Data dependence of approximate current-vortex sheets near the onset of instability

0.21 MB·

by  Alessandro Morando

DATA DEPENDENCE OF APPROXIMATE CURRENT-VORTEX SHEETS
NEAR THE ONSET OF INSTABILITY
ALESSANDROMORANDO,PAOLOSECCHI,ANDPAOLATREBESCHI
6
1
0 Abstract. Thepaperisconcernedwiththefreeboundaryproblemfor2Dcurrent-vortexsheetsinideal
2 incompressible magneto-hydrodynamics near the transition point between the linearized stability and
instability. Inordertostudythedynamicsofthediscontinuityneartheonsetoftheinstability,Hunter
n
andThoo[4]haveintroducedanasymptoticquadraticallynonlinearintegro-differentialequationforthe
a
amplitudeofsmallperturbations oftheplanardiscontinuity.
J
The local-in-timeexistence of smooth solutions to the Cauchy problem for the amplitude equation

4 wasshownin[8,7]. Inthepresentpaperweprovethecontinuousdependenceinstrongnormofsolutions
1 ontheinitialdata. Thiscompletes theproofofthewell-posednessoftheproblemintheclassicalsense
ofHadamard.

]
P
A
h. 1. Introduction
t
a In the present paper we consider the following equation
m
1
ϕ µϕ = H[φ2] +φϕ , φ=H[ϕ], (1)
[ tt xx xx xx
− (cid:18)2 (cid:19)x
1 where the unknown is the scalar function ϕ = ϕ(t,x), where t denotes the time, x R is the space
v variable and H denotes the Hilbert transform with respect to x, and µ is a constant. H∈unter and Thoo
4
[4] have derived this asymptotic equation in order to study the dynamics of 2D current-vortex sheets in
7
6 ideal incompressible magneto-hydrodynamics, near the transition point between the linearized stability
3 and instability.
0 Equation(1) is an integro-differentialevolution equation in one space dimension, with quadratic non-
.
1 linearity. This is a nonlocal equation of order two: in fact, it may also be written as
0
ϕ µϕ = [H;φ]∂ φ +H[φ2] , (2)
6 tt− xx x x x x
1 where [H;φ]∂x is a pseudo-differentialoperator(cid:0)of order zero. This a(cid:1)lternative form (2) shows that (1) is
:
v an equation of second order, due to a cancelation of the third order spatial derivatives appearing in (1).
i Equation (1) also admits the alternative spatial form
X
r ϕtt (µ 2φx)ϕxx+ [ϕ]=0, (3)
a − − Q
where
[ϕ]:= 3[H; φ ]φ [H; φ]φ . (4)
x xx xxx
Q − −
The alternative form (3) puts in evidence the difference µ 2φ which has a meaningful role. In fact it
x
−
can be shown that the linearized operator about a given basic state is elliptic and (1) is locally linearly
ill-posed in points where
µ 2φ <0.
x
−
On the contrary, in points where
µ 2φ >0 (5)
x
−
Date:January15,2016.
2010 Mathematics Subject Classification. 35Q35,76E17,76E25, 35R35,76B03.
Key words and phrases. Magneto-hydrodynamics, incompressiblefluids, current-vortex sheets, interfacial stability and
instability.
The authors are supported by the national research project PRIN 2012 “Nonlinear Hyperbolic Partial Differential
Equations,DispersiveandTransportEquations: theoretical andapplicativeaspects”.
1
2 A.MORANDO,P.SECCHI,ANDP.TREBESCHI
the linearized operator is hyperbolic and (1) is locally linearly well-posed, see [4]. In this case we can
think of (3) as a nonlinear perturbation of the wave equation.
Thelocal-in-timeexistenceofsmoothsolutionstotheCauchyproblemfor(1),undertheabovestability
condition (5), was shown in [8, 7]. In the present paper we prove the continuous dependence in strong
normofsolutionsontheinitialdata. Thiscompletestheproofofthewell-posednessof (1)intheclassical
sense of Hadamard, after existence and uniqueness.
As written in Kato’s paper [5], this part may be the most difficult one, when dealing with hyperbolic
problems. OurmethodissomehowinspiredbyBeira˜odaVeiga’sperturbationtheoryforthecompressible
Euler equations [1, 2], and its application to the problem of convergence in strong norm of the incom-
pressible limit, see [3, 9]. Instead of directly estimating the difference between the given solution and
the solutions to the problems with approximatinginitial data,the main idea is to use a triangularization
with the more regular solution to a suitably chosen close enough problem.
Let us consider the initial value problem for the equation (1) supplemented by the initial condition
ϕ =ϕ(0), ∂ ϕ =ϕ(1), (6)
|t=0 t |t=0
for sufficiently smooth initial data ϕ(0), ϕ(1) satisfying the stability condition
µ 2φ(0) >0, φ(0) :=H[ϕ(0)].
