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Pooley, Benjamin C. and Robinson, James C.. (2016) An Eulerian–Lagrangian form for the
Euler equations in Sobolev spaces. Journal of Mathematical Fluid Mechanics .
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AN EULERIAN-LAGRANGIAN FORM FOR THE EULER EQUATIONS IN SOBOLEV SPACES
BENJAMIN C. POOLEY AND JAMES C. ROBINSON
ABSTRACT. In 2000 Constantin showed that the incompressible Euler equations can be written in an “Eulerian-Lagrangian” form which involves the back-to-labels map (the in-verse of the trajectory map for each fixed time). In the same paper a local existence result is proved in certain H¨older spacesC1,µ.
We review the Eulerian-Lagrangian formulation of the equations and prove that given initial data inHsforn≥2ands > n
2 + 1, a unique local-in-time solution exists on then-torus that is continuous intoHsandC1intoHs−1. These solutions automatically haveC1trajectories.
The proof here is direct and does not appeal to results already known about the classical formulation. Moreover, these solutions are regular enough that the classical and Eulerian-Lagrangian formulations are equivalent, therefore what we present amounts to an alterna-tive approach to some of the standard theory.
1. INTRODUCTION
We study a reformulation (following Constantin [2]) of the incompressible Euler equa-tions on a domainTn :=
Rn/2πZnin the absence of external forcing. The Euler equations
model the flow of an incompressible inviscid fluid and are (classically) formulated in terms of a divergence-free vector fieldu(i.e.∇ ·u= 0) as follows:
(1) ∂u
∂t + (u· ∇)u+∇p= 0
wherepis a scalar potential representing internal pressure (as opposed to physical pressure at a boundary). The divergence-free condition reflects the incompressibility constraint.
In two and particularly in three dimensions, these equations continue to be of great in-terest; some recent surveys include [5, 8, 18]. As an illustration of the challenge posed by these equations we note that unlike the Navier–Stokes equations where global weak solutions have been known to exist since 1934 due to Leray [12], existence of global weak solutions of the Euler equations (on periodic domains) was not proved until 2011 by Wiede-mann [17], following the work of DeLellis and Sz´ekelyhidi [7]. On the spatial domainR3,
more regular local solutions (u∈C0([0, T];Hs)∩C1([0, T];Hs−1)withs >5/2) have been known to exist since the 1970s due to Kato et al, see for example [10, 11].
In the study of the Navier–Stokes equations, results such as those found in [15] motivate us to approach the classical equations of fluid mechanics from a more Lagrangian view-point. In that paper, Robinson and Sadowski show that ifuis a suitable weak solution of the Navier–Stokes equations in 3D in the sense of Caffarelli, Kohn and Nirenberg [1], then almost every particle trajectory is unique andC1in time. The arguments there are based
on the fact that almost all trajectories avoid the set of points(x, t)where singularities could develop using the fact that the set of such points has box-counting dimension at most5/3. Constantin has studied a form for the Euler equations that involves both the classical velocity field and the so called back-to-labels mapAwhich is defined to be the inverse of
Date: October 29, 2014.
2010Mathematics Subject Classification. Primary: 76B03, 35Q35. Secondary: 35Q31. Key words and phrases. Euler equations, Eulerian-Lagrangian formulation.
the trajectory mapXat each timet. More precisely, for an evolving vector fieldudefined onTn×[0, T], the trajectory map solves
(2)
dX
dt (y, t) =u(X(y, t), t)
X(y,0) =y
for eachy ∈ Tn. Ifuis divergence-free and sufficiently regular thenX is well defined
andX(·, t)is bijective for eacht. In this case we can define the back-to-labels mapAby setting
(3) A(·, t) :=X−1(·, t),
where we considerX as a mapX(·, t) :Tn →
Tnfor eacht∈[0, T]. For the
Eulerian-Lagrangianform, as we shall continue to call it, Constantin [2] proved local existence and uniqueness results in certain H¨older spaces onR3for solutions that are periodic, or satisfy
suitable decay conditions.
As Yudovich [18] has noted, a similar combination of Eulerian and Lagrangian ap-proaches was used to investigate the Euler equations in H¨older spaces, by G¨unther and Lichtenstein independently, as early as the 1920s ([13], [9]).
First we will review the Eulerian-Lagrangian formulation and discuss how it is formally equivalent to the usual Euler equations. We then turn to the main topic of this paper which is the proof of an existence and uniqueness result for the Eulerian-Lagrangian formulation inC0([0, T];Hs(Tn))withs > n2 + 1in dimensionn≥2. The proof is self contained,
in the sense that it neither appeals to results about the classical Euler equations, nor to the problem in H¨older spaces.
