  In this article, we consider a class of the contact discontinuity for the
full compressible Euler equations, namely the entropy wave, where the velocity
is continuous across the interface while the density and the entropy can have
jumps. The nonlinear stability of entropy waves is a longstanding open problem
in multi-dimensional hyperbolic conservation laws. The rigorous treatments are
challenging due to the characteristic discontinuity nature of the problem
(G.-Q. Chen and Y.-G. Wang in \textit{Nonlinear partial differential
equations}, Volume 7 of \textit{Abel Symp.}(2012)). In this article, we
discover that the Taylor sign condition plays an essential role in the
nonlinear stability of entropy waves. By deriving the evolution equation of the
interface in the Eulerian coordinates, we relate the Taylor sign condition to
the hyperbolicity of this evolution equation, which reveals a stability
condition of the entropy wave. With the optimal regularity estimates of the
interface, we can derive the a priori estimates without loss of regularity.
