WITH ROUGH MOMENTUM PROFILE
CLAUDE BARDOS, FRANC¸ OIS GOLSE, PETER MARKOWICH, AND THIERRY PAUL
Abstract. Consider in the phase spaceRNx⇥RN⇠ a probability density carried by the graph of a vector fieldUinonRN, i.e. a Radon measure of the form µin = ⇢in(x) (⇠ Uin(x)). Let t be a Hamiltonian flow on RN ⇥RN. In this paper, we study the structure of the transported measure µ(t) :=
t#µinand of its integral in the⇠variable denoted⇢(t). In particular, we (a) provide estimates on the number of folds in supp(µ(t)) = t( graph ofUin), (b) establish a decomposition ofµ(t) into a “regular” component whose integral in the⇠variable is absolutely continuous with respect to the Lebesgue measure LN, (c) discuss the possibility of atoms for the measure⇢(t) and (d) construct an example in which ⇢(t) is singular with respect to LN and di↵use. We conclude our study by explaining how our results can be applied to the classical limit of the Schr¨odinger equation by using the formalism of Wigner measures.
Our results hold for initial momentum profilesUinless regular thanC1, for which the usual notion of caustic is not relevant. The proofs of these results is based on the area formula of geometric measure theory.
1. Introduction
The subject of this article is the propagation of a certain class of positive Radon measures by Hamiltonian flows.
LetH ⌘H(x,⇠) be a Hamiltonian of classC2onT⇤RN =RNx ⇥RN⇠ . Assume that the system of Hamilton’s equations
(1)
(X˙t= r⇠H(Xt,⌅t), X0(x,⇠) =x ,
⌅˙t = rxH(Xt,⌅t), ⌅0(x,⇠) =⇠, generates a global flow
(2) t(x,⇠) = (Xt(x,⇠),⌅t(x,⇠)).
Letµinbe a monokinetic measure onRNx ⇥RN⇠ i.e. a Radon measure of the form (3) µin(x,⇠) =⇢in(x) Uin(x)(⇠)
whereUinis a vector field onRN and⇢in2L1(RN). Define
(4) µ(t) = t#µin
1991Mathematics Subject Classification. 81Q20, 81S30, 35Q40, 35L03, 28A75.
Key words and phrases. Wigner measure, Liouville equation, Schr¨odinger equation, WKB method, Caustic, Area formula, Coarea formula .
1
the push-forward ofµinunder t. Equivalently,µ(t)2Cb(R+;w M(RN⇥RN)) is the unique weak solution of the Liouville equation
(5)
(@tµ+{H, µ}= 0, µt=0=µin, where{·,·}designates the Poisson bracket
{f, g}=r⇠f·rxg rxf ·r⇠g .
Our purpose is to study the structure ofµ(t) and to deduce from it some information on its support
supp(µ(t))⇢⇤t:= t({(y, Uin(y))|y2RN}).
While⇤tis the image under tof the graph ofUin, it is not in general the graph of a vector field onRN for all values oft.
When Uin =rSin is a smooth gradient field on RN, then ⇤t is the union of graphs ofx7! rxSj(t, x), whereSj is a solution of the Hamilton-Jacobi equation (6) @tS(t, x) +H(x,rxS(t, x)) = 0
defined on some open set ofRt⇥RN. The graphs of rxSj are glued along sub- manifolds of⇤twhere the restriction of the canonical projection
⇧: T⇤RN =RNx ⇥RN⇠ 3(x,⇠)7!x2RN
is not smooth: see §8 in [5] and §46-47 in [6]. A natural question is to compute, or at least estimate, the number of solutionsSj of the Hamilton-Jacobi equation needed to obtain⇤t.
Equivalently, the restriction to⇤tof the canonical projection⇧is in general not one-to-one for allt. The question above reduces to estimating the numberN(t, x) of points in ⇤t whose image under ⇧ is x. Thus, the function N describes the number of folds in⇤tinduced by the Hamiltonian dynamics ton the graph ofUin
— even whenUinis not a gradient field.
