scholarly journals Asymptotic expansion in Gevrey spaces for solutions of Navier–Stokes equations

2017 ◽  
Vol 104 (3-4) ◽  
pp. 167-190 ◽  
Author(s):  
Luan T. Hoang ◽  
Vincent R. Martinez
Keyword(s):  
Asymptotic Expansion ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Gevrey Spaces

scholarly journals Steady streaming in a channel with permeable walls

2013 ◽  
Vol 25 (1) ◽  
pp. 65-82
Author(s):  
KONSTANTIN ILIN
Keyword(s):  
Asymptotic Expansion ◽  
Viscous Fluid ◽  
Steady Flow ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Steady Streaming ◽  
Stokes Layer ◽  
Permeable Walls ◽  
Time Injection

We study steady streaming in a channel between two parallel permeable walls induced by oscillating (in time) injection/suction of a viscous fluid at the walls. We obtain an asymptotic expansion of the solution of the Navier–Stokes equations in the limit when the amplitude of normal displacements of fluid particles near the walls is much smaller than both the width of the channel and the thickness of the Stokes layer. It is shown that the steady part of the flow in this problem is much stronger than the steady flow produced by vibrations of impermeable boundaries. Another interesting feature of this problem is that the direction of the steady flow is opposite to what one would expect if the flow was produced by vibrations of impermeable walls.

scholarly journals Sobolev regular solutions for the incompressible Navier–Stokes equations in higher dimensions: asymptotics and representation formulae

2020 ◽  
Author(s):  
Weiping Yan ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
Asymptotic Expansion ◽  
Stokes Equations ◽  
Higher Dimensions ◽  
Finite Energy ◽  
Explicit Representation ◽  
Navier Stokes ◽  
Regular Solutions ◽  
Smooth Bounded Domain ◽  
Navier Stokes Equations ◽  
Expansion Formulae

Abstract In this paper, we consider the steady incompressible Navier–Stokes equations in a smooth bounded domain $$\Omega \subset \mathbb R^n$$ Ω ⊂ R n with the dimension $$n\ge 3$$ n ≥ 3 . We first establish asymptotic expansion formulae of Sobolev regular finite energy solutions in $$\Omega$$ Ω . In the second part of this paper, explicit representation formulae of Sobolev regular solutions are showed in the regular polyhedron $$\Omega :=[0,T]^n$$ Ω : = [ 0 , T ] n .

Viscous profiles of vortex patches

2013 ◽  
Vol 14 (1) ◽  
pp. 1-68 ◽  
Author(s):  
Franck Sueur
Keyword(s):  
Asymptotic Expansion ◽  
Initial Data ◽  
Euler Equations ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Vanishing Viscosity ◽  
Navier Stokes Equations ◽  
Vanishing Viscosity Limit ◽  
Viscosity Limit ◽  
Sharp Variation

AbstractWe deal with the incompressible Navier–Stokes equations with vortex patches as initial data. Such data describe an initial configuration for which the vorticity is discontinuous across a hypersurface. We give an asymptotic expansion of the solutions in the vanishing viscosity limit which exhibits an internal layer where the fluid vorticity has a sharp variation. This layer moves with the flow of the Euler equations.

scholarly journals Long-time asymptotic expansions for Navier-Stokes equations with power-decaying forces

2019 ◽  
Vol 150 (2) ◽  
pp. 569-606 ◽  
Author(s):  
Dat Cao ◽  
Luan Hoang
Keyword(s):  
Asymptotic Expansion ◽  
Body Force ◽  
Stokes Equations ◽  
Three Dimensional ◽  
The Body ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Gevrey Space ◽  
Long Time ◽  
The Individual

AbstractThe Navier-Stokes equations for viscous, incompressible fluids are studied in the three-dimensional periodic domains, with the body force having an asymptotic expansion, when time goes to infinity, in terms of power-decaying functions in a Sobolev-Gevrey space. Any Leray-Hopf weak solution is proved to have an asymptotic expansion of the same type in the same space, which is uniquely determined by the force, and independent of the individual solutions. In case the expansion is convergent, we show that the next asymptotic approximation for the solution must be an exponential decay. Furthermore, the convergence of the expansion and the range of its coefficients, as the force varies are investigated.

scholarly journals Non-stationary Navier–Stokes equations in 2D power cusp domain

2021 ◽  
Vol 10 (1) ◽  
pp. 982-1010
Author(s):  
Konstantin Pileckas ◽  
Alicija Raciene
Keyword(s):  
Asymptotic Expansion ◽  
Singular Point ◽  
Stokes Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Navier Stokes ◽  
Existence Of A Solution ◽  
Navier Stokes Equations ◽  
Stationary Navier Stokes Equations

Abstract The initial boundary value problem for the non-stationary Navier-Stokes equations is studied in 2D bounded domain with a power cusp singular point O on the boundary. The case of the boundary value with a nonzero flow rate is considered. In this case there is a source/sink in O and the solution necessary has infinite energy integral. In the first part of the paper the formal asymptotic expansion of the solution near the singular point is constructed. The justification of the asymptotic expansion and the existence of a solution are proved in the second part of the paper.

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