For a bounded domain \(\Omega \subset {\mathbb {R}}^n\) with smooth boundary, we explicitly calculate the first two coefficients of the asymptotic expansion for the integral of the trace of the Stokes semigroup \(e^{-t S}\) as \(t\rightarrow 0^+\). These coefficients (i.e., spectral invariants) provide precise information for the volume of the domain \(\Omega \) and the surface area of the boundary \(\partial \Omega \) by the spectrum of the Stokes problem. As an application, we show that an n-dimensional ball is uniquely determined by its Stokes spectrum among all Euclidean bounded domains with smooth boundary.

对于具有光滑边界的有界域\（\ Omega \ subset {\ mathbb {R}} ^ n \），我们为Stokes半群\（e ^ {-t S} \）作为\（t \ rightarrow 0 ^ + \）。这些系数（即频谱不变性）通过斯托克斯问题的频谱为域\（\ Omega \）的体积和边界\（\ partial \ Omega \）的表面积提供了精确的信息。作为一个应用，我们表明，n维球由其Stokes谱唯一地确定于所有具有光滑边界的欧几里德有界域中。