Another natural question is to study the structure of the push-forward⇢(t) of µ(t) under the canonical projection ⇧ — equivalently, of the first marginal of the measureµ(t) in the product spaceRNx ⇥RN⇠ . As we shall see, both questions are intimately related.
The mathematical problem described above appears in a great variety of con- texts. It appeared first in the theory of geometric optics, in the works of Fermat and Huygens: see for instance chapter VII in [15], chapter III in [11] and chapter 12
§2 in [13]. It appears in the classical limit of quantum mechanics: see for instance chapter VII in [16] or [25]— we shall give more details on this case below.
In both examples above, the fact thatUinis a gradient field is important. There are however other types of physical models leading to the same mathematical prob- lem even when Uin is not a gradient field. Indeed, the Maxwell distribution with density⇢, bulk velocityU and temperature✓, i.e.
⇢(x)
(2⇡✓)N/2e |⇠ U(x)|2/2✓
converges weakly to the monokinetic measure⇢(x) U(x)(⇠) as✓!0+. The propa- gation of this class of measures by the flow t generated by the free Hamiltonian H(x,⇠) := 12|⇠|2, i.e. t(x,⇠) = (x+t⇠,⇠) can be viewed as the kinetic theory of
pressureless gases, and appears for instance in a cosmological model due to Zel- dovich [27, 9, 8, 7]. When Uin is a gradient field, the Liouville equation (5) can therefore be viewed as a kinetic formulation of the Hamilton-Jacobi equation (6).
While the classical mathematical theory of geometric optics or of the semi- classical limit of quantum mechanics is centered on the geometry of⇤tin the case where both the Hamiltonian H andUin=rSinare smooth, our approach of the mathematical problem stated above is centered on the propagation of the monoki- netic measure⇢(x) Uin(x)(⇠) by the Hamiltonian flow. Besides, our analysis on the propagation problem is focussed on mathematical methods and results in which the initial momentum profile Uin is not required to be everywhere di↵erentiable
— so that ⇤t is not even a C1-manifold. Likewise, the Hamiltonian nature of the dynamics is of limited importance in our analysis, and there is no need for the mo- mentum profileUinto be a gradient — or, equivalently, for⇤tto be a Lagrangian submanifold ofT⇤RN.
However, we assume that the vector field generating the dynamics is at least of class C1 — or equivalently that the Hamiltonian H is at least of class C2 on RNx ⇥RN⇠ — so that the existence, uniqueness and regularity of the flow tresults from the classical Cauchy-Lipschitz theory. The classical limit of quantum dynam- ics with rough potentials, for which the existence and uniqueness of the classical Hamiltonian flow does not follow from the Cauchy-Lipschitz theory, has been re- cently studied in [2]. Our viewpoint in the present paper is di↵erent and in some sense complementary: we focus our attention to the special class of monokinetic measures and to their propagation by smooth Hamiltonian flows, but obtain de- tailed information on the structure of the propagated measureµ(t) even for rough initial momentum profilesUin.
The outline of the paper is as follows. In section 2 are gathered our assumptions on the Hamiltonian H with some elementary estimates on the flow t that are crucial in the sequel. Section 3 is focussed on the problem of estimating the number of folds in the support ⇤t of the propagated measure µ(t), while the structure of µ(t) itself is studied in section 5. Section 4 gathers together a few examples showing that the results in section 3 are sharp. In section 6, we study di↵erent exceptional sets that appear naturally in connection with the structure of the projected measure
⇢(t) := ⇧#µ(t), and explain how these sets are related to the traditional notion of “caustic” —- introduced by Tschirnhaus in the 17th century in the context of geometric optics). Section 7 discusses various applications of the theory presented in sections 3-5, with an emphasis on the classical limit of quantum mechanics.
2. On the Hamiltonian flow
Let H ⌘ H(x,⇠)2 R be a C2 function on RN ⇥RN satisfying the following assumptions: there exists>0, and a functionh2C(R;R+) that is sublinear at infinity, i.e.
h(r)
r !0 asr!+1 such that
(7)
|r⇠H(x,⇠)|(1 +|⇠|)
|rxH(x,⇠)|h(|x|) +|⇠|
|r2H(x,⇠)|
for all (x,⇠)2RN⇥RN.
