scholarly journals The geometric invariants for the spectrum of the Stokes operator

2021 ◽  
Author(s):  
Genqian Liu
Keyword(s):  
Asymptotic Expansion ◽  
Surface Area ◽  
Smooth Boundary ◽  
Stokes Problem ◽  
Stokes Operator ◽  
Bounded Domains ◽  
Spectral Invariants ◽  
Precise Information ◽  
Geometric Invariants ◽  
Dimensional Ball

AbstractFor 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.

Convection in a horizontally heated sphere

2001 ◽  
Vol 438 ◽  
pp. 1-10 ◽  
Author(s):  
G. D. McBAIN
Keyword(s):  
Asymptotic Expansion ◽  
Body Force ◽  
Stokes Problem ◽  
Creeping Flow ◽  
Zeroth Order ◽  
First Order ◽  
Smooth Body ◽  
Spherical Fluid ◽  
Rayleigh Numbers ◽  
Spanwise Flow

Natural convection in horizontally heated spherical fluid-filled cavities is considered in the low Grashof number limit. The equations governing the asymptotic expansion are derived for all orders. At each order a Stokes problem must be solved for the momentum correction. The general solution of the Stokes problem in a sphere with arbitrary smooth body force is derived and used to obtain the zeroth-order (creeping) flow and the first-order corrections due to inertia and buoyancy. The solutions illustrate the two mechanisms adduced by Mallinson & de Vahl Davis (1973, 1977) for spanwise flow in horizontally heated cuboids. Further, the analytical derivations and expressions clarify these mechanisms and the conditions under which they do not operate. The inertia and buoyancy effects vanish with the Grashof and Rayleigh numbers, respectively, and both vanish if the flow is purely vertical, as in a very tall and narrow cuboid.

THE STOKES PROBLEM IN ℝn: AN APPROACH IN WEIGHTED SOBOLEV SPACES

1999 ◽  
Vol 09 (05) ◽  
pp. 723-754 ◽  
Author(s):  
F. ALLIOT ◽  
C. AMROUCHE
Keyword(s):  
Asymptotic Expansion ◽  
Sobolev Spaces ◽  
Vector Fields ◽  
Stokes Problem ◽  
Weighted Sobolev Spaces ◽  
Explicit Representation ◽  
Solenoidal Vector ◽  
Regularity Results

We prove some existence, uniqueness and regularity results for the solutions to the Stokes problem in ℝn, n≥2 in weighted Sobolev spaces [Formula: see text]. This framework enables us to characterise for which data the problem has solutions with prescribed decay or growth at infinity. Moreover, we obtain an explicit representation as well as an asymptotic expansion of the solution for non-smooth decaying data. We also establish the density of smooth solenoidal vector fields in the subspace of [Formula: see text] such that div v=0.

scholarly journals On the cubic NLS on 3D compact domains

2013 ◽  
Vol 13 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Fabrice Planchon
Keyword(s):  
Schrödinger Equation ◽  
Boundary Conditions ◽  
Bounded Domain ◽  
Schrodinger Equation ◽  
Dirichlet Boundary ◽  
Smooth Boundary ◽  
Dirichlet Boundary Conditions ◽  
Well Posedness ◽  
Bounded Domains ◽  
Bilinear Estimates

AbstractWe prove bilinear estimates for the Schrödinger equation on 3D domains, with Dirichlet boundary conditions. On non-trapping domains, they match the ${ \mathbb{R} }^{3} $ case, while on bounded domains they match the generic boundaryless manifold case. We obtain, as an application, global well-posedness for the defocusing cubic NLS for data in ${ H}_{0}^{s} (\Omega )$, $1\lt s\leq 3$, with $\Omega $ any bounded domain with smooth boundary.

scholarly journals Topological asymptotic formula for the 3D non-stationary Stokes problem and application

2020 ◽  
Vol Volume 32 - 2019 - 2020 ◽  
Author(s):  
Maatoug Hassine ◽  
Rakia Malek
Keyword(s):  
Asymptotic Expansion ◽  
Stokes System ◽  
Stokes Problem ◽  
Three Dimensional ◽  
Detection Algorithm ◽  
International Audience ◽  
Flow Domain ◽  
Topological Asymptotic Expansion ◽  
Small Obstacle ◽  
Theoretical Results

International audience This paper is concerned with a topological asymptotic expansion for a parabolic operator. We consider the three dimensional non-stationary Stokes system as a model problem and we derive a sensitivity analysis with respect to the creation of a small Dirich-let geometric perturbation. The established asymptotic expansion valid for a large class of shape functions. The proposed analysis is based on a preliminary estimate describing the velocity field perturbation caused by the presence of a small obstacle in the fluid flow domain. The obtained theoretical results are used to built a fast and accurate detection algorithm. Some numerical examples issued from a lake oxygenation problem show the efficiency of the proposed approach. Ce papier porte sur l'analyse de sensibilité topologique pour un opérateur parabolique. On considère le problème de Stokes instationnaire comme un exemple de modèle et on donne une étude de sensibilité décrivant le comportement asymptotique de l'opérateur relativement à une petite perturbation géométrique du domaine. L'analyse présentée est basée sur une estimation du champ de vitesse calculée dans le domaine perturbé. Les résultats de cette étude ont servi de base pour développer un algorithme d'identification géométrique. Pour la validation de notre approche, on donne une étude numérique pour un problème d'optimisation d'emplacement des injecteurs dans un lac eutrophe. Des exemples numériques montrent l'efficacité de la méthode proposée

Weak Solutions of the Navier–Stokes Equations

2006 ◽  
Author(s):  
Jean-Yves Chemin ◽  
Benoit Desjardins ◽  
Isabelle Gallagher ◽  
Emmanuel Grenier
Keyword(s):  
Unbounded Domain ◽  
Mathematical Analysis ◽  
Weak Solutions ◽  
Spectral Properties ◽  
Stokes Equations ◽  
Stokes Operator ◽  
Navier Stokes ◽  
Bounded Domains ◽  
Navier Stokes Equations ◽  
Compactness Result

The mathematical analysis of the incompressible Stokes and Navier–Stokes equations in a possibly unbounded domain Ω of Rd (d = 2 or 3) is the purpose of this chapter. Notice that no regularity assumptions will be required on the domain Ω. Because of the compactness result stated in Theorem 1.3, page 27, the case of bounded domains will be different (in fact slightly simpler) than the case of general domains. The study of the spectral properties of the Stokes operator previously defined relies on the study of its inverse, which is in fact much easier. We shall restrict ourselves here to the case of the homogeneous Stokes operator which is adapted to the case of a bounded domain.

Maximal Lp -Lq regularity to the Stokes problem with Navier boundary conditions

2017 ◽  
Vol 8 (1) ◽  
pp. 743-761 ◽  
Author(s):  
Hind Al Baba
Keyword(s):  
Boundary Conditions ◽  
Stress Tensor ◽  
Parabolic Problem ◽  
Stokes Problem ◽  
Stokes Operator ◽  
Fractional Powers ◽  
Navier Boundary Conditions ◽  
Regularity Results

Abstract We prove in this paper some results on the complex and fractional powers of the Stokes operator with slip frictionless boundary conditions involving the stress tensor. This is fundamental and plays an important role in the associated parabolic problem and will be used to prove maximal L^{p} - L^{q} regularity results for the non-homogeneous Stokes problem.

Sign in / Sign up

Export Citation Format

Share Document