Representation of volume-preserving maps induced by solenoidal vector fields

1985 ◽  
Vol 28 (3) ◽  
pp. 1005 ◽  
Author(s):  
A. Thyagaraja ◽  
F. A. Haas
Keyword(s):  
Vector Fields ◽  
Solenoidal Vector ◽  
Volume Preserving

scholarly journals Poloidal-toroidal decomposition of solenoidal vector fields in the ball

2019 ◽  
Vol 22 (3) ◽  
pp. 74-95
Author(s):  
S. G. Kazantsev ◽  
V. B. Kardakov
Keyword(s):  
Vector Fields ◽  
Solenoidal Vector

Estimates of the Distance to the Set of Solenoidal Vector Fields and Applications to A Posteriori Error Control

2015 ◽  
Vol 15 (4) ◽  
pp. 515-530 ◽  
Author(s):  
Sergey Repin
Keyword(s):  
Error Control ◽  
Vector Fields ◽  
Bounded Operator ◽  
Posteriori Error ◽  
Stability Theorem ◽  
A Posteriori ◽  
Solenoidal Vector ◽  
Bounded Lipschitz Domain ◽  
Incompressible Viscous Fluids ◽  
The Stability

AbstractThe paper is concerned with computable estimates of the distance between a vector-valued function in the Sobolev space$W^{1,\gamma }(\Omega ,\mathbb {R}^d)$(where${\gamma \in (1,+\infty )}$and Ω is a bounded Lipschitz domain in ℝd) and the subspace${S^{1,\gamma }(\Omega ,\mathbb {R}^d)}$containing all divergence-free (solenoidal) vector functions. Derivation of these estimates is closely related to the stability theorem that establishes existence of a bounded operator inverse to the operator${\operatorname{div}}$. The constant in the respective stability inequality arises in the estimates of the distance to the set${S^{1,\gamma }(\Omega ,\mathbb {R}^d)}$. In general, it is difficult to find a guaranteed and realistic upper bound of this global constant. We suggest a way to circumvent this difficulty by using weak (integral mean) solenoidality conditions and localized versions of the stability theorem. They are derived for the case where Ω is represented as a union of simple subdomains (overlapping or non-overlapping), for which estimates of the corresponding stability constants are known. These new versions of the stability theorem imply estimates of the distance to${S^{1,\gamma }(\Omega ,\mathbb {R}^d)}$that involve only local constants associated with subdomains. Finally, the estimates are used for deriving fully computable a posteriori estimates for problems in the theory of incompressible viscous fluids.

Magnetfeldbeschreibung mit verallgemeinerten poloidalen und toroidalen Skalaren

1972 ◽  
Vol 27 (8-9) ◽  
pp. 1167-1172 ◽  
Author(s):  
Gerhard Gerlich
Keyword(s):  
Vector Field ◽  
Magnetic Fields ◽  
Vector Fields ◽  
Solenoidal Vector ◽  
Integrability Conditions ◽  
Solenoidal Vector Field ◽  
Complicated Geometry

Abstract Representation of Magnetic Fields by Generalized poloidal and Toroidal Scalars Every solenoidal vector field can be represented by unique poloidal and toroidal scalars. This description is especially appropriate to the geometry of a sphere. A generalization which can be applied to a more or less complicated geometry could be elaborated by means of transforming integrability conditions of space into integrability conditions of surfaces. This formalism enables us to give simple proofs of other important representations of vector fields by two scalars (magnetic coordinates, complex-lamellar fields).

Piecewise Solenoidal Vector Fields and the Stokes Problem

1990 ◽  
Vol 27 (6) ◽  
pp. 1466-1485 ◽  
Author(s):  
Garth A. Baker ◽  
Wadi N. Jureidini ◽  
Ohannes A. Karakashian
Keyword(s):  
Vector Fields ◽  
Stokes Problem ◽  
Solenoidal Vector
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