Lemma 2.1. Any HamiltonianH 2C2(RN⇥RN)satisfying (7) generates a global Hamiltonian flow ton RN⇥RN. The map
R⇥RN⇥RN 3(t, x,⇠)7! t(x,⇠)2RN ⇥RN is of class C1. Moreover, for each⌘>0, there existsC⌘ >0such that
sup
|t|T|Xt(x,⇠) x|C⌘(1 +|⇠|) +⌘|x| for eachx,⇠2RN, and
|D t(x,⇠) IdRN⇥RN|e|t| 1 for allt2R.
Proof. By (7), one has the following a priori estimates, with the notation (2).
First
|Xt(x,⇠) x| Z t
0 |r⇠H( s(x,⇠))|dst+ Z t
0 |⌅s(x,⇠)|ds and
|⌅t(x,⇠)||⇠|+ Z t
0 |rxH( s(x,⇠))|ds
|⇠|+ Z t
0 |⌅s(x,⇠)|ds+ Z t
0
h(Xs(x,⇠))ds . By Gronwall’s inequality, for all 0st
(8) |⌅s(x,⇠)|
✓
|⇠|+ Z t
0
h(X⌧(x,⇠))d⌧
◆ es,
so that
|Xt(x,⇠) x|t+ Z t
0
esds
✓
|⇠|+ Z t
0
h(X⌧(x,⇠))d⌧
◆
t+et
✓
|⇠|+ Z t
0
h(X⌧(x,⇠))d⌧
◆ . Sincehis sublinear at infinity, we have, for everyR >0
(9) h(r)1[0,R](r) sup
0rRh(r) +1(R,+1)(r)rsup
r>R
h(r)
r ⌘MR+rmR, where
MR= sup
0rRh(r) andmR= sup
r>R
h(r) r , so that
mR!0 asR!+1. Therefore
|Xt(x,⇠) x|(+MRet)t+et|⇠|+mRet Z t
0 |Xs(x,⇠)|ds
(+MRet)t+et|⇠|+mRet|x|+mRet Z t
0 |Xs(x,⇠) x|ds .
By Gronwall’s inequality, (10)
|Xt(x,⇠) x|((+MRet)t+et|⇠|+mRet|x|)etmRet
tetmRet+MRtet(+mRet)+|⇠|et(+mRet)+mR|x|et(+mRet). The same estimates hold for T t0 after substituting|t|to t.
In view of (10)-(8), for each (x,⇠)2RN ⇥RN, the trajectory (x,⇠)7! t(x,⇠) cannot escape to infinity in finite time, and is therefore globally defined.
Besides, sincemR!0 asR!+1, the estimate (10) obviously implies the first inequality in the lemma with
⌘:=mReT(+mReT) and C⌘ := (1 +T+MRT)eT(+mReT).
Since H 2 C2(RN ⇥RN), the map (t, x,⇠) 7! t(x,⇠) is of class C1 on its domain of definition R⇥RN ⇥RN. Di↵erentiating the Hamilton equations (1) with respect to the initial condition, one finds that
(DX˙ t= +r2x,⇠H( t)·DXt+r2⇠,⇠H( t)·D⌅t, D⌅˙ t = r2x,xH( t)·DXt r2x,⇠H( t)·D⌅t, so that
|D t IdRN⇥RN| Z |t|
0 |D s|ds .
The second inequality in the lemma follows from Gronwall’s inequality. ⇤ 3. On the number of folds in ⇤t
LetUin2C(RN;RN) satisfy the condition
(11) |Uin(y)|
|y| !0 as |y|!0.
The present section is focussed on the structure of the set⇤tin the introduction, defined as
(12) ⇤t:= t({(y, Uin(y))|y2RN}), t2R. Consider the map
(13) Ft: RN 3y7!Ft(y) =Xt(y, Uin(y))2RN.
Whenever Uin is di↵erentiable aty, the map Ft is also di↵erentiable aty by the chain rule; for any suchy, define absolute value of the Jacobian determinant (14) Jt(y) =|det(DFt(y))|.