2. THEEULERIAN-LAGRANGIAN FORM OF THE EQUATIONS
The Eulerian-Lagrangian form of the Euler equations comprises the following system:
(4) ∂tA+ (u· ∇)A= 0,
(5) u=P((∇A)∗v),
(6) ∂tv+ (u· ∇)v= 0.
Given an initial divergence-free velocityu0for the classical equations, we choose initial conditions for the above system as follows:
(7) A(x,0) =x,
(8) u(x,0) =v(x,0) =u0(x).
We use the notationPfor the Leray projector onto the space of divergence-free functions.
For a matrixM,M∗denotes the transposed matrix. The vector fieldvis called thevirtual velocityand represents the initial velocity transported by the flow.
It will often be convenient to treatAas a perturbation of the identity map onTn. In this
case we use the notationη(x, t) :=A(x, t)−xand replace (4) and (7) with the equations (9) ∂tη+ (u· ∇)η+u= 0, η(x,0) = 0
respectively. We do this because the identity map (hence A) does not have sufficient Sobolev regularity when considered as a function on the torus with values inRn(i.e.
with-out accounting for the topology of the target torus ).
Proposition 1. Letn≥2, consideru∈C1((0, T)×
Tn), withu(0)∈ C1(Tn). Ifuis
divergence-free and satisfies (1) for some p, with spatially periodic boundary conditions thenA∈C1((0, T)×
Tn;Tn)andusatisfies (5) withv(x, t) =u0(A(x, t)).
Proof. From the regularity assumptions on uand periodicity of the domain we deduce
that the trajectoriesX(y,·) ∈ C2(0, T)and∇X(y,·) ∈ C1(0, T)for ally ∈
Tn, we
also haveX,∂X∂t ∈C1((0, T)×Tn). It follows from the divergence-free condition that
det∇X ≡ 1, so X is volume preserving and locally injective, hence bijective, given that Tn has finite volume. By the inverse function theorem we see thatA exists and is
an element ofC1((0, T)×Tn). We now have enough regularity to make the following
calculations rigorous.
From (1) and (2) we obtain
∂2X
∂t2 (y, t) =−∇p(X(y, t), t),
which is of course just a Lagrangian interpretation of the Euler equations. Settingp˜(y, t) =
p(X(y, t), t)this becomes
∂2X
∂t2 =−((∇X)
∗)−1
∇p˜(y, t).
Multiplying through by(∇X)∗and changing the order of differentiation yields
(10) ∂
∂t ∂Xj
∂t ∂Xj
∂yi
= ∂
∂yi "
−p˜+1 2
∂X ∂t
2#
fori= 1, . . . , n, where there is an implicit sum overj = 1, . . . , nandXj,yidenote the
components inRn ofX,yrespectively. Integrating (10) in time, multiplying the
corre-sponding vector equation by(∇A)∗and evaluating atA(x, t)gives
(11) u(x, t) =∂X
∂t (A(x, t), t) = (∇A)
∗u0(A(x, t))− ∇n
where
n(x, t) =
Z t
0
˜
p(A(x, t), s)−1
2
∂X
∂t (A(x, t), s) 2
ds.
As gradients lie in the kernel of the Leray projector, applyingPto (11) shows thatusatisfies
(5) as required. Note thatv(x, t) =u0(A(x, t))satisfies (6), hence solutions to the Euler equations indeed solve the Eulerian-Lagrangian form.
The converse is a little more technical.
Proposition 2. Lets > n2 + 1andu,v,η ∈ C0([0, T];Hs)∩C1([0, T];Hs−1)satisfy
(5), (6), (8) and (9). Then for somep∈C0([0, T];Hs)usolves (1).
Proof. SinceHs−1(
Tn) ,→ L∞(Tn)is an algebra, we have that iff, g ∈ Hs−1 (scalar
valued) then
∂xi(f g) = (∂xif)g+f(∂xig)
as an equlity ofL2functions, fori= 1,2, . . . , n. Therefore, denoting the material
deriva-tive byDt:=∂t+ (u· ∇), forf, g∈C0([0, T];Hs−1)∩C1([0, T];Hs−2)we have
(12) Dt(f g) = (Dtf)g+f(Dtg).
Moreover, iff ∈Hs,
(u· ∇)∇f =∇((u· ∇)f)−(∇u)∗∇f.
