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In the present manuscript, we formulate and prove rigorously, necessary and sufficient conditions for all kinds of separation of variables that a solution of the irrotational Stokes equation may exhibit, in any orthogonal axisymmetric system, namely: simple separation and R-separation. These conditions may serve as a road map for obtaining the corresponding solution space of the irrotational Stokes equation, in any orthogonal axisymmetric coordinate system. Additionally, we investigate how the inversion of the coordinate system, with respect to a sphere, affects the type of separation. Specifically, we prove that if the irrotational Stokes equation separates variables in an axisymmetric coordinate system, then it R-separates variables in the corresponding inverted coordinate system. This is a quite useful outcome since it allows the derivation of solutions for a problem, from the knowledge of the solution of the same problem in the inverted geometry and vice-versa. Furthermore, as an illustration, we derive the eigenfunctions of the irrotational Stokes equation governing the flow past oblate spheroid particles and inverted oblate spheroidal particles.

For Stokes flow in non spherical geometries, when separation of variables fails to derive closed form solutions in a simple product form, analytical solutions can still be obtained in an almost separable form, namely in semiseparable form, R-separable form or R-semiseparable form. Assuming a stream function Ψ, the axisymmetric viscous Stokes flow is governed by the fourth order elliptic partial differential equation E4Ψ = 0 where E4 = E2oE2 and E2 is the irrotational Stokes operator. Depending on the geometry of the problem, the general solution is given in one of the above separable forms, as series expansions of particular combinations of eigenfunctions that belong to the kernel of the operator E2. In the present manuscript, we provide a review of the methodology and the general solutions of the Stokes equations, for almost any axisymmetric system of coordinates, which are given in a ready to use form. Furthermore, we present necessary and sufficient conditions that are serving as criterion for identifying the kind of the separation the Stokes equation admits, in each axisymmetric coordinate system. Additionally, as an illustration of the usefulness of the obtained analytical solutions, we demonstrate indicatively their application to particular Boundary Value Problems that model medical problems.

In this paper, we establish spectral inequalities on measurable sets of positive Lebesgue measure for the Stokes operator, as well as an observability inequalities on space-time measurable sets of positive measure for non-stationary Stokes system. Furthermore, we provide their applications in the theory of shape optimization and time optimal control problems.

The algebraic decay for the total kinetic energy of weak solutions of the viscous Boussinesq system in an exterior domain is given by means of the spectral analysis method of the Stokes operator, which gives a positive answer to the L 2 -decay question in [11] .

In this paper, we first investigate some basic properties of the linear non-local Stokes operator, such as the structure of the Oseen tensor , maximal regularity and so on. By the special structure of the spherically symmetric -stable Levy processes, we can decompose as where is smooth for with decaying as . Combining this decomposition with the Fourier multipliers theorem, we establish the maximal regularity theory of the non-local Stokes operator. Secondly, as an important application to the decomposition (0.1), we derive the precise asymptotic behavior at infinity for the small self-similar solution of the Navier–Stokes equations with a fractional diffusion.

We study in this paper the $$L^2$$ -decay to the solutions of the Navier–Stokes system with Navier-type boundary conditions in a domain $$\Omega $$ not necessarily simply connected. Since under this condition the Stokes operator has a non-trivial kernel we consider the solution lying in the orthogonal of that kernel. We also compare the decay rate of a weak solution to the Navier–Stokes problem with the solution to the linear problem emanating from the same initial data.

In this paper, we consider some two phase problems of compressible and incompressible viscous fluids’ flow without surface tension under the assumption that the initial domain is a uniform Wq2−1/q domain in RN (N≥2). We prove the local in the time unique existence theorem for our problem in the Lp in time and Lq in space framework with 2&lt, p&lt, ∞ and N&lt, q&lt, ∞ under our assumption. In our proof, we first transform an unknown time-dependent domain into the initial domain by using the Lagrangian transformation. Secondly, we solve the problem by the contraction mapping theorem with the maximal Lp-Lq regularity of the generalized Stokes operator for the compressible and incompressible viscous fluids’ flow with the free boundary condition. The key step of our proof is to prove the existence of an R-bounded solution operator to resolve the corresponding linearized problem. The Weis operator-valued Fourier multiplier theorem with R-boundedness implies the generation of a continuous analytic semigroup and the maximal Lp-Lq regularity theorem.