− x
For the sake of convenience, in the paper the unknown ϕ = ϕ(t,x) is a scalar function of the time
t R+ and the space variable x, rangingon the one-dimensionaltorus T (that is ϕ is periodic in x). For
∈
all notation we refer to the following Section 2.
In [7] we prove the following existence theorem.
Theorem 1. Let s 3 be a real number. Assume ϕ(0) Hs(T), ϕ(1) Hs−1(T), and let ϕ(0), ϕ(1) have
≥ ∈ ∈
zero spatial mean. Given 0<δ <µ, there exist 0<R 1, and constants C >0, C >0 such that, if
1 2
≤
ϕ(0) 2 + ϕ(1) 2 <R2, (7)
k x kH2(T) k kH2(T)
there exists a unique solution ϕ C(I ;Hs(T)) C1(I ;Hs−1(T)) of the Cauchy problem (1), (6) defined
0 0
∈ ∩
on the time interval I =[0,T ), where
0 0
−1/2
T =C ϕ(0) 2 + ϕ(1) 2 . (8)
0 1 k x kH2(T) k kH2(T)
The solution ϕ has zero spatial mean an(cid:16)d satisfies, for all t I , (cid:17)
0
∈
µ 2φ δ, (9)
x
− ≥
ϕ(t) 2 + ϕ (t) 2 C ϕ(0) 2 + ϕ(1) 2 . (10)
k kHs(T) k t kHs−1(T) ≤ 2 k kHs(T) k kHs−1(T)
(cid:16) (cid:17)
Notice that the size of the existence time interval depends on the H3,H2 norms of the initial data,
for all s 3. Every solution, with the regularity as in the statement of Theorem 1, has the additional
≥
regularity
ϕ C2(I ;Hs−2(T)),
0
∈
following from equation (1) and suitable commutator estimates, see [7], Corollary 16.
1.1. The main result. Now we state the main resultof this paper about the continuous dependence in
strong norm of solutions on the initial data.
Theorem 2. Let s 3 be a real number and µ>δ >0. Let us consider ϕ(0) Hs(T), ϕ(1) Hs−1(T)
≥ ∈ ∈
and sequences ϕ(n0) n∈N Hs(T), ϕ(n1) n∈N Hs−1(T), where all functions ϕ(0),ϕ(1),ϕ(n0),ϕ(n1) have
{ } ⊂ { } ⊂
zero spatial mean. Assume that
ϕ(0) ϕ(0) strongly in Hs(T), ϕ(1) ϕ(1) strongly in Hs−1(T), asn + .
n → n → → ∞
Let T > 0 and set I = [0,T]. Assume that there exists a unique solution ϕ C(I;Hs(T))
C1(I;Hs−1(T)) of problem (1), (6) with initial data ϕ(0),ϕ(1), and that for all n th∈ere exist uniqu∩e
solutions ϕ C(I;Hs(T)) C1(I;Hs−1(T)) of (1), (6) with initial data ϕ(0),ϕ(1). All solutions satisfy
n n n
∈ ∩
(9), (10) on I (with corresponding initial data in the r.h.s.) for a given constant C independent of n.
2
DATA DEPENDENCE OF APPROXIMATE CURRENT-VORTEX SHEETS 3
Then
ϕ ϕ stronglyin C(I;Hs(T)) C1(I;Hs−1(T)), asn + . (11)
n
→ ∩ → ∞
Here we are assuming that all solutions ϕ,ϕ are defined on the same time interval I = [0,T], where
n
T is arbitrarily given, neither necessarily small nor given by (8).
Using the a priori estimate (10) and standard arguments, it is rather easy to show the continuous
dependence of solutions on the initial data in the topology of C(I;Hs−ε(T)) C1(I;Hs−1−ε(T)), for all
∩
smallenoughε>0. Instead,inTheorem2weprovethe continuousdependencepreciselyinthe topology
of C(I;Hs(T)) C1(I;Hs−1(T)), i.e. in the same function space where we show the existence. From
∩
Theorem 1 and Theorem 2 we obtain that the initial value problem (1), (6) is well-posed in Hs in the
classical sense of Hadamard.
The rest of the paper is organized as follows. In Section 2 we introduce some notation and give some
preliminary technical results. In Section 3 we prove our main Theorem 2.
2. Notations and preliminary results
2.1. Notations. Inthe paperwedenotebyC genericpositiveconstants,thatmayvaryfromlinetoline
or even inside the same formula.
Let T denote the one-dimensionaltorus defined as T:=R/(2πZ). As usual, all functions defined on T
can be considered as 2π periodic functions on R.