Hence the classical commutation relation
(13) Dt∇f =∇Dtf−(∇u)∗∇f
Sinceusatisfies (5), we may write
(14) u(x, t) =v+ (∇η)∗v− ∇n
for some real-valuedn. Then by (12) and (13) the following calculations are justified:
(15)
Dtu= Dtv+ (Dt∇η)∗v+ (∇η)∗Dtv−Dt∇n
= (∇Dtη)∗v−(∇u)∗(∇η)∗v− ∇Dtn+ (∇u)∗∇n
=−(∇u)∗[v+ (∇η)∗v− ∇n]− ∇Dtn
=−(∇u)∗u− ∇Dtn
=−∇p
wherep= 12|u|2+ Dtn.
3. ANEXISTENCE ANDUNIQUENESSTHEOREM
Forr≥0, we will use the notationHrvariously for scalar or vector valued functions
inHr(
Tn)(componentwise), where this does not cause ambiguity. We will often consider
functions in spaces of the formC0([0, T]; (Hs(
Tn))n). To simplify notation we define
Σs(T)(usually denotedΣs) forT ≥0ands≥0by
Σs(T) :=C0([0, T]; (Hs(Tn))n).
We consider the natural norm onΣs:
kukΣs = sup
t∈[0,T]
ku(t)kHs.
The aim of the rest of this paper is to prove the following theorem.
Theorem 1. Ifn≥2,s > n2+ 1andu0∈Hsis divergence free then there existsT >0, such that the system (4–6) with initial conditions (7) and (8) has a unique solutionA, u, v
such that η, u, v ∈ Σs(T)∩C1([0, T];Hs−1)whereη(x, t) = A(x, t)−x. Moreover A∈C1([0, T]×Tn)as a map into the torus.
We will prove this by constructing a contracting iteration scheme using the equations (5),(6) and (9). More precisely, givenu∈Σs(T)we findv, η ∈Σs∩C1([0, T]×Tn),
solutions of
∂tη+ (u· ∇)η=−u, η(0, x) = 0
and
∂tv+ (u· ∇)v= 0, v(0, x) =u0(x).
We then construct the next iterate ofu, using
u0=P[(∇A)∗v]
and show thatu7→u0is a contraction on a certain subset ofΣs.
In the case of H¨older spaces, Constantin constructed an iteration scheme that was in-stead a contraction with respect toA. This involves controlling differences between can-didate virtual velocities (v1andv2, say) in terms of the difference between the respective
back-to-labels maps (A1andA2). This can be achieved, using the fact thatvi =u0(Ai)
is a solution to (6). In the H¨older setting this is a natural way to proceed, however, relying on thisa posterioriknowledge about the solution introduces an extra technicality when we work in Sobolev spaces. For this reason we will proceed as described above, relying only ona prioriestimates. Following the proof, we shall see how the argument differs if the contraction is with respect toA, in particular we get an alternative proof under the additional assumption thats∈Z.
Lemma 1. Fors > n2 there exists C1 > 0 such that ifu ∈ Hsand v ∈ Hs+1 then B(u, v)∈Hsand
(16) kB(u, v)kHs ≤C1kukHskvkHs+1.
This is really just the fact that Hs is a Banach algebra. For the second lemma the
assumption thatuis divergence-free allows us to “save a derivative” by means of the iden-tities
(B(u,(−∆)r/2v),(−∆)r/2v)L2 = 0
forr∈[0, s].
Lemma 2. Ifs > n2 + 1there existsC2 >0 such that foru ∈ Hs,v ∈ Hs+1withu
divergence-free we have
(17) |(B(u, v), v)Hs| ≤C2kukHskvk2Hs.
We use the following shorthand for closed balls inΣs: BM =Bk·kΣs(0, M),
i.e. BM is the closed unit ball centred at the origin of radiusM > 0with respect to the
normk · kΣs. Where ambiguity could arise we writeBM(T)for the closed ball inΣs(T). Lemma 3. Ifs > n
2 + 1and η, v ∈ Σs(T)thenP[(∇η)
∗v] ∈ Σ
s and there exists a
constantC3>0(independent ofη,v,tandT) such that for fixedt,
(18) kP[(∇η)∗v]kHr ≤C3kηkHskvkHr,
wherer = sorr = s−1. Furthermore, there existsC30 > 0 such that for anyM >
0 and T > 0, the following bounds hold uniformly with respect tot ∈ [0, T] for any
η1, η2, v1, v2∈BM(T):
(19) kP[(∇η1)∗v1−(∇η2)∗v2]kX≤C30M(kη1−η2kX+kv1−v2kX).
whereXisL2(
Tn)orHs−1.
