Affiliations: Département de Mathématiques, UMR CNRS 8628, Bât. 425, Université Paris‐Sud, F‐91405 Orsay Cedex, France E‐mail: [email protected]‐psud.fr | Département de Mathématiques, Institut Galilée, UMR 7539 CNRS, Université Paris‐Nord, F‐93430 Villetaneuse, France E‐mail: [email protected] | CEA, Centre d'Études de Saclay, DM 25, F‐91191 Gif‐sur‐Yvette Cedex, France
Abstract: In this paper, we study the spectrum of the Rayleigh equation, which models the Rayleigh–Taylor instability in a fluid of variable density ρ(x), where ρ(x) goes to ρ− at −∞ and ρ+ at +∞: -h^{2}{\frac{\mathrm{d}}{\mathrm{d}x}}\bigl(\rho(x){\frac{\mathrm{d}u}{\mathrm{d}x}}\bigr)+(\rho(x)+\delta \rho'(x))u=0,\quad u\in L^{2}(\mathbb{R} ). The behavior of the smallest value δ(h) of δ>0 for which there exists a nontrivial solution u is investigated in terms of h for the two regimes h→0 (called the semi‐classical regime) and h→+∞. The so‐called growth rate of the Rayleigh–Taylor instability is $\sqrt{h/\delta(h)}$. When ρ−ρ−∈L2($\mathbb{R} $−) and ρ−ρ+∈L2($\mathbb{R} $+), we prove that δ(h)/h→(ρ++ρ−)/(ρ−−ρ+), as h→+∞, generalizing the result of Lord Rayleigh [23]. We then investigate the expansion of δ(h)/h in terms of 1/h and properties of ρ. In particular, we identify the number of terms n of the expansion in function of the behavior of ρ(x)−ρ± at ±∞, extending the results of Cherfils, Lafitte and Raviart [6].
