Vortex like finite energy asymptotic profiles for isentropic compressible flows

Miguel Rodrigues

Description

Niveau: Supérieur, Doctorat, Bac+8
Vortex-like finite-energy asymptotic profiles for isentropic compressible flows L.Miguel Rodrigues Institut Fourier, U.M.R. C.N.R.S. 5582 Universite de Grenoble I B.P. 74 38402 Saint-Martin-d'Heres, France Abstract Bidimensional incompressible viscous flows with well-localised vorticity are well-known to develop vortex structures. The purpose of the present paper is to recover the asymptotic profiles describing these phenomena for homogeneous finite-energy flows as asymptotic profiles for near-equilibrium isentropic com- pressible flows. This task is performed by extending the sharp description of the asymptotic behaviour of near-equilibrium compressible flows obtained by David Hoff and Kevin Zumbrun [8] to the case of finite-energy vortex-like solu- tions. Mathematics subject classification (2000).. 76N99, 35B40, 35M20, 35Q30. Keywords. Isentropic compressible Navier-Stokes equations, near-equilibrium, long-time asymptotic profiles, vortex, hyperbolic-parabolic composite-type, ar- tificial viscosity approximation. Introduction The present paper is focused on the long-time asymptotic behaviour of viscous bidimensional flows. When no exterior force is applied the flow is expected to re- turn to equilibrium, namely to a state of constant density and zero velocity.

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• vortex


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• finite-energy flows


• energy asymptotic


• flows mass


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Sujets

• Rodrigues
• Vortex

Informations

• Publié par: mijec
• Nombre de lectures: 36
• Langue: English

Extrait

Vortex-like nite-energy asymptotic pro les for
isentropic compressible o ws
L.Miguel Rodrigues
Institut Fourier, U.M.R. C.N.R.S. 5582
Universite de Grenoble I
B.P. 74
38402 Saint-Martin-d’Heres, France
[e-mail]
Abstract
Bidimensional incompressible viscous o ws with well-localised vorticity are
well-known to develop vortex structures. The purpose of the present paper is
to recover the asymptotic pro les describing these phenomena for homogeneous
nite-energy o ws as asymptotic pro les for near-equilibrium isentropic com-
pressible o ws. This task is performed by extending the sharp description of
the asymptotic behaviour of near-equilibrium compressible o ws obtained by
David Ho and Kevin Zumbrun [8] to the case of nite-energy vortex-like solu-
tions.
Mathematics subject classi cation (2000).. 76N99, 35B40, 35M20, 35Q30.
Keywords. Isentropic compressible Navier-Stokes equations, near-equilibrium,
long-time asymptotic pro les, vortex, hyperbolic-parabolic composite-type, ar-
ti cial viscosity approximation.
Introduction
The present paper is focused on the long-time asymptotic behaviour of viscous
bidimensional o ws. When no exterior force is applied the o w is expected to re-
turn to equilibrium, namely to a state of constant density and zero velocity. Our
purpose is thus to determine asymptotic pro les for the return to equilibrium.
The motion of the considered o ws may be described by the time evolution
of the pair ( ; u), = (t; x) > 0 being the density eld of the uid and
2u = u (t; x) 2 R the velocity eld. The main purpose of the paper is to
prove that for some initial data near equilibrium one recovers for isentropic
compressible o ws the same asymptotic pro les as in the constant-density case.
Therefore let us begin introducing the constant-density pro les we are interested
in.
When the density is constant, , mass conservation and a force balance?
1for Newtonian uids lead to the Navier-Stokes evolution equations
)
div u = 0
(1)
@ ( u) + (u r) ( u) = 4 u r pt ? ?
where > 0 is the shear Lame viscosity coe cien t and p = p (t; x)2 R is the
pressure eld of the uid. In order to make the former equations compatible
the pressure must be determined (up to a constant) by the elliptic equation

4 p = div (u r) u : (2)?
In this bidimensional incompressible context, it may seem more natural and it
is often more convenient to work with the curl of the velocity rather than with
the velocity itself. The evolution of the vorticity ! = @ u @ u obeys1 2 2 1
@ ! + u r ! = 4 ! (3)t
where = = is the kinematic viscosity and the velocity u is recovered by the?
Biot-Savart law,
Z ?1 (x y) 2u(x) = ! (y) dy ; x2 R ; (4)
22 2 jx yjR
?with (z ; z ) = ( z ; z ), which we also denote u = K ?!, K being called1 2 2 1 BS BS
the Biot-Savart kernel. Note that in terms of Fourier transforms the Biot-Savart
law becomes
?i 2
ub() = !b () ; 2 R : (5)
2jj
Concerning the widely-developed literature about the (homogeneous) Navier-
Stokes equations, the reader is referred to some advanced entering gates such
as the following books [2], [12], [13], [17], and to the more vorticity-focused
review [1].
Flows with constant density and initially well-localised vorticity are well-
known to develop vortex-like structures. In the compressible case we shall re-
cover near equilibrium this kind of behaviour.
For instance it is proved in [7] that any solution ! of (3) with integrable
initial datum ! satis es in Lebesgue spaces0
11 Gplim t k !(t) ! (t)k = 0 ; (6)p
t!1
1 1 G2 qlim t k u(t) u (t)k = 0 ; (7)q
t!1
for any 1 p1 and any 2 < q1, where
r
1 x xG G Gp p! (t; x) = G ; u (t; x) = v (8)
t t t t
with pro les
? 1 2 1 2 j j =4 G j j =4G() = e ; v () = 1 e ; (9)
24 2 jj
2and 2 R is such that the initial velocity circulations at in nit y coincide at
the initial time t = 0,
Z
= ! (x) dx : (10)0
2R
GActually, for any = 0, the vorticity ! is a (self-similar) solution of (3)
with initial datum a Dirac mass | centred at the origin and of weight = ;
the corresponding o w is called Oseen vortex. Thus when the circulation is non
zero equalities (6) and (7) show that the o w behaves asymptotically as a single
vortex, whereas when = 0 it returns to equilibrium faster than a single vortex
does.
However nite energy o ws have zero circulation. Indeed, as is easily derived
from (5), to obtain an integrable vorticity and a square-integrable velocity one
must ask for the vorticity to be of zero mean. To consider nite energy o ws
we must thus investigate pro les decaying faster. Yet it is well-known, see [6]
(combined with [7, Proposition 1.5]) for instance, that if the initial vorticity !0R
3=2is such that (1 +jj) ! is square-integrable and ! = 0 then for any0 2 0R
index 1 p1
3 1 ; 1 22 plim t k !(t) ! (t)k = 0 (11)p
t!1
where
; F F1 2 1 2! (t; x) = ! (t; x) + ! (t; x) (12)1 2
with for i = 1; 2
1 xFi! (t; x) = p F p (13)i3=2 t t
i
F () = @ G () = G () (14)i i
2
and is such thati
Z
= x ! (x) dx : (15)i i 0
2R
; 1 2Observe that ! is not a solution of equation (3) but only of its linearisation
around equilibrium, a heat equation. However equality (11) is easily seen to
apply also to some o ws with nite measures as initial vorticities, such as those
of initial vorticity
1 1
+ : (16)(