Proof. For continuity intoHs−1 we use the fact that Hs−1 is a Banach algebra. More
precisely, we see that
(20) kP[(∇η1)
∗v1−(∇η2)∗v2]k
Hs−1≤Ckη1−η2kHskv1+v2kHs−1
+Ck∇η1+∇η2kHs−1kv1−v2kHs−1,
whereC >0is independent of theηiandvi. The key step in the proof of (18) whenr=s
is that ifη, v∈C2then for someq∈Hs, ∂xiP[(∇η)
∗v] =∂
xi(∂xjηkvk)−∂xi∂xjq
=∂xj(∂xiηkvk)−∂xiηk∂xjvk+∂xjηk∂xivk−∂xi∂xjq
where sums are taken implicitly overk. The left-hand side is already divergence-free so projecting again removes the gradient terms and yields
(21) ∂xiP[(∇η) ∗v] =
P[(∇η)∗∂xiv−(∇v) ∗∂
xiη].
By continuity, this still holds if we only have η, v ∈ Hs. A calculation similar to (20) applied to (21) yields continuity with respect to theHsnorm as claimed.
The inequalities (18) forr=s−1andr=sare obtained by taking theHs−1norms
ofP[(∇η)∗v]and (21) respectively.
To prove (19), we again use the fact thatPremoves gradients. Indeed forf,g ∈ Hs,
we have
(22) P((∇f)∗g) =P(∇(f·g)−(∇g)∗f) =−P((∇g)∗f).
The next lemma gives uniform bounds on theHsnorms of solutions to the transport
equations (4) and (6). We will consider the following system:
(23)
∂tf+ (u· ∇)f =g f(0) =f0
wheref, g: [0, T]×Tn→Rnanduis divergence free.
Lemma 4. Lets >n2 + 1and fixf0∈Hs,g∈Σ
s. Ifu∈Σsis non-zero and divergence
free then there exists a unique solutionf to (23). Furthermore, the solution f ∈ Σs∩
C1([0, T];Hs−1)∩C1([0, T]×Tn)and there existsC4>0(from Lemma 2) such that if
r, t∈[0, T]we have:
(24) kf(t)kHs≤
kf(r)kHs+
kgkΣs
C4kukΣs
exp(C4|t−r|kukΣs)−
kgkΣs
C4kukΣs
.
Proof. By the method of characteristics we obtain a solutionf ∈ C1([0, T]×
Tn). The
formal argument that follows motivates our consideration of the regularity off. Taking the
Hsproduct of (23) withfyields
1 2
d dtkfk
2
Hs =−(B(u, f), f)Hs+ (f, g)Hs.
By Lemma 2, there existsC >0such that for allt∈[0, T],
(25) 1 2
d dtkf(t)k
2
Hs ≤Cku(t)kHskf(t)kH2s+kg(t)kHskf(t)kHs.
Now (24) follows from Gronwall’s inequality. In the caser > t, this argument is applied to the time-reversed equation, that is, using the fact that for fixedr,−f(r−t)is transported by−u(r−t)with forcingg(r−t).
To properly justify this we can proceed by a Galerkin method. For eachN ∈Nwe find a solution to the system
(26)
∂tfN+PNB(uN, fN) =gN fN(r) =PNf(r),
on[r, T], wherePN denotes truncation up to Fourier modes of orderN (in space),uN :=
PNuandgN :=PNg. The estimate (24) applies tofN so by a standard argument using the Aubin-Lions lemma we obtain a weak solution h ∈ L∞(r, T;Hs)such that∂
th ∈ L∞(r, T;Hs−1), henceh ∈ C0([0, T];Hs−1). Using the divergence free property we
obtain uniqueness of solutionsh∈L2(r, T;H1)with time derivative∂
th∈L2(r, T;L2).
Indeed, ifhand˜hare two such solutions it follows from (23) that d
dskh−
˜
hk2 L2 = 0.
Thereforef =h, i.e. this weak solution agrees with ourC1classical solution on[r, T].
We now prove (24) in the caser≤t. SincefN →finL2(r, T;Hs−1), we may choose
a dense countable subset{tk}∞
k=1⊂[r, T]such thatfN(tk)→f(tk)inHs−1asN→ ∞
for eachk. The formal argument above is valid on the truncated system, thus
(27) kfN(tk)kHs ≤
kPNf(r)kHs+
kgkΣs
CkuNkΣs
exp(C|tk−r|kukΣs)−
kgNkΣs
CkukΣs
.