We also introduce the set
(15) C :={(t, x)2R⇥RN|Ft 1({x})\Jt 1({0})6=?}, Ct:={x2RN|(t, x)2C}.
For want of a better terminology and by analogy with geometric optics,C will be referred to as the “caustic” set.
Proposition 3.1. Under the conditions above, for eacht2R, the mapFtis proper and onto. Moreover
(16) sup
|t|T
|Ft(y) y|
|y| !0 as|y|! 1.
The proof of this proposition and the next one will use the following topological argument.
Lemma 3.2. Let g : RN ! RN be a continuous map satisfying the following condition: for someR >0
(g(x)|x)>0 for allx2RN such that |x|=R . Then
a) there existsx2RN such that|x|R andg(x) = 0;
b) if gis of classC1 onRN and0is a regular value ofg, theng 1({0})\B(0, R) is finite and #(g 1({0})\B(0, R))is odd.
Proof. Consider the homotopyG2C([0,1]⇥RN;RN) defined by G(t, x) =tx+ (1 t)g(x).
One has
G(t, x)6= 0 whenevert2[0,1] and|x|=R .
Indeed, G(1, x) =x6= 0 if|x|=R > 0; besides, ift2 [0,1[ and G(t, x) = 0, one has
g(x) = t
1 tx so that (g(x)|x) = t
1 t|x|2= t
1 tR2<0 for allx2RN such that|x|=R, which contradicts our assumption.
By the homotopy invariance of the degree (see Properties 7, 8 and Theorem 12.7 in chapter 12,§A of [24])
d(g, B(0, R),0) =d(I, B(0, R),0) = 1.
This implies a). Moreover, ifg is of classC1 onRN and 0 is a regular value ofg, all the elements of g 1({0}) are isolated points by the implicit function theorem, sog 1({0})\B(0, R) is finite. Besides (see Property 2 in chapter 12, §A of [24])
d(g, B(0, R),0) = X
x2g 1({0})\B(0,R)
sign(detDg(x)) = 1. Therefore, there exists an integerm2Nsuch that
#{x2B(0, R)|g(x) = 0 and det(Dg(x))>0}=m+ 1
#{x2B(0, R)|g(x) = 0 and det(Dg(x))<0}=m
so that #(g 1({0})\B(0, R)) = 2m+ 1, which proves b). ⇤ Proof of Proposition 3.1. By the first inequality in Lemma 2.1 and the condition (11 on Uin, for each⌘>0, one has
|y|!lim+1 sup
|t|T
|Ft(y) y|
|y| ⌘, so that (16) holds.
Because of (16), the continuous mapFtsatisfies
(17) (Ft(y) x|y) =|y|2+o(|y|2) as|y|!+1
so that Ft is onto by applying Lemma 3.2 to the mapg : y 7!Ft(y) x. On the other hand
|Ft(y)|!+1 as|y|!+1
so thatFt is proper. ⇤
By Proposition 3.1, we know that, for each (t, x) 2 R⇥RN, the equation Ft(y) =xhas at least one solution y2RN whenUin is a continuous vector field sublinear at infinity, i.e. satisfying (11). In the next proposition, we study the numberN(t, x) of solutions of this equation in the case whereUinis of classC1 at least. Equivalently,N(t, x) is the number of intersections of the manifold⇤t with Tx⇤RN '{x}⇥RN. Therefore, the integer-valued functionN measures the number of folds in the manifold⇤tresulting from the interaction of the Hamiltonian flow
twith the initial profileUin.
Proposition 3.3. [Smooth case] Assume that (11) holds for Uin 2C1(RN,RN) and that
|DUin(y)|=O(1) as|y|!+1. a) For each t2R, one hasLN(Ct) = 0.
b) The setC is closed inR⇥RN.
c) For each(t, x)2R⇥RN\C, the setFt 1({x})is finite, and henceforth denoted by
{yj(t, x), j= 1, . . . ,N(t, x)}.
The integer N is a constant function of (t, x) in each connected component of R⇥RN \C and, for each j 1, the map yj is of class C1 on each connected component ofR⇥RN \C whereN j.
d) There existsa <0< bsuch thatC\((a, b)⇥RN) =?andN = 1on(a, b)⇥RN. e)N(t, x)is odd for each(t, x)2R⇥RN \C.
