate solutions corresponding to large wave numbers associated with \(\tilde{y}\), and we discuss the existence or the non-existence of such normal modes. The results depend on the value of the dimensionless growth rate Γ compared with two relevant mathematical parameters, namely σ p (p standing for the model) and some effective semiclassical parameter h.
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Helffer, B., Lafitte, O. The Semiclassical Regime for Ablation Front Models. Arch Rational Mech Anal 183, 371–409 (2007). https://doi.org/10.1007/s00205-006-0006-8
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DOI: https://doi.org/10.1007/s00205-006-0006-8