Hence, passing to a subsequence offN for eachkwith a diagonalisation argument, we may assume that for allk,fN(tk)converges weakly inHsasN → ∞. Moreover, by the choice of the pointstkand uniqueness of weak limits, we must havefN(tk)* f(tk)in
Hs. Taking thelim infof (27) with respect toN → ∞yields
(28) kf(tk)kHs ≤
kf(r)kHs+
kgkΣs
CkukΣs
exp(C|tk−r|kukΣs)−
kgkΣs
CkukΣs
To prove (24) and the weak continuity of f intoHs we will use the fact that a weakly
convergent sequence inHs−1that is also bounded inHsmust converge weakly inHsto
the same limit by the Banach–Alaoglu theorem. Indeed ifxk * xinHs−1is bounded in Hsthen any subsequence admits a further subsequence converging weakly inHstoxby
the uniqueness of weak limits.
From this, (24) follows by the density of{tk}and the continuity offintoHs−1. Indeed,
in the caset ≥ r, for any subsequence(tk`)∞`=1 ⊂ (tk)∞k=1 such thattk` → twe have
f(tk`)* f(t)inHs. Applying (28) attk` and taking thelim infas`→ ∞yeilds (24) at timet. Fort < rthe required bounds are obtained in the same way from the time-reversed version of (26).
We have shown that kf(t)kHs in bounded uniformly, not merely almost everywhere.
Therefore for any fixedτ ∈[0, T]and any sequence{τk} ⊂ [0, T]such thatτk →τ we
deduce, by the continuity intoHs−1, thatf(τ
k)* f(τ)inHs. This says thatf is weakly
continuous intoHs.
To see thatf ∈Σsit is therefore enough to show thatkf(t)kHs is continuous. This is
the case since for allr, t∈[0, T], (24) gives bounds of the form
(kf(r)kHs+α)e−β|t−r|−α≤ kf(t)kHs≤(kf(r)kHs+α)eβ|t−r|−α
for time independent constantsα, β >0, where the first inequality comes from (24) with
randtinterchanged.
The fact thatf ∈C1([0, T];Hs−1)follows from the fact that∂
tf ∈Σs−1which can
be seen from the regularity of the other terms in (23).
Lemma 5. Fors > n/2 + 1fixu1,u2∈Σsandf0∈Hs. Letg1=g2= 0orgi=−ui
fori= 1,2. Iff1,f2are the solutions of (23) corresponding tou1,u2,g1,g2respectively, then in the case thatg1=g2= 0, there existsC5>0depending only onssuch that
(29) kf1(t)−f2(t)kL2 ≤C5kf1+f2kΣsku1−u2kΣ0t
for allt∈[0, T]. In the case thatgi=−uifori= 1,2we instead have (30) kf1(t)−f2(t)kL2 ≤(C5kf1+f2kΣs+ 1)ku1−u2kΣ0t
Proof. Using the anti-symmetry of(B(u1−u2,·),·)L2 we have, fort∈[0, T],
d
dtkf1−f2k 2
L2≤ |(B(u1−u2, f1+f2), f1−f2)L2|+ 2|(g1−g2, f1−f2)|
≤Ckf1+f2kHsku1−u2kL2kf1−f2kL2+ 2kg1−g2kΣ0kf1−f2kL2
≤Ckf1+f2kΣsku1−u2kΣ0kf1−f2kL2+ 2kg1−g2kΣ0kf1−f2kL2
WhereCdepends on the embeddingHs−1,→L∞. Formally dividing bykf1−f2k
L2and
integrating the resulting inequality gives (29) or (30) depending on the choice ofg1andg2. Justifying this last step is straightforward.
We are now in a position to prove the main result.
Proof of Theorem 1. Fixs > n/2 + 1and letC3,C4be the constants in (18), (24) (from
Lemmas 3 and 4) respectively. FixM >ku0kHsandT >0so that
exp(C4T M)ku0kHs
C 3
C4[exp(C4T M)−1] + 1
≤M.
Letu∈BM(T)be a divergence free function and letηbe the solution of (23) for the flow
uwith initial dataη0= 0and forcingg=u. Letvbe the solution for initial datav0=u0
withg= 0. DefineSu:=P[(∇η)∗v+v], then by Lemmas 3 and 4,
(31) kSu(t)kHs ≤exp(C4tM)ku0kHs
C 3
C4[exp(C4tM)−1] + 1
for all t ∈ [0, T]. HenceS : BM(T) → BM(T). Note thatSu(·,0) = u0 even if u(·,0)6=u0.