Proof. IfUin2C1(RN), the mapFtis of classC1fromRN toRN, being the com- position ofC1maps. SinceCtis the set of critical values ofFt, one hasLN(Ct) = 0 by Sard’s Theorem (in the equal dimension case), which proves a).
Pick any sequence (tn, xn)2Csuch that (tn, xn)!(t, x) inR⇥RN asn! 1. By definition of C, there exists yn such that Ftn(yn) = xn. Since the sequences tn and xn converge, they are both bounded. Let T = supn|tn|; assume that the sequenceyn is unbounded; if so there exists a subsequenceynlsuch that|ynl|! 1. By (16)
|xnl ynl|=|Ftnl(ynl) ynl| sup
|t|T|Ft(ynl) ynl|=o(|ynl|)
so that|xnl|⇠|ynl|asnl!+1. Since this contradicts the fact that the sequence xn is bounded, we conclude that the sequence yn is bounded. Therefore, there exists a convergent subsequenceynk ofyn; cally its limit asnk !+1. Passing to the limit in both relations
Ftnk(ynk) =xnk andJtnk(ynk) =|det(rFtnk(ynk))|= 0, we conclude that (t, x, y) satisfies
Ft(y) =x andJt(y) = 0 and therefore that (t, x)2C, which proves b).
For (t, x) 2 R⇥RN \C, all the solutions of the equation Ft(y) x = 0 are isolated by the implicit function theorem. The set of all such points,Ft 1({x}), is therefore compact sinceFtis proper. By the implicit function theorem, the integer N is a locally constant function of (t, x)2R⇥RN\C, and is therefore constant on each connected component of R⇥RN \C. Let j2N⇤, and let⌦be a connected component of R⇥RN \C; by the implicit function theorem yj 2 C1(⌦). This proves c).
Assume inf{t >0|Ct6=?}= 0. Then, there exists (tn, xn, yn) such that tn !0+, Ftn(yn) =xn, andJtn(yn) = 0.
Assume that a subsequenceynk of the sequence yn is bounded. Up to extraction of a subsequence, one can assume that ynk !y, so that 0 = Jtnk(ynk) !J0(y).
But since Ft = IdRN, one has J0(y) = 1. Therefore|yn| ! +1. By the second inequality in Lemma 2.1
|DxXtn(yn, Uin(yn)) IdRN|e|tn| 1,
|D⇠Xtn(yn, Uin(yn))·DUin(yn)|=O⇣
e|tn| 1⌘ , so that
0 =Jtn(yn) =|det(DxXtn(yn, Uin(yn)) +D⇠Xtn(yn, Uin(yn))·DUin(yn))|
!|det(IdRN)|= 1 as n! 1.
Thus the assumptiontn!0 leads to a contradiction. Therefore, inf{t >0|Ct6=?}=b >0.
By the same token,
sup{t <0|Ct6=?}=a <0.
Thus (a, b)⇥RN is contained in the connected component of{0}⇥RN inR⇥RN\C.
SinceF0= IdRN, one hasN(0, x) = 1 for allx2RN, and sinceN is constant on each connected component of R⇥RN \C, one concludesN = 1 on (a, b)⇥RN, which proves d).
If (t, x)2R⇥RN \C, the point xis a regular value ofFt. SinceFt is proper, Ft 1({x}) is compact, and therefore bounded. PickR >0 such that
Ft 1({x})⇢B(0, R).
By (17) and Lemma 3.2 b) applied to the mapg: y7!Ft(y) x
#Ft 1({x}) = #(Ft 1({x})\B(0, R)) is odd,
which proves e). ⇤
WhenUinis not of classC1, the arguments used to prove Proposition 3.3 are no longer valid. However one can still obtain some information on the numberN(t, x) of solutionsyof the equationFt(y) =xby a completely di↵erent method, involving the area — or co-area formula.
