Standing Solitary Euler-Korteweg Waves are Unstable

0.08 MB·English

by  Johannes Höwing

STANDING SOLITARY EULER-KORTEWEG WAVES ARE UNSTABLE
JOHANNESHO¨WING
1. The result
The Euler-Korteweg system is given by the equations
3
1
V −U = 0,
0 t y
2 (1) 1
U +p(V) = −(κ(V)V + (κ(V)) V ) ,
t y yy y y y
n 2
a
with κ(V) > 0. System (1), and notably its solitary waves
J

6
V v v v∗
1 (x,t) = (x−ct) with (±∞) = ,
(cid:18)U(cid:19) (cid:18)u(cid:19) (cid:18)u(cid:19) (cid:18)u∗(cid:19)

]
P have been intensely studied by Benzoni-Gavage et al. in [1, 2, 3].
A
Definition 1. [3] A traveling wave (v,u) of (1) is called orbitally stable if for each ε > 0,
h. there exists a δ > 0 such that for any solution (V,U) ∈ (v,u)+C([0,T);H3(R)×H2(R)) of
at (1), closeness at initial time,
m
[ k(V,U)(·,0)−(v,u)(·)kH1×L2 < δ

2 implies closeness at any time
v
7 infk(V,U)(·,t)−(v,u)(·+σ)kH1×L2 < ε for all t > 0.
6 σ∈R
7
The following is the point of this short note.
2
. Theorem 1. All non-trivial standing solitary Euler-Korteweg waves are not orbitally stable.
1
0
3
2. The proof
1
:
v Forfixedbasestatev∗,thesolitarywaveshomoclinictov∗occurinfamilies(uc,vc)parametrized
Xi by their speed c. The proof of Theorem 1 is based on
r Lemma 1. [1] A solitary wave (uc∗,vc∗) is orbitally unstable if the moment of instability
a
∞
m(c) = κ(v)v′2 dξ
Z−∞
is not convex at c = c∗.
Theorem 1 follows from
Lemma 2. m′′(0) < 0.
Date: January 13, 2013.
1
2 JOHANNESHO¨WING
To prove Lemma 2, we recall that with
1 df(v)
(2) F(v,c) = −f(v)+f(v∗)−p(v∗)(v−v∗)+ c2(v−v∗)2, − = p(v),
2 dv
the profile equation
1 ∂F(v,c)
′′ ′ ′
κ(v)v + (κ(v)) v = −
2 ∂v
possesses (cf. [3]) a first integral given by
1
(3) I(v,v′) = κ(v)v′2 +F(v,c).
2
As
vm(c)
′
m(c) = 2 κ(v)v dv
Z
v∗
′
with v∗,vm(c) > v∗ consecutive zeros of F(·,c). Since I(v,v ) ≡ 0 along solutions, we have
vm(c)
m(c) = 2 (κ(v))1/2 (−2F(v,c))1/2 dv
Z
v∗
= 4 (vm(c)−v∗)1/2 κ(v (c)−w2) 1/2 −2F(v (c)−w2,c) 1/2 w dw,
m m
Z
0 (cid:0) (cid:1) (cid:0) (cid:1)
where w := (v (c)−v)1/2 (cf. [6]). The first derivative of m is
m
d
m′(c) = 4 κ(v (c)−w2) 1/2 −2F(v (c)−w2,c) 1/2 w dw
m m
Z dcn o
(cid:0) (cid:1) (cid:0) (cid:1)
κ (v (c)−w2)v′ (c)
= 4 v m m −2F(v (c)−w2,c) 1/2 w dw +
m
Z 2(κ(v (c)−w2))1/2
m (cid:0) (cid:1)
κ(v (c)−w2) 1/2 −F (v (c)−w2,c)v′ (c)−F (v (c)−w2,c)
m v m m c m
+4 w dw
Z (cid:0) (cid:1) (cid:0)(−2F(v (c)−w2,c))1/2 (cid:1)
m
which, due to
∂ κ (v)(−2F(v,c))1/2 κ(v)1/2F (v,c)
(κ(v)(−2F(v,c)))1/2 = v − v ,
∂v (cid:16) (cid:17) 2(κ(v))1/2 (−2F(v,c))1/2
simplifies to
κ(v (c)−w2) 1/2F (v (c)−w2,c)
′ m c m
m(c) = −4 w dw.
Z (cid:0) (−2F(v (cid:1)(c)−w2,c))1/2
m
The second derivative of m can be written in the form
vm(c) A(v,c)+B(v,c)
′′
m (c) = 2 dv
Zv∗ (κ(v))1/2(−2F(v,c))3/2
′
with A(v,c) = F(v,c)κ (v)v (c)F (v,c) and
v m c
′ ′
B(v,c) = κ(v) (v−v∗) 2F(v,c) (v−v∗)+2cvm(c) −c(v −v∗) Fv(v,c)vm(c)+Fc(v,c) .
(cid:0) (cid:0) (cid:0) (cid:1) (cid:0) (cid:1)(cid:1)(cid:1)
As F (v,0) = 0= A(v,0) and
c
B(v,0) = 2κ(v)(v −v∗)2F(v,0) < 0 for all v ∈ (v∗,vm(0)),
STANDING SOLITARY EULER-KORTEWEG WAVES ARE UNSTABLE 3
′′
we indeed have m (0) < 0.
Remark. Certain of the results by Zumbrun [7] on the Bona-Sachs model (κ(V) ≡ 1
and p(V) = −V + Vq with q ≥ 2) and by De Bouard [4] on a Gross-Pitaevskii model
(κ(V) = 1/(4V4) and p(V) = α/V2−β/V3 with α,β > 0) appear as interesting special cases
of Theorem 1.
I obtained parts of this result during my doctoral dissertation under the supervision of Hein-
rich Freistu¨hler; I am very grateful to him for helpful discussions.
References
[1] S. Benzoni-Gavage, Spectral transverse instability of solitary waves in Korteweg fluids, J. Math. Anal.
Appl.361, 2010, 338–357.
[2] S. Benzoni-Gavage, R. Danchin, S. Descombes, Well-posedness of one-dimensional Korteweg Models,
Electron. J. Differential Equations 59, 2006, 1–35.
[3] S. Benzoni-Gavage, R. Danchin, S. Descombes, D. Jamet, Structure of Korteweg models and stability of
diffusive interfaces, Interfaces Free Bound. 7, 2005, 371–414.
[4] A.De Bouard, Instability of Stationary Bubbles, SIAMJ. Math. Anal. 26, 1995, 566–582.
[5] M. Grillakis, J. Shatah, W. Strauss, Stability Theory of Solitary Waves in the Presence of Symmetry, I,
J. Funct.Anal. 74, 1987, 160–197.
[6] J.H¨owing,Stabilityoflarge- andsmall-amplitudesolitarywaves inthe generalized Korteweg-de Vriesand
Euler-Korteweg/Boussinesq equations, J. Differential Equations 251, 2011, 2515–2533.
[7] K.Zumbrun,Asharpstabilitycriterionforsoliton-typepropagatingphaseboundariesinKorteweg’smodel,
Z. Anal. Anwend.27, 2008, 11–30.
E-mail address: [email protected]
University of Hamburg, Department of Mathematics, Germany