We next show thatS is a contraction onBM(T)in the L2 norm ifT is sufficiently
small. For u1,u2 ∈ BM(T)we constructvi andηi fromui as above fori = 1,2with
v1(·,0) =v2(·,0) =u0. Now
(32)
kSu1−Su2kL2 ≤Cakη1−η2kL2+Cbkv1−v2kL2
≤(Cckv1+v2kΣs+Cdkη1+η2kΣs+Ce)Tku1−u2kΣ0
≤C(u0, M, T)ku1−u2kΣ0,
whereCa, . . . , Cedenote various constants arising from the application of Lemmas 3, 4 and 5. Keeping careful track of the constants shows that C(u0, M, T) is given by the formula
(33)
C(u0, M, T) := 2T
C5(C30M+ 1)ku0kHs+
C30C5M C4
exp(C4T M)
+C30M 1
2−
C5 C4
Where C30,C4,C5 are the constants from Lemmas 3, 4 and 5 respectively. Taking the supremum of (32) with respect totand choosingT >0small enough, we see thatSis a contraction in the required sense.
We conclude that S has a unique accumulation point u, in the closure of BM with respect tok · kΣ0. SinceBM(T)is convex and closed inΣsit is weakly closed, henceu∈ BM(T)is a fixed point ofS. A fixed point ofS, along with associated back-to-labels map and virtual velocity, clearly give a solution to the Eulerian-Lagrangian formulation of the Euler equations with the required regularity. The contraction argument gives uniqueness inBM(T)and it remains to prove that we have uniqueness inΣs(T).
Since S is a contraction on BM(Te) for anyTe ∈ (0, T], we have by continuity of
ku(t)kHs, that ifu0,A0andv0also satisfy (4–6) withu0∈Σs(T), thenu(t) =u0(t)when
0≤t≤min(T,inf{r:ku0(r)kHs=M}).
Now we know that for allk ∈Nthere existsTk ≤ T such thatS is a contraction on
BM+1/k(Tk)and we may assumeTk →T ask→ ∞. By the previous observation, this
means thatuis the unique solution inΣs(T−ε)for allε >0, hence by continuityuis the
unique solution inΣsas required.
The proof thatu∈C1([0, T];Hs−1)uses the same trick as Lemma 3 to save a spatial derivative (we have only shown that∇ηt∈Hs−2, which might otherwise limit the regu-larity ofu). By definitionu=P[(∇η)∗v+v]. We use (22) from the proof of Lemma 3.
Precisely we have
1
hku(t+h)−u(t)−hP[(∇η(t))
∗∂tv(t) +∂tv(t) + (∇v(t))∗∂tη(t)]k
Hs−1
≤ 1
2hkP[(∇η(t+h) +∇η(t))
∗(v(t+h)−v(t)−h∂tv)]k
Hs−1
+ 1
2hkP[(∇v(t+h) +∇v(t))
∗(η(t+h)−η(t)−h∂tη)]k
Hs−1
+1
2kP[(∇η(t+h)− ∇η(t))
∗∂tv(t)]k
Hs−1
+1
2kP[(∇v(t+h)− ∇v(t))
∗∂
tη(t)]kHs−1
+ 1
SinceHs−1is an algebra andη, v ∈ C0([0, T];Hs)∩C1([0, T];Hs−1), the right-hand
side vanishes ash→0. Thereforeu∈C1([0, T];Hs−1)and
∂tu=P[(∇η(t))∗∂tv(t) +∂tv(t)(∇v(t))∗∂tη(t)].
4. ANALTERNATIVEITERATION
Here we exhibit an alternative proof of existence and uniqueness for (4–6), which is based on contractions with respect toArather than u. The extra technicality in this ap-proach is contained in the following lemma, which is proved in an appendix. We will denote the identity map onTnbyιand use the correspondence between mapsTn →Rn
andTn→Tnwithout comment.
Lemma 6. Lets∈Zwiths > n2 + 1and fixf, g∈H
s. Ifg+ιis a volume preserving
map thenf ◦(g+ι)∈Hsand
(34) kf◦(g+ι)kHs≤C6kfkHs(kgkHs+ (2π)n)s
for someC6>0depending only onsand the constants from some Sobolev embeddings.
This allows us to write a second proof of existence and uniqueness of solutions inΣs
fors > n/2 + 1in the cases∈Z.
