The geometric invariants for the spectrum of the Stokes operator

For a bounded domain $$\Omega \subset {\mathbb {R}}^n$$ Ω ⊂ 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 $$e^{-t S}$$ e - t S as $$t\rightarrow 0^+$$ t → 0 + . These coefficients (i.e., spectral invariants) provide precise information for the volume of the domain $$\Omega $$ Ω and the surface area of the boundary $$\partial \Omega $$ ∂ Ω 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.

Time decay rates for the coupled modified Navier-Stokes and Maxwell equations on a half space

<abstract><p>This paper is concerned with time decay rates of the strong solutions of an incompressible the coupled modified Navier-Stokes and Maxwell equations in a half space $ \mathbb{R}^3_+ $. With the use of the spectral decomposition of the Stokes operator and $ L^p-L^q $ estimates developed by Borchers and Miyakawa ^{[<xref ref-type="bibr" rid="b2">2</xref>]}, we study the $ L^2 $-decay rate of strong solutions.</p></abstract>

On the K-th Power of Stokes Operator

Stokes operators, are well known partial differential operators of elliptic type, which are often used in Applied Mathematics. Stokes equation describes the irrotational, axisymmetric creeping flow and Stokes bi-stream equation denotes the rotational one, where Necessary and sufficient conditions for the separability and the R-separability of the equation have been proved recently. Moreover, the 0-eigenspace and the generalized 0-eigenspace of the operator have been derived in several coordinate systems. Specifically, the spherical coordinate system is employed in many problems taking into account that in many engineering applications, the solutions in spherical geometry seem to be adequate for solving a problem. In the present manuscript, it is shown that equation admits a solution of the form where are solutions of Stokes equation and r is the radial spherical variable. Additionally, we obtain the kernel of the k-th power of the Stokes operator, in the spherical geometry for every

Computing the q-Numerical Range of Differential Operators

A linear operator on a Hilbert space may be approximated with finite matrices by choosing an orthonormal basis of thez Hilbert space. In this paper, we establish an approximation of the q-numerical range of bounded and unbounnded operator matrices by variational methods. Application to Schrödinger operator, Stokes operator, and Hain-Lüst operator is given.