Assume thatUin2C(RN;RN) satisfies (11) and that its gradient (in the sense of distributions)DUinsatisfies the condition
(18) @lUkin ⌦2LN,1(⌦) for each bounded open ⌦⇢RN,
for allk, l= 1, . . . , N. We recall that a measurable functionf : ⌦!Rbelongs to the Lorentz spaceLN,1(⌦) if
Z 1 0
LN({x2⌦| |f(x)| }) 1/Nd <1.
By Theorem B in [14], the vector fieldUin is di↵erentiable a.e. on RN. LetE be theLN-negligible set defined as
(19) E:={y2RN|Uinis not di↵erentiable aty}. By the chain rule, the absolute value of the Jacobian determinant (20) Jt(y) =|det(DxXt(y, Uin(y)) +D⇠Xt(y, Uin(y))DUin(y))|
is defined for all (t, y)2R⇥(RN \E). Henceforth, the notationJt 1({0}) designates the set
(21) Jt1({0}) :={y2RN\E|Jt(y) = 0}, and we shall also consider the set
(22) Zt:=Jt1({0})[E .
Theorem 3.4. [Rough case] Assume that Uin 2 C(RN;RN) satisfies (11) and (18).
a) for allt2R, the functionJt2L1loc(RN);
b) for all t2R, there are finitely many solutions y of the equation
(23) Ft(y) =x
for a.e. x2RN — in other words,N(t, x)is finite for a.e. x2RN, for allt2R;
c) for each boundedB⇢RN, denote by NB(t, x)the number of solutionsy2B of (23); then, wheneverB is bounded, for eacht2R and eachn2N, one has
LN {x2RN| NB(t, x) n} 1 n
Z
B
Jt(y)dy
1
neN|t|k1 + (1 e |t|)kDUinkNLN(B); d) lett2R; then for a.e. x2RN, all solutionsy of (23) satisfy
y2RN \E andJt(y)>0 ; e) for all T >0
H1({(t, y)2[ T, T]⇥RN|Ft(y) =x})<+1 for a.e. x2RN.
Remarks.
a) By the first statement in Proposition 3.1, N(t, x) = NRN(t, x) 1 for all (t, x)2R⇥RN.
b) Even in the smooth case, i.e. assuming in addition that Uin 2C1(RN), state- ment c) in Theorem 3.4 provides information on the number of folds of⇤t, that is the image under t of the graph ofUin, which seems to be new at the time of this writing.
Proof. By the second estimate in Lemma 2.1
(24) |DxXt(y, Uin(y)) +D⇠Xt(y, Uin(y))·DUin(y)|e|t|+ (e|t| 1)|DUin(y)| so that
(25) Jt(y) =|det(DxXt(y, Uin(y)) +D⇠Xt(y, Uin(y))·DUin(y))|
eN|t|(1 + (1 e |t|)|DUin(y)|)N
by Hadamard’s inequality. Since Uin satisfies (18) and since LN,1(⌦) ⇢ LN(⌦), this inequality implies statement a).
Since the mapFtis proper by Proposition 3.1, the set Kt,R=Ft 1(B(0, R)) is compact for eachR >0. By the area formula (see Theorem 3.4 in [20] and Theorem A in [14]) Z
RN
#(Ft 1({x})\Kt,R)dx= Z
Kt,R
Jt(y)dy <+1.
Therefore #Ft 1({x})<1for a.e. x2 B(0, R); since this is true for all R2N, one concludes that #Ft 1({x})<1for a.e. x2RN, so that b) holds.
Let B be a measurable subset of RN; applying again the area formula shows
that Z
RNNB(t, x)dx= Z
B
Jt(y)dy . By the Bienaym´e-Chebyshev inequality, for eachn 1
LN {x2RN| NB(t, x) n} 1 n
Z
B
Jt(y)dy ,
which is precisely the first inequality in c). The second inequality follows from (25) and H¨older’s inequality.
Letn= 1 andZt:=Jt 1({0})[E. LetB=Zt\Kt,R; the setB is measurable
and bounded. Then Z
B
Jt(y)dy= 0
sinceJt(y) = 0 for ally2B\E andLN(E) = 0. Applying c) shows that LN({x2RN| |x|R andNZt(t, x) 1}) = 0,
which entails d) by monotone convergence, lettingR2N tend to infinity.