All functions f :T −C can be expanded in terms of Fourier series as
→
f(x)= f(k)eikx,
k∈Z
X
b
where f(k) are the Fourier coefficients defined by
k∈Z
n o
b f(k):= 1 f(x)e−ikxdx, k Z. (12)
2π T ∈
Z
For positive realnumbers s, Hbs =Hs(T) denotes the Sobolev space oforder s on T, defined to be the
set of functions f :T C such that
→
f 2 := k 2s f(k)2 <+ , (13)
k kHs h i | | ∞
k∈Z
X
b
where it is set
k :=(1+ k 2)1/2. (14)
h i | |
The function defines a norm on Hs, associated to the inner product
Hs
k·k
(f,g) := k 2sf(k)g(k),
Hs
h i
k∈Z
X
which turns Hs into a Hilbert space. For s =0 one has Hb0 =bL2. The L2 norm will simply be denoted
by . From (13) we have
k·k
f = ∂ sf ,
Hs x
k k kh i k
where ∂ s is the Fourier multiplier of symbol k s, defined by
x
h i h i
\∂ su(k)= k su(k), k Z.
x
h i h i ∀ ∈
We will work with functions with zero spatial mean, so that f(0) = 0. For these functions, we easily
b
obtain from (13) the Poincar´einequality
b
kfkHs ≤√2kfxkHs−1 ∀s≥1. (15)
ForT >0andj N,wedenotebyCj([0,T]; )thespaceofj timescontinuouslydifferentiablefunctions
f :R . ∈ X
→X
4 A.MORANDO,P.SECCHI,ANDP.TREBESCHI
2.2. Preliminary results. Let us consider the Cauchy problem
ψ (µ 2φ )ψ =F, t I, x T,
tt x xx
− − ∈ ∈ (16)
(ψ|t=0 =ψ(0), ∂tψ|t=0 =ψ(1) x T,
∈
with unknown the scalar function ψ = ψ(t,x), and where φ = H[ϕ] is the Hilbert transform of a given
function ϕ, sufficiently smooth, with zero mean and such that
µ 2φ δ t I, x T, (17)
x
− ≥ ∈ ∈
for given constants µ>δ >0. Let us define
1/2
(t):= ψ (t) 2+ (µ 2φ )ψ (t)2dx .
t x x
E k k T − | |
(cid:18) Z (cid:19)
Lemma 3. Letϕ C(I;H3) C1(I;H2)begiven and satisfying (17). For all ψ(0) H1, ψ(1) L2, both
∈ ∩ ∈ ∈
functions with zero mean, and F L1(I;L2) there exists a unique solution ψ C(I;H1) C1(I;L2), of
∈ ∈ ∩
(16), with zero mean and such that
d
dtE ≤C(kϕtkH2 +kϕkH3)E +kFk, (18)
for every t I.
∈
Proof. To obtain (18) we multiply the equation in (16) by ψ and integrate over T. Integrating by parts
t
gives
1 d
ψ 2+ (µ 2φ )ψ 2dx =2 φ ψ ψ dx φ ψ 2dx+ Fψ dx.
t x x xx t x xt x t
2dt k k T − | | T − T | | T
(cid:18) Z (cid:19) Z Z Z
Then, by the Cauchy-Schwarzinequality, the Sobolev imbedding H1 ֒ L∞ and the estimate
→
H[ϕ] ϕ , ϕ Hs, s R, (19)
Hs Hs
k k ≤k k ∀ ∈ ∈
we obtain
1 d
ψ 2+ (µ 2φ )ψ 2dx C( ϕ + ϕ ) ψ 2+ ψ 2 + F ψ . (20)
2dt k tk T − x | x| ≤ k tkH2 k kH3 k tk k xk k kk tk
(cid:18) Z (cid:19)
(cid:0) (cid:1)
From (17) we obtain the estimate
1 1/2 1
ψ (µ 2φ )ψ 2dx , (21)
x x x
k k≤ √δ T − | | ≤ √δ E
(cid:18)Z (cid:19)
and, substituting it in (20), we obtain the bound
1 d
ψ 2+ (µ 2φ )ψ 2dx
t x x
2dt k k T − | |
(cid:18) Z (cid:19)
C( ϕ + ϕ ) ψ 2+ (µ 2φ )ψ 2dx + F ψ ,
t H2 H3 t x x t
≤ k k k k k k T − | | k kk k
(cid:18) Z (cid:19)
with a new constant C also depending on δ. Since ψ , the above inequality implies
t
k k≤E
d
C( ϕ + ϕ ) + F , (22)
dtE ≤ k tkH2 k kH3 E k k
that is (18). Applying the Gronwall lemma to (22) gives the a priori estimate
t
(t) eCTmaxτ∈I(kϕtkH2+kϕkH3) (0)+ F dτ ,
E ≤ E k k
(cid:26) Z0 (cid:27)
DATA DEPENDENCE OF APPROXIMATE CURRENT-VORTEX SHEETS 5
which yields
ψ (t) 2+ ψ (t) 2 1/2
t x
k k k k
(cid:0) (cid:1) 1/2 T
≤CeCTmaxτ∈I(kϕtkH2+kϕkH3)((cid:16)kψ(1)k2+(µ+2kϕ(0)kH2)kψx(0)k2(cid:17) +Z0 kFkdτ), (23)
forallt I,withC alsodependingonδ. Using(23),theexistenceofonesolutionisobtainedbystandard
∈
arguments, e.g. see [6]. By linearity of the problem, the difference of any two solutions with the same
data satisfies (23) with zero right-hand side. This shows that the solution is defined up to an additive
constant. Thus, requiringthe solutionto have zero meangives one uniquely defined solution. Since such
a solution has zero mean, we can apply the Poincar´einequality (15) and obtain from (23)
ψ (t) 2+ ψ(t) 2 1/2
k t k k kH1
(cid:0) ≤CeCTm(cid:1)axτ∈I(kϕtkH2+kϕkH3)((cid:16)kψ(1)k2+(µ+2kϕ(0)kH2)kψx(0)k2(cid:17)1/2+Z0T kFkdτ).