Fixu0∈HsandM >0and supposeη ∈BM(T)for someT >0such thatη(t) +ι
is volume-preserving for all t ∈ [0, T]. Define uandv viav = u0◦(η+ι)andu =
P[(∇η)∗v+v]. Constructη0, the iterate ofηby solving
∂tη0+ (u· ∇)η0 =−u, η0(x,0) = 0.
By Lemmas 3, 4 and 6 we have
kη0kΣs≤
1
C4[exp(C4C6(C3M + 1)(M+ (2π) n)s
ku0kHsT)−1].
Hence forTsmall enough, we may assumeη0∈BM(T)and since∇ ·u= 0we also have thatη0+ιis volume preserving.
Now suppose thatη1,η2∈BM(T)and letη10,η02be the respective iterates then
kη01−η20kΣ0≤2(C5M+ 1)(C
0
3M+ (C
0
3M + 1)CLip)Tkη1−η2kΣ0,
by Lemmas 3 and 5. HereCLip is the Lipschitz constant ofu0. It follows that, for small enoughT, this iteration procedure is a contraction onBM(T)in theL2norm. Existence
and uniqueness of solutions now follows using the same steps as in the previous method.
5. CONCLUSIONS
We have seen that Constantin’s proof of local well-posedness for the Eulerian-Lagrangian formulation of the Euler equations inC1,µcan be adapted to prove analogous results in the
corresponding Sobolev spaces, Hs for s > n/2 + 1, directly. This involved different
estimates, which may seem more familiar to some readers. We have given two different iteration schemes to deduce well-posedness using these estimates; iterating with respect to
uis natural in this setting and leads to a fairly clean proof, whereas iterating with respect to the Lagrangian coordinateAinvolves estimates on the compositions of Sobolev functions, which are proved in Appendix A. It would be interesting to investigate these composition estimates further and extend them to non-integer Sobolev spaces, for example.
Robinson and Sadowski [15] have shown that in the case of the Navier–Stokes equa-tions, almost every Lagrangian trajectory is well-defined andC1, even for suitable weak solutions. This suggests it may be reasonable to study Eulerian-Lagrangian formulations for diffusive systems. For example, calculations analogous to the derivation above suggest that the Navier–Stokes equations can be formulated as
with
∂tv+ (u· ∇)v−((∇A)∗)−1∆(∇A)∗v= 0,
however obtaining results using such formulations has proved difficult, so far.
Constantin [3, 4] has put forward an Eulerian-Lagrangian form in the viscous case, where diffusive terms appear in the equations for the back-to-labels map and the virtual velocity. Ideally we would be able to make a meaningful study of formulations where the back-to-labels map retains its physical meaning.
Alternatively, if one formally considers the equation satisfied by(∇A)∗v, one arrives at a formulation in magnetization variables. We recently showed that this leads to an interesting model system for Navier–Stokes, which is globally well-posed inH1/2in 3D
[14].
APPENDIXA. COMPOSITIONS INHs
In this appendix we prove Lemma 6, which gives bounds on the compositionsHs
func-tions with certain volume-preserving locallyHsfunctions wheres∈
Zwiths > n2.
To begin with we consider gi ∈ Hsand multi indices βi with|βi| ∈ [1, s] for i =
1, . . . , `. We call p ∈ [1,∞]admissiblefor(βi)1≤i≤` if there exists a constantC > 0
independent of(gi)1≤i≤`such that
(35) ` Y i=1
Dβigi
Lp ≤C ` Y i=1
kgikHs.
Of coursepis admissible if there existq1, . . . , q` ∈[1,∞)such thatHs−|βi| ,→ Lqi for
eachiand
` X i=1 1 qi =1 p,
orp=∞andqi =∞for alli. We may assume, without loss of generality that there are constantsk1andk2with0≤k1≤k2≤`such that
s− |βi| ∈[0, n/2)for1≤i≤k1 s− |βi|=n/2fork1+ 1≤i≤k2 s− |βi|> n/2fork2+ 1≤i≤`
So we have
k1 Y i=1 Dβigi
Lp ≤C k1 Y i=1
kgikHs
for
1
p∈ "k1
X
i=1
n−2(s− |βi|) 2n ,
k1 2 # . Moreover k2 Y
i=k1+1 Dβig
i Lp ≤C k2 Y
i=k1+1
kgikHs
forp∈[2,∞). Lastly,
` Y
i=k2+1 Dβigi
L∞ ≤C ` Y
i=k2+1
kgikHs.