Consider next the continuous map
F : [ T, T]⇥RN 3(t, y)7!F(t, y)2RN.
In view of (16), |F(t, y)|! 1 as |y|!+1 uniformly int 2[ T, T]. Therefore, the setKR:=F 1(B(0, R)) is compact for eachR >0. Then, for eacht2[ T, T] and eachy2RN\E, the JacobianDF(t, y) is the column-wise partitioned matrix
DF(t, y) = [V(t, y), M(t, y)], with
V(t, y) =r⇠H( t(y, Uin(y))) and
M(t, y) :=DxXt(y, Uin(y)) +D⇠Xt(y, Uin(y))DUin(y). Therefore,
DF(t, y)DF(t, y)T =V(t, y)V(t, y)T +M(t, y)M(t, y)T
so that, by the co-area formula (Theorem 1.3 in [21]) Z
RN
H1(F 1({x})\KR)dx
= Z
KR
q
det(V(t, y)V(t, y)T +M(t, y)M(t, y)T)dtdy .
By Lemma 2.1, (t, x,⇠)7! t(x,⇠) is of classC1onR⇥RN⇥RN, so that the map (t, y)7!V(t, y) is continuous on R⇥RN, and therefore bounded on the compact KR. On the other hand, by (24)
sup
|t|T|M(t, y)|eT + (eT 1)|DUin(y)|2LNloc(RN), sinceUin satisfies (18). Denoting
KR0 :={y2RN| there exists t2[ T, T] s.t. (t, y)2KR}
that is compact in RN (being the projection of the compact KR on the second factor inR⇥RN), one has
kV VT +M MTkN/2LN/2(KR)2N/2 1kVkNL1(KR)LN+1(KR) + 2N/2TkMkNLN(KR0)
<1.
ThereforeH1(F 1({x})\KR)<+1is finite for a.e. x2B(0, R), and since this is true for all R2N, one concludes thatH1(F 1({x}))<+1 for a.e. x2RN,
which is statement e). ⇤
Theorem 3.4 suggests considering the sets
(26) Ct0={x2RN|Ft 1({x}) is infinite} for allt2R.
On the other hand, the definition of the caustic (15) in the smooth case should be modified as follows when the continuous vector fieldUinis not of class C1 but satisfies (11) and (18):
(27)
Ct:={x2RN|Ft 1({x})\Zt6=?}, C := [
t2R
Ct,
whereZtis defined in (22). Notice that this definition coincides with (15) whenever Uin is di↵erentiable everywhere. However we do not know whether C or Ct are closed wheneverUinis not of classC1.
The first part of statement c) in Proposition 3.3 is equivalent to the inclusion
(28) Ct0⇢Ct for allt2R.
Statements b) and d) in Theorem 3.4 can be recast as follows (29) LN(Ct0) = 0 and LN(Ct) = 0, for allt2R.
This is the analogue of statement a) in Proposition 3.3 in the case of a rough Uin. Notice that (29) is a consequence of the area formula while statement a) in Proposition 3.3 follows from Sard’s theorem (see Remark 2.97 on pp. 103–104 in [3], discussing the relation between Sard’s theorem and the co-area formula).
4. Some examples
While the notion of causticCnaturally occurs in studying the geometry of⇤tin the smooth case, its relation to the number of solutionsyof the equationFt(y) =x is slightly less obvious, as shown by the following examples. In all these examples, the Hamiltonian is
H(x,⇠) = 12⇠2 generating the free flow
t: R⇥R3(x,⇠)7!(x+t⇠,⇠)2R⇥R. Thus
Ft: R3y7!y+tUin(y)2R.
The first example below shows that the setFt 1({x}) may be finite even ifx2Ct. In other words, it may happen that the inclusion (28) is strict for allt2R.
Example 1. Set N = 1, and let Uin be real analytic on R and satisfy (11).