for all t I, which shows that ψ C(I;H1) C1(I;L2). (cid:3)
∈ ∈ ∩
The next lemma concerns the regularity of the solution ψ to (16).
Lemma 4. Let r 2 and s max 3,r . Let ϕ C(I;Hs) C1(I;H2) satisfying (17). For all
≥ ≥ { } ∈ ∩
ψ(0) Hr, ψ(1) Hr−1 with zero mean, and F L2(I;Hr−1) there exists a unique solution ψ
∈ ∈ ∈ ∈
C(I;Hr) C1(I;Hr−1) of (16), with zero mean and such that
∩
ψ(t) 2 + ψ (t) 2
k kHr k t kHr−1
≤CeCTmaxτ∈I(1+kϕtkH2+kϕkHs) kψ(1)k2Hr−1 +(µ+2kϕ(0)kH2)kψ(0)k2Hr +kFk2L2(0,t;Hr−1) (24)
n o
for every t I.
∈
Proof. Weapply ∂ r−1 totheequationin(16),multiplyby ∂ r−1ψ andintegrateoverT. Integrating
x x t
h i h i
by parts gives
1 d
ψ 2 + (µ 2φ ) ∂ r−1ψ 2dx
2dt k tkHr−1 T − x |h xi x|
(cid:18) Z (cid:19)
=2 φ ∂ r−1ψ ∂ r−1ψ dx φ ∂ r−1ψ 2dx
xx x t x x xt x x
T h i h i − T |h i |
Z Z (25)
2 [ ∂ r−1;φ ]ψ ∂ r−1ψ dx+ ∂ r−1F ∂ r−1ψ dx
x x xx x t x x t
− T h i h i Th i h i
Z Z
4
= I .
k
k=1
X
We estimate each term of this sum.
Estimate of I . We apply the Cauchy-Schwarz inequality, the Sobolev imbedding H1 ֒ L∞ and the
1
→
estimate (19):
|I1|≤2kφxxkL∞kh∂xir−1ψtkkh∂xir−1ψxk≤CkϕxkH2kψtkHr−1kψxkHr−1. (26)
Estimate of I . In a similar way we obtain
2
|I2|≤kφxtkL∞kh∂xir−1ψxk2 ≤CkϕtkH2kψxk2Hr−1. (27)
Estimate of I . First of all we write
3
I 2 [ ∂ r−1;φ ]ψ ∂ r−1ψ . (28)
3 x x xx x t
| |≤ kh i kkh i k
For the estimate of the commutator we need to distinguish between the different values of r.
6 A.MORANDO,P.SECCHI,ANDP.TREBESCHI
i) If r>5/2 we apply the estimate (69) for commutators with τ =r 1,σ =r 2>1/2 and (19) to
− −
obtain
k[h∂xir−1;φx]ψxxk≤C(kφxkHr−1 +kφxxkH1)kψxxkHr−2 (29)
≤C(kϕxkHr−1 +kϕxkH2)kψxkHr−1 ≤CkϕxkHs−1kψxkHr−1.
ii) If 2 r <5/2 we apply (70) with τ =r 1,σ =3 r >1/2 and (19) to get
≤ − −
k[h∂xir−1;φx]ψxxk≤C(kφxkH2kψxxk+kφxxkH1kψxxkHr−2) (30)
C ϕx H2( ψx H1 + ψx Hr−1) C ϕx Hs−1 ψx Hr−1.
≤ k k k k k k ≤ k k k k
iii) Finally if r =5/2 we apply (71) with τ =r 1 and (19) to obtain
−
k[h∂xir−1;φx]ψxxk≤C(kφxkH2kψxxkH1/2 +kφxxkH1kψxxkHr−2) (31)
≤CkϕxkH2kψxkH3/2 ≤CkϕxkHs−1kψxkHr−1.
