A stream-function-vorticity variational formulation for the exterior Stokes problem in weighted Sobolev spaces

1992 ◽  
Vol 15 (5) ◽  
pp. 345-363 ◽  
Author(s):  
V. Girault ◽  
J. Giroire ◽  
A. Sequeira
Keyword(s):  
Stream Function ◽  
Sobolev Spaces ◽  
Variational Formulation ◽  
Stokes Problem ◽  
Weighted Sobolev Spaces

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 Variational formulations of steady rotational equatorial waves

2020 ◽  
Vol 10 (1) ◽  
pp. 534-547
Author(s):  
Jifeng Chu ◽  
Joachim Escher
Keyword(s):  
Linear Stability ◽  
Stream Function ◽  
Water Waves ◽  
Variational Formulation ◽  
Energy Functional ◽  
Equatorial Waves ◽  
Second Variation ◽  
Governing Equations ◽  
The Second Variation ◽  
Variational Formulations

Abstract When the vorticity is monotone with depth, we present a variational formulation for steady periodic water waves of the equatorial flow in the f-plane approximation, and show that the governing equations for this motion can be obtained by studying variations of a suitable energy functional 𝓗 in terms of the stream function and the thermocline. We also compute the second variation of the constrained energy functional, which is related to the linear stability of steady water waves.

scholarly journals Pseudo-monotonicity and degenerated or singular elliptic operators

1998 ◽  
Vol 58 (2) ◽  
pp. 213-221 ◽  
Author(s):  
P. Drábek ◽  
A. Kufner ◽  
V. Mustonen
Keyword(s):  
Sobolev Spaces ◽  
Differential Operators ◽  
Weighted Sobolev Spaces ◽  
Type Inequality ◽  
Elliptic Operators ◽  
Hardy Type Inequality ◽  
Hardy Type ◽  
Elliptic Differential Operators

Using the compactness of an imbedding for weighted Sobolev spaces (that is, a Hardy-type inequality), it is shown how the assumption of monotonicity can be weakened still guaranteeing the pseudo-monotonicity of certain nonlinear degenerated or singular elliptic differential operators. The result extends analogous assertions for elliptic operators.

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