Combining these observations we see thatpis admissible if
(36) 1
p∈ k1 X
i=1
n−2(s− |βi|)
2n , `
2
#
or ifk1=k2thenpis still admissible if (37) 1 p = k1 X i=1
n−2(s− |βi|) 2n ,
furthermorep=∞is admissible ifk1=k2= 0.
Note that ifp∈[1,∞]is admissable andfi :Tn →Rnare linear maps then we have
(rather crudely) (38) ` Y i=1
Dβi(gi+fi)
Lp ≤C ` Y i=1
kgikHs+kfikop(2π)n/qi.
In the proof of the lemma below, we will need the fact that ifs > n2 andP`i=1|βi| ≤
s then p = 2 is admissible for (βi)1≤i≤`. Furthermore, we will need to show that if s > n/2 + 1then there exists an admissiblep > s−n` and thatp = ∞is admissible if
s=` > n/2 + 1.
For the first claim, note that ifk1 = 0 ork1 = 1 thenp = 2is clearly admissible. Otherwise, if1< k1≤`ands > n/2, we have the following calculation:
(39)
k1 X
i=1
n−2(s− |βi|)≤k1n−2k1s+ 2s= (k1−1)(n−2s) +n < n
sop= 2is admissible. For the second claim, observe that ifs > n/2 + 1then
(40)
k1 X
i=1
n−2(s− |βi|)<2 k1 X
i=1
|βi| −2k1≤2(s−k1)−2 ` X
i=k1+1
|βi| ≤2(s−`),
where the middle inequality uses the assumption thatP`i=1|βi| ≤s. Hence there exists an
admissible valuep > n
s−`, ifs−` >0. Ifs=`then necessarily,|βi|= 1fori= 1, . . . , `
hencep=∞is admissible by (37).
Lemma 6. Lets∈Zwiths > n2 + 1and fixf, g∈H
s. Denote the identity map on
Tn
byι. Ifg+ιis a volume preserving map thenf◦(g+ι)∈Hs(Tn)and
(41) kf◦(g+ι)kHs ≤CkfkHs(kgkHs+ (2π)n)s
for someC >0depending only onsand the constants from some Sobolev embeddings.
Proof. For eachk∈N, consider functionsfk, gk∈C∞(Tn;Rn)such thatfk →finHs
andgk→ginHs. Without loss of generality we assume that||det∇(gk(x) +x)| −1|< 1
k+1 holds uniformly inx.
Now by the chain and Leibniz rules, we see that for a multi-index γ with|γ| ≤ s,
Dγ(fk◦(gk+ι))is a (weighted) sum with summands of the form
(42) ((Dαfk)◦(gk+ι))
` Y
i=1
Dβi(gri
k +xri),
where` =|α| ≤ |γ|andP`i=1|βi|=|γ|. Heregki denotes theith vector component of gk. We seek to bound terms of the form (42) inL2using the preceding observations.
Since Dαfk ∈ Hs−` andgk +ι is “almost volume preserving” it can be seen that (Dαf
k)◦(gk+ι)∈Lq if
1
q ∈ 1
2 −
s−` n ,
1 2
withs−`∈(0, n/2]or
1
q =
1 2−
s−` n
To bound (42) inL2therefore, we need to check that there is an admissiblepsuch that,
1
p ∈
0,s−` n
.
and thatp=∞is admissible ifs=`. This follows from the claims we proved before the statement of the lemma.
Now we see that
kfk◦(gk+ι)kHs ≤C
p
1 + 1/kkfkkHs(kgkkHs+ (2π)n)s
whereCdepends only on Sobolev embeddings and some combinatorics. Sincefkandgk
converge we may assume thatfk ◦(gk+ι)converges weakly inHs. Thus the lemma is proved if we can show thatfk◦(gk+ι)→f◦(g+ι)inL2for example. This is indeed the case:
kf ◦(g+ι)−fk◦(gk+ι)kL2
≤ kf◦(g+ι)−f◦(gk+ι)kL2+kf◦(gk+ι)−fk◦(gk+ι)kL2
≤CLipkg−gkkL2+ p
1 + 1/kkf−fkkL2,
where we make use of the fact thatf ∈Hsis Lipschitz sinces > n/2 + 1and denote by
CLipthe Lipschitz constant off.
ACKNOWLEDGEMENTS
We would like to thank the anonymous referees for their helpful comments. BCP was supported by an EPSRC Doctoral Training Grant.
JCR was supported by an EPSRC Leadership Fellowship, grant EP/G007470/1.
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MATHEMATICSINSTITUTE, UNIVERSITY OFWARWICK, COVENTRY, CV4 7AL, UK E-mail address:[email protected]