Recalling (28), from (29)–(31) we have obtained, for all values of r 2,
≥
|I3|≤CkϕxkHs−1kψxkHr−1kψtkHr−1. (32)
Estimate of I . The Cauchy-Schwarz inequality gives
4
|I4|≤kh∂xir−1Fkkh∂xir−1ψtk=kFkHr−1kψtkHr−1. (33)
Estimating the right-hand side of (25) by (26), (27), (32), (33) yields
d
ψ 2 + (µ 2φ ) ∂ r−1ψ 2dx
dt k tkHr−1 T − x |h xi x|
(cid:18) Z (cid:19)
≤C(kϕtkH2 +kϕxkHs−1) kψtk2Hr−1 +kψxk2Hr−1 +2kFkHr−1kψtkHr−1
≤C(1+kϕtkH2 +kϕxk(cid:0)Hs−1) kψtk2Hr−1 +kψx(cid:1)k2Hr−1 +kFk2Hr−1,
≤C(1+kϕtkH2 +kϕxkHs−1) kψtk2H(cid:0)r−1 + T(µ−2φx)|h∂x(cid:1)ir−1ψx|2dx +kFk2Hr−1, (34)
(cid:18) Z (cid:19)
where in the last inequality we have used (17). Applying the Gronwall lemma to (34) gives
ψ (t) 2 + (µ 2φ ) ∂ r−1ψ (t)2dx
k t kHr−1 T − x |h xi x |
Z
t
≤eCTmaxτ∈I(1+kϕtkH2+kϕxkHs−1)(cid:26)kψ(1)k2Hr−1 +ZT(µ−2φx(0))|h∂xir−1ψx(0)|2dx+Z0 kFk2Hr−1dτ(cid:27),
which yields, by (17) and the Poincar´e inequality,
ψ (t) 2 + ψ(t) 2
k t kHr−1 k kHr
t
≤CeCTmaxτ∈I(1+kϕtkH2+kϕkHs) kψ(1)k2Hr−1 +(µ+2kϕ(0)kH2)kψ(0)k2Hr + kFk2Hr−1dτ , (35)
(cid:26) Z0 (cid:27)
for all t I, with C also depending on δ, that is (24). (35) provides the a priori estimate for ψ in
C(I;Hr)∈ C1(I;Hr−1). (cid:3)
∩
Nowweconsiderthe quadraticoperator [ϕ],definedin(4). Firstofallwerecalltheestimateproved
Q
in [7].
Proposition 5. There exists a positive constant C suchthat for every real s 1 and for all ϕ Hs H3
≥ ∈ ∩
kQ[ϕ]kHs−1 ≤CkϕxkH2kϕxkHs−1. (36)
Proof. For the proof see [7]. (cid:3)
Next we give an estimate for the difference of values of [ϕ].
Q
Lemma 6. The quadratic operator [ϕ], defined in (4), satisfies
Q
[ϕ] [ϕ˜] C( ϕ + ϕ˜ ) (ϕ ϕ˜) (37)
H3 H3 x
kQ −Q k≤ k k k k k − k
for all functions ϕ,ϕ˜ H3.
∈
DATA DEPENDENCE OF APPROXIMATE CURRENT-VORTEX SHEETS 7
Proof. We denote φ=H[ϕ], φ˜=H[ϕ˜], δϕ=ϕ ϕ˜, δφ=φ φ˜. Then we can write
− −
[ϕ] [ϕ˜]= 3[H; φ ]φ [H; φ]φ +3[H; φ˜ ]φ˜ +[H; φ˜]φ˜
x xx xxx x xx xxx
Q −Q − −
= 3[H; δφ ]φ 3 H; φ˜ δφ [H; δφ]φ [H; φ˜]δφ
x xx x xx xxx xxx
− − − −
4 h i
= J .
k
k=1
X
Now we estimate each term of the sum.
Estimate of J . We apply estimate (68), with p=s=0, and (19):
1
J1 C δφx φxx H1 C δϕx ϕ H3. (38)
k k≤ k kk k ≤ k kk k
Estimate of J . We apply estimate (68), with p=2,s=0, (19) and the Poincar´einequality:
2
J C φ˜ δφ C ϕ˜ δϕ . (39)
2 xxx H1 H3 x
k k≤ k kk k ≤ k k k k
Estimate of J . We apply estimate (67), with s=1, (19) and the Poincar´e inequality:
3
J3 C δφ H1 φxxx C δϕx ϕ H3. (40)
k k≤ k k k k≤ k kk k
Estimate of J . We apply estimate (68), with p=3,s=0, (19) and the Poincar´einequality:
4
J C φ˜ δφ C ϕ˜ δϕ . (41)
4 xxx H1 H3 x
k k≤ k kk k ≤ k k k k
Collecting (38)–(41) gives (37).
(cid:3)
More generally we apply (68) with different choices of p to estimate the Hs−1-norm of J J and
1 4
−
obtain
Lemma 7. Let s 3. The quadratic operator [ϕ], defined in (4), satisfies
≥ Q
kQ[ϕ]−Q[ϕ˜]kHs−1 ≤C(kϕkHs +kϕ˜kHs)kϕ−ϕ˜kHs (42)
for all functions ϕ,ϕ˜ Hs.
