Existence and stability of curved multidimensional detonation fronts Nicola CostanzinoKris JenssenGregory LyngMark Williams 76L05multidimensional detonation frontsZND profilesspectral stabilitynonlinear stability The rigorous study of spectral stability for ZND detonations was begun by J.J. Erpenbeck in \emph{Stability of steady-state equilibrium detonations}, Physics of Fluids \textbf{5} (1962), 604--614. He used a normal mode analysis to define a stability function $V(\lambda, \eta)$, whose zeros in $\mbox{\upshape Re} \lambda > 0$ correspond to multidimensional perturbations of a steady planar profile that grow exponentially with time. In [J.J. Erpenbeck, \emph{Detonation stability for disturbances of small transverse wavelength}, Physics of Fluids \textbf{9} (1966), 1293-1306], he was able to prove that for large classes of steady ZND profiles, unstable zeros of $V$ always exist in the high frequency regime, \emph{even when} the von Neumann shock, regarded as a gas dynamical shock, is uniformly stable in the sense (later) defined by Majda; subsequent numerical work has shown that unstable zeros usually exist in the medium frequency regime as well.\par In this paper we begin a rigorous study of the implications for \emph{nonlinear stability} of the spectral instabilities just described. We show that in spite of the existence of unstable zeros of $V(\lambda, \eta)$, one can prove the finite (but arbitrarily long) time existence of slightly curved, nonsteady, multidimensional detonation fronts for ideal polytropic gases in both the ZND and Chapman-Jouguet models. In the ZND case we show that this nonlinear stability problem is actually governed by a different stability function, $\Delta_{\mathrm{ZND}}(\hat{\lambda}, \hat{\eta})$, which turns out to coincide with the high frequency limit of $V(\lambda, \eta)/ |\lambda, \eta|$ in $\mbox{\upshape Re} \hat{\lambda} > 0$. Moreover, the above nonlinear stability result for ideal polytropic gases holds more generally in any situation where $\Delta_{\mathrm{ZND}}(\hat{\lambda}, \hat{\eta})$ is bounded away from zero in $\mbox{\upshape Re} \hat{\lambda} > 0$. We also revisit the argument of Erpenbeck (1966) in order to simplify and complete some of the analysis in the proof of the main result there. Indiana University Mathematics Journal 2007 text pdf 10.1512/iumj.2007.56.2972 10.1512/iumj.2007.56.2972 en Indiana Univ. Math. J. 56 (2007) 1405 - 1462 state-of-the-art mathematics http://iumj.org/access/