Inverse Source Problem for the Stokes System

An initial-boundary-value problem is considered for the one-dimensional diffusion equation with a general convolutional derivative in time and nonclassical boundary conditions. We are concerned with the inverse source problem of recovery of a space-dependent source term from given final time data. Generalized eigenfunction expansions are used with respect to a biorthogonal pair of bases. Existence, uniqueness and stability estimates in Sobolev spaces are established.

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Relative Decay Conditions on Liouville Type Theorem for the Steady Navier–Stokes System

Journal of Mathematical Fluid Mechanics ◽

10.1007/s00021-020-00549-9 ◽

2021 ◽

Vol 23 (1) ◽

Author(s):

Dongho Chae

Keyword(s):

Stokes System ◽

Type Theorem ◽

Navier Stokes ◽

Liouville Type ◽

Liouville Type Theorem

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Error analysis of higher order trace finite element methods for the surface Stokes equation

Journal of Numerical Mathematics ◽

10.1515/jnma-2020-0017 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Thomas Jankuhn ◽

Maxim A. Olshanskii ◽

Arnold Reusken ◽

Alexander Zhiliakov

Keyword(s):

Finite Element ◽

Stokes System ◽

Parametric Representation ◽

Higher Order ◽

Optimal Order ◽

Helmholtz Instability ◽

Surface Stability ◽

Instability Problem ◽

Convergence Results ◽

Optimal Order Convergence

AbstractThe paper studies a higher order unfitted finite element method for the Stokes system posed on a surface in ℝ3. The method employs parametric Pk-Pk−1 finite element pairs on tetrahedral bulk mesh to discretize the Stokes system on embedded surface. Stability and optimal order convergence results are proved. The proofs include a complete quantification of geometric errors stemming from approximate parametric representation of the surface. Numerical experiments include formal convergence studies and an example of the Kelvin--Helmholtz instability problem on the unit sphere.

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To what extent is cross-diffusion controllable in a two-dimensional chemotaxis-(Navier–)Stokes system modeling coral fertilization

Calculus of Variations and Partial Differential Equations ◽

10.1007/s00526-021-02039-w ◽

2021 ◽

Vol 60 (4) ◽

Author(s):

Wei Wang ◽

Minghua Zhang ◽

Sining Zheng

Keyword(s):

Stokes System ◽

System Modeling ◽

Navier Stokes ◽

Two Dimensional ◽

Cross Diffusion

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Asymptotic analysis of an optimal control problem for a viscous incompressible fluid with Navier slip boundary conditions

Asymptotic Analysis ◽

10.3233/asy-211685 ◽

2021 ◽

pp. 1-21

Author(s):

Claudia Gariboldi ◽

Takéo Takahashi

Keyword(s):

Optimal Control ◽

Optimal Control Problem ◽

Boundary Conditions ◽

Control Problem ◽

Stokes System ◽

Navier Stokes ◽

Slip Boundary Conditions ◽

Navier Slip ◽

Slip Boundary ◽

Navier Slip Boundary Conditions

We consider an optimal control problem for the Navier–Stokes system with Navier slip boundary conditions. We denote by α the friction coefficient and we analyze the asymptotic behavior of such a problem as α → ∞. More precisely, we prove that if we take an optimal control for each α, then there exists a sequence of optimal controls converging to an optimal control of the same optimal control problem for the Navier–Stokes system with the Dirichlet boundary condition. We also show the convergence of the corresponding direct and adjoint states.

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