∈
3. Proof of Theorem 2
By assumption the sequence ϕ(n0) n∈N is uniformly bounded in Hs, and the sequence ϕ(n1) n∈N is
{ } { }
uniformly bounded in Hs−1. Because of the uniform a prioriestimate (10), fromnow on we may assume
that the sequence of solutions ϕn n∈N is uniformly bounded in C(I;Hs) C1(I;Hs−1) on the common
{ } ∩
time interval I =[0,T], i.e. there exists K >0 such that
ϕ (t) 2 + (ϕ ) (t) 2 C ϕ(0) 2 + ϕ(1) 2 K, t I, n. (43)
k n kHs k n t kHs−1 ≤ 2 k n kHs k n kHs−1 ≤ ∀ ∈ ∀
Notice that the similar bound holds as well fo(cid:16)r the solution ϕ. (cid:17)
FromnowonT willbe usuallyincludedin the genericconstantC, butsometimes not, whenweprefer
to emphasize its presence.
In the next proposition we prove the convergence of ϕn n∈N in a weaker topology than in our main
{ }
Theorem 2.
Proposition 8. Let s 3. Under the assumptions of Theorem 2 the sequence of solutions ϕn n∈N
≥ { }
converges to ϕ strongly in C(I;Hs−1) C1(I;Hs−2).
∩
Proof. We take the difference of equation (3) for ϕ and ϕ and get
n
(ϕ ϕ ) (µ 2φ )(ϕ ϕ ) = [ϕ]+ [ϕ ]+2(φ φ )ϕ ,
n tt x n xx n n,x x n,xx
− − − − −Q Q −
(where φ =(φ ) ,ϕ =(ϕ ) ) which has the form of (16) with
n,x n x n,xx n xx
ψ =ϕ ϕ ,
n
−
F = [ϕ]+ [ϕ ]+2(φ φ )ϕ .
n n,x x n,xx
−Q Q −
Applying (37) gives
[ϕ] [ϕ ] C( ϕ + ϕ ) ψ . (44)
n H3 n H3 x
kQ −Q k≤ k k k k k k
8 A.MORANDO,P.SECCHI,ANDP.TREBESCHI
We also have
2(φn,x φx)ϕn,xx 2 φn,x φx ϕn,xx L∞ C ψx ϕn H3. (45)
k − k≤ k − kk k ≤ k kk k
Thus, from (44), (45) we obtain
F C( ϕ H3 + ϕn H3) ψx . (46)
k k≤ k k k k k k
From Lemma 3, (21), (46) we get
d
C( ϕ + ϕ ) +C( ϕ + ϕ ) ψ C , (47)
dtE ≤ k tkH2 k kH3 E k kH3 k nkH3 k xk≤ E
wherewehaveusedthe uniformboundedness(43)forϕ andϕ. ApplyingGronwall’slemmato(47)and
n
using (9) again yields
ψ (t) 2+ ψ (t) 2 CeCT ψ (0) 2+ ψ (0) 2 t I,
t x t x
k k k k ≤ k k k k ∈
that is (cid:0) (cid:1)
(ϕ ϕ ) (t) 2+ (ϕ ϕ ) (t) 2 CeCT ϕ(1) ϕ(1) 2+ (ϕ(0) ϕ(0)) 2 t I,
k − n t k k − n x k ≤ k − n k k − n xk ∈
(cid:16) (cid:17)
which gives the strong convergence of ϕ to ϕ in C(I;H1) C1(I;L2), when passing to the limit as
n
∩
n + . Recallthat,sinceweareworkingwithfunctionswithzerospatialmean,thePoincar´einequality
→ ∞
(15) holds. By interpolation and the uniform boundedness (43) we get
(ϕ ϕ ) (t) 2 + (ϕ ϕ )(t) 2
k − n t kHs−2 k − n kHs−1
2
C (ϕ ϕ ) (t) 1−1/(s−1) (ϕ ϕ ) (t) 1/(s−1)+ (ϕ ϕ )(t) 1−1/(s−1) (ϕ ϕ )(t) 1/(s−1)
≤ k − n t kHs−1 k − n t kL2 k − n kHs k − n kH1
(cid:16) (cid:17)
C (ϕ ϕ ) (t) 2/(s−1)+ (ϕ ϕ )(t) 2/(s−1) , t I.
≤ k − n t kL2 k − n kH1 ∈
Finally, passing to the limit as n + in(cid:16)the above inequality gives the thesis. (cid:17) (cid:3)
→ ∞
Remark 9. Obviously, by a similar argument with a finer interpolation we could prove the strong con-
vergence of ϕ to ϕ in C(I;Hs−ε) C1(I;Hs−1−ε), for all small enough ε>0. However, this is useless
n
∩
for the following argument.