With the help of Stokes operator S2 squeezed state (also called polarization squeezed state (PSS)) of 795-nm light, rubidium-87 (87Rb) atomic magnetometer based on Faraday rotation has been implemented and characterized.The PSS of Stokes operator S2 of 795-nm light has been prepared by means of coherently combining the polarization coherent state (PCS) of a linearly p-polarized bright 795-nm light beam and a linearly s-polarized squeezed vacuum state (SVS) generated by a 397.5-nm ultraviolet laser pumped sub-threshold optical parametric oscillator (OPO) with a PPKTP bulk crystal inside the OPO cavity.PSS with a squeezing level of -3.7 has been achieved around the analysis frequency of 10 kHz. At different transitions of D1 line, various frequency detuning, and reasonable atomic vapor cells temperature, Faraday rotation has been measured and compared.To decrease absorption (scattering) losses and the back-action from atomic spin noise to the probe beams polarization noise for maintaining the quantum properties of PSS of Stokes operator S2 of 795-nm light, we had to run our magnetometer with 87Rb vapor cells temperature below 60, at which the PSS was almost destroyed.The sensitivities of magnetic field measurement were characterized via measuring signal-to-noise ratio of the alternating current (AC) calibrated magnetic field signal with a balanced polarimeter. Under the conditions of the atomic number density of 5.8*1010 /cm3 and the probe beam with a detuning of - 400 MHz relative to the 5S1/2 (Fg=2) - 5P1/2 (Fe=1) transition of 87Rb D1 line, a typical sensitivity of 19.5 pT/Hz1/2 has been achieved employing PSS of Stokes operator S2 as the probe, compared with a sensitivity of 28.3 pT/Hz1/2 using PCS as the probe.We preliminarily demonstrated that the quantum-enhanced sensitivity in a Faraday-rotation-based 87Rb atomic magnetometer with the help of PSS of 795-nm light., Comment: 8 pages, 5 figures J. Opt. 23 (2021) 085202

We study global well-posedness for the inhomogeneous Navier–Stokes equations on $${\mathbb {R}}^n$$ , $$n\ge 2$$ , with initial velocity in endpoint critical Besov spaces $$B^{-1+n/q}_{q,\infty }({\mathbb {R}}^n)$$ , $$n\le qq$$ .

A linear operator on a Hilbert space may be approximated with finite matrices by choosing an orthonormal basis of the Hilbert space. In this paper, we found an approximation of the S-numerical range of bounded and unbounded operator matrices by variation methods. Applications to Hain-L st operator and Stokes operator are given.

We derive basic properties of Triebel-Lizorkin-Lorentz spaces important in the treatment of PDE. For instance, we prove Triebel-Lizorkin-Lorentz spaces to be of class $\mathcal{HT}$, to have property $(\alpha)$, and to admit a multiplier result of Mikhlin type. By utilizing these properties we prove the Laplace and the Stokes operator to admit a bounded $H^\infty$-calculus. This is finally applied to derive local strong well-posedness for the Navier-Stokes equations on corresponding Triebel-Lizorkin-Lorentz ground spaces.

This paper concerns Hodge-Dirac operators D = d + δ acting in L p (Ω, Λ) where Ω is a bounded open subset of R n satisfying some kind of Lipschitz condition, Λ is the exterior algebra of R n , d is the exterior derivative acting on the de Rham complex of differential forms on Ω, and δ is the interior derivative with tangential boundary conditions. In L 2 (Ω, Λ), δ = d * and D is self-adjoint, thus having bounded resolvents (I + itD) −1 t∈R as well as a bounded functional calculus in L 2 (Ω, Λ). We investigate the range of values p H < p < p H about p = 2 for which D has bounded resolvents and a bounded holomorphic functional calculus in L p (Ω, Λ). On domains which we call very weakly Lipschitz, we show that this is the same range of values as for which L p (Ω, Λ) has a Hodge (or Helmholz) decomposition, being an open interval that includes 2. The Hodge-Laplacian ∆ is the square of the Hodge-Dirac operator, i.e. −∆ = D 2 , so it also has a bounded functional calculus in L p (Ω, Λ) when p H < p < p H. But the Stokes operator with Hodge boundary conditions, which is the restriction of −∆ to the subspace of divergence free vector fields in L p (Ω, Λ 1) with tangential boundary conditions , has a bounded holomorphic functional calculus for further values of p, namely for max{1, p H S } < p < p H where p H S is the Sobolev exponent below p H , given by 1/p H S = 1/p H + 1/n, so that p H S < 2n/(n + 2). In 3 dimensions, p H S < 6/5. We show also that for bounded strongly Lipschitz domains Ω, p H < 2n/(n + 1) < 2n/(n − 1) < p H , in agreement with the known results that p H < 4/3 < 4 < p H in dimension 2, and p H < 3/2 < 3 < p H in dimension 3. In both dimensions 2 and 3, p H S < 1 , implying that the Stokes operator has a bounded functional calculus in L p (Ω, Λ 1) when Ω is strongly Lipschitz and 1 < p < p H .

Consider the (Navier–) Stokes system on an exterior domain with moving boundary and Dirichlet boundary conditions. In 2003 Saal proved that the Stokes operator in a domain with moving boundary has the property of maximal regularity provided that the operator is invertible. Hence his result can be applied if the domain is bounded or by adding a shift to the Stokes operator if the domain is unbounded or the time interval is finite. In this paper, we will generalize his result to a result global in time if the reference domain is an exterior domain. Finally, we will apply this result to the Navier–Stokes equations to obtain a global in time existence theorem for small data.

The problem of time-periodic solutions is considered for a system of partial differential equations with the Stokes operator in the main part with coefficients rapidly oscillating by time in the case of multiple degeneration of the averaged stationary operator. The results concerning the existence and uniqueness of these solutions are proved and their asymptotic expansions are constructed and justified.