Therefore Ft is real analytic on R for each t 2 R. For each x 2 R, the set Ft 1({x}) is the set of zeros of the analytic function y 7! Ft(y) x. Since Uin satisfies (11), one hasFt(y)⇠y as|y|! 1, so that, for eacht2R, the function Ft is not a constant. Therefore the zeros of y 7! Ft(y) xare isolated for each (t, x)2R⇥R. Equivalently, the set Ft 1({x}) consists of isolated points. On the other hand, for each (t, x)2R2, the setFt 1({x}) is compact sinceFtis proper by Proposition 3.1. Therefore the setFt 1({x}) is finite for all (t, x)2R2.
The next example shows that the set Ft 1({x}) may be infinite for infinitely many timest— in this case, for alltin a nonempty, open interval ofR.
Example 2. SetN = 1, and letUin be defined by Uin(z) :=
(tanh(z) sin(ln|z|) ifz >0,
0 ifz= 0.
ClearlyUin2C1(R⇤) and, for allz2R⇤, one has
(Uin)0(z) = (1 tanh2(z)) sin(ln|z|) +tanh(z)
z cos(ln|z|). Observe that (Uin)0(z) =p
2 sin(ln|z|+⇡4) +O(z2) does not have a limit forz!0, so thatUin2/C1(R). On the other hand
sup
z2R|Uin(z)|= 1 and sup
z6=0|(Uin)0(z)|2
so thatUinis Lipschitz continuous onRand the mapFtis proper onR.
Letx= 0; for eacht such that|t|>1, the setFt 1({0}) is infinite.
Indeed, the setFt 1({0}) obviously containsz = 0; besides, Ft 1({0}) contains also a decreasing sequenceyn!0 asn!+1, which satisfies
sin(ln|yn|)! 1/t as n! 1. ThereforeFt 1({0}) is countably infinite whenever|t|>1.
Notice that, in example 2, wheneverT >1
{(t, y)2[ T, T]⇥R|Ft(y) = 0}= ([ T, 1][[1, T])⇥({0}[{yn|n 1}) so that
H1({(t, y)2[ T, T]⇥R|Ft(y) = 0}) = +1.
In other words, the set of pointsxfor which
H1({(t, y)2[ T, T]⇥R|Ft(y) =x}) = +1,
whileL1-negligible by statement e) of Theorem 3.4, may be non empty.
There is another interesting observation in connection with Example 2. Observe that 02Ct0 and that
Ft 1({0}) ={0}[{yn|n 1}, whenever|t|>1. Nevertheless, for eachn 1
(Uin)0(yn) = sin(ln|yn|) + cos(ln|yn|) +O(|yn|2)! 1 t +
r
1 1
t2 6= 1 t if|t|>1, andFtis not di↵erentiable aty= 0. Thusy= 0 is in the exceptional set E where the Lipschitz continuous functionUinis not di↵erentiable, so that 02Ct
with the definition (27) although
Ft 1({0})\Jt 1({0}) =?,
so that 0 would not belong to Ct had we kept the classical definition (15) in the case of non everywhere di↵erentiableUinprofiles.
The next two examples show that Ft 1({x}) can even be uncountably infinite, even for a smooth profileUin.
Example 3. SetN = 1, and letUin be defined by Uin(z) :=
8>
<
>:
+ 1 ifz < 1, z if|z|1, 1 ifz >1. Consider the equation
Ft(y) :=y+tUin(y) =x
with unknowny; fort <1, its solution is unique and given by
y= 8>
><
>>
:
x t ifx < t 1, x
1 t if|x|1 t , x+t ifx >1 t . Fort= 1, the solution is
y= 8>
<
>:
x t ifx <0, any y2[ 1,1] ifx= 0, x+ 1 ifx >0. Fort >1, the solution is
y= 8>
><
>>
:
x t ifx <1 t ,
x t , x
1 t andx+t if 1 txt 1,
x+t ifx > t 1.
NowFt 1({x}) is finite for allx2Rwhenevert6= 1, whileF11({0}) = [ 1,1].
Example 4. In the previous example,Uinis Lipschitz continuous but not of class C1. Yet the same phenomenon can be observed by smoothing Uin near z =±1.
RegularizeUinand obtainU✏in2C1(R) so that
supp(U✏in Uin)⇢[ 1 ✏, 1 +✏][[1 ✏,1 +✏], and U✏inUinon [ 1 ✏, 1 