Now we take one spatial derivative of (3) and get
(ϕ ) (µ 2φ )(ϕ ) = [ϕ] 2φ ϕ ,
x tt− − x x xx −Q x− xx xx
which has the form of (16) with
ψ =ϕ , F = [ϕ] 2φ ϕ .
x −Q x− xx xx
Using Proposition 5, a Moser-type estimate, the Sobolev imbedding and (19) we compute
F Hs−2 [ϕ] Hs−1 +2 φxxϕxx Hs−2
k k ≤kQ k k k
≤C(kϕxkH2kϕxkHs−1 +kφxxkL∞kϕxxkHs−2 +kφxxkHs−2kϕxxkL∞) (48)
C ϕ ϕ ,
H3 Hs
≤ k k k k
and we deduce that F L∞(I;Hs−2). Moreover,the initial values of ϕ , (ϕ ) are ϕ(0) Hs−1, ϕ(1)
x x t x x
∈ ∈ ∈
Hs−2, respectively.
We are going to introduce regularized approximations of the data ϕ(0), ϕ(1), F. Given any ε > 0, let
x x
us take functions Ψ(0) Hs,Ψ(1) Hs−1 with zero mean, and Fε L2(I;Hs−1) such that
ε ε
∈ ∈ ∈
kΨ(ε0)−ϕ(x0)kHs−1 +kΨ(ε1)−ϕ(x1)kHs−2 +kFε−FkL2(I;Hs−2) ≤ε. (49)
Let us consider the Cauchy problem with regularized data
Ψε (µ 2φ )Ψε =Fε, t I, x T,
tt− − x xx ∈ ∈ (50)
(Ψε|t=0 =Ψ(ε0), ∂tΨε|t=0 =Ψ(ε1) x∈T.
Again this problem has the form (16).
DATA DEPENDENCE OF APPROXIMATE CURRENT-VORTEX SHEETS 9
Lemma 10. Let s 3. For any ε>0, the Cauchy problem (50) has a unique solution Ψε C(I;Hs)
≥ ∈ ∩
C1(I;Hs−1) and
Ψε 2 + Ψε 2
k kC(I;Hs) k tkC(I;Hs−1)
≤CeCTmaxτ∈I(1+kϕtkH2+kϕkHs) kΨ(ε1)k2Hs−1 +(µ+2kϕ(0)kH2)kΨ(ε0)k2Hs +kFεk2L2(I;Hs−1) . (51)
n o
Proof. The result follows from Lemma 4 with r=s. (cid:3)
Now we estimate the difference Ψε ϕ , which solves the linear problem
x
−
(Ψε ϕ ) (µ 2φ )(Ψε ϕ ) =Fε F, t I, x T,
x tt x x xx
− − − − − ∈ ∈ (52)
((Ψε ϕx)|t=0 =Ψ(ε0) ϕ(x0), ∂t(Ψε ϕx)|t=0 =Ψ(ε1) ϕ(x1) x T.
− − − − ∈
Lemma 11. Let s 3. For any ε>0, the difference Ψε ϕ satisfies the estimate
x
≥ −
k(Ψε−ϕx)(t)kHs−1 +k(Ψε−ϕx)t(t)kHs−2 ≤Cε ∀t∈I. (53)
Proof. We apply Lemma 4 with r =s 1 and get
−
Ψε ϕ 2 + (Ψε ϕ ) 2
k − xkC(I;Hs−1) k − x tkC(I;Hs−2)
≤CeCTmaxτ∈I(1+kϕtkH2+kϕkHs) kΨ(ε1)−ϕ(x1)k2Hs−2
n
+(µ+2 ϕ(0) ) Ψ(0) ϕ(0) 2 + Fε F 2 . (54)
k kH2 k ε − x kHs−1 k − kL2(I;Hs−2)
Thus (53) follows from (49). o (cid:3)
Finally we estimate the difference between ϕ =(ϕ ) and Ψε. The difference of the corresponding
n,x n x
problems reads
(Ψε ϕ ) (µ 2φ )(Ψε ϕ ) =Gn,ε, t I, x T,
n,x tt n,x n,x xx
− − − − ∈ ∈
(Ψε ϕ ) =Ψ(0) ϕ(0), (55)
 n,x |t=0 ε n,x
− −
∂t(Ψε ϕn,x)|t=0 =Ψ(ε1) ϕ(n1,)x x T.
− − ∈
where we have set
Gn,ε =Fε Fn+2(φ φ )Ψε , Fn = [ϕ ] 2φ ϕ , (56)
− n,x− x xx −Q n x− n,xx n,xx
ϕ(0) =(ϕ(0)) , ϕ(1) =(ϕ(1)) .
n,x n x n,x n x
Lemma12. Lets 3. Foranyε>0,thereexistsM(ε)>0suchthat,for anynthedifferenceΨε ϕ
n,x
≥ −
satisfies the estimate
(Ψε ϕ )(t) 2 + (Ψε ϕ ) (t) 2
k − n,x kHs−1 k − n,x t kHs−2
t
C ε2+ ϕ(1) ϕ(1) 2 + ϕ(0) ϕ(0) 2 + ϕ ϕ 2 dτ +TM(ε) ϕ ϕ 2
≤ k − n kHs−1 k − n kHs k − nkHs k n− kC(I;Hs−1)
n (cid:12)(cid:12)Z0 (cid:12)(cid:12) (o57)
(cid:12) (cid:12)
(cid:12) (cid:12)
for any t I.
