The geometric invariants for the spectrum of the Stokes operator

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