∈
Proof. We apply Lemma 4 with r =s 1 and obtain for every t I
− ∈
(Ψε ϕ )(t) 2 + (Ψε ϕ ) (t) 2
k − n,x kHs−1 k − n,x t kHs−2
≤CeCTmaxτ∈I(1+kϕn,tkH2+kϕnkHs) kΨ(ε1)−ϕ(n1,)xk2Hs−2
n
+(µ+2 ϕ(0) ) Ψ(0) ϕ(0) 2 + Gn,ε 2 . (58)
k n kH2 k ε − n,xkHs−1 k kL2(0,t;Hs−2)
o
10 A.MORANDO,P.SECCHI,ANDP.TREBESCHI
First of all, recalling the uniform boundedness w.r.t. n (43), we may write (58) as
(Ψε ϕ )(t) 2 + (Ψε ϕ ) (t) 2
k − n,x kHs−1 k − n,x t kHs−2
C Ψ(1) ϕ(1) 2 + Ψ(0) ϕ(0) 2 + Gn,ε 2 . (59)
≤ k ε − n,xkHs−2 k ε − n,xkHs−1 k kL2(0,t;Hs−2)
n o
From the definition in (56) we have, for every τ [0,t],
∈
kGn,εkHs−2 ≤kFε−FnkHs−2 +2k(φn,x−φx)ΨεxxkHs−2
≤kFε−FkHs−2 +kF −FnkHs−2 +Ckϕn−ϕkHs−1kΨεkHs.
Integrating this inequality in τ between 0 and t gives
t t t
Gn,ε 2 dτ 3 Fε F 2 dτ +3 F Fn 2 dτ +CTM(ε) ϕ ϕ 2
k kHs−2 ≤ k − kHs−2 k − kHs−2 k n− kC(I;Hs−1)
(cid:12)Z0 (cid:12) (cid:12)Z0 (cid:12) (cid:12)Z0 (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (60)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
where we have denoted
M(ε):=CeCTmaxτ∈I(1+kϕtkH2+kϕkHs) kΨ(ε1)k2Hs−1 +(µ+2kϕ(0)kH2)kΨ(ε0)k2Hs +kFεk2L2(I;Hs−1) ,
n o
that is the right-hand side of (51). On the other hand, for all τ we have
kF −FnkHs−2 ≤kQ[ϕ]−Q[ϕn]kHs−1 +2kφxxϕxx−φn,xxϕn,xxkHs−2. (61)
From (42) we have
kQ[ϕ]−Q[ϕn]kHs−1 ≤C(kϕkHs +kϕnkHs)kϕ−ϕnkHs. (62)
Moreover,since Hs−2 is an algebra we can estimate
2kφxxϕxx−φn,xxϕn,xxkHs−2
≤Ckφxx−φn,xxkHs−2kϕxxkHs−2 +Ckφn,xxkHs−2kϕxx−ϕn,xxkHs−2
C( ϕ + ϕ ) ϕ ϕ . (63)
Hs n Hs n Hs
≤ k k k k k − k
From (61)–(63) and the uniform boundedness (43) we obtain
kF −FnkHs−2 ≤Ckϕ−ϕnkHs ∀τ ∈I, (64)
and substituting it in (60) gives
t t t
Gn,ε 2 dτ 3 Fε F 2 dτ +C ϕ ϕ 2 dτ +CTM(ε) ϕ ϕ 2 .
k kHs−2 ≤ k − kHs−2 k − nkHs k n− kC(I;Hs−1)
(cid:12)Z0 (cid:12) (cid:12)Z0 (cid:12) (cid:12)Z0 (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (65)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
Finally, from (49), (59), (65) we get
(Ψε ϕ )(t) 2 + (Ψε ϕ ) (t) 2
k − n,x kHs−1 k − n,x t kHs−2
t
C ε2+ ϕ(1) ϕ(1) 2 + ϕ(0) ϕ(0) 2 + ϕ ϕ 2 dτ +TM(ε) ϕ ϕ 2
≤ k − n kHs−1 k − n kHs k − nkHs k n− kC(I;Hs−1)
for all tn I, that is (57). (cid:12)(cid:12)(cid:12)Z0 (cid:12)(cid:12)(cid:12) o(cid:3)
∈ (cid:12) (cid:12)
Proof of Theorem 2. Adding (53), (57), and applying the Poincar´einequality gives
(ϕ ϕ )(t) 2 + (ϕ ϕ ) (t) 2
k − n kHs k − n t kHs−1
t
C ε2+ ϕ(1) ϕ(1) 2 + ϕ(0) ϕ(0) 2 + ϕ ϕ 2 dτ +TM(ε) ϕ ϕ 2
≤ k − n kHs−1 k − n kHs k − nkHs k n− kC(I;Hs−1)
n (cid:12)(cid:12)Z0 (cid:12)(cid:12) o
(cid:12) (cid:12)
(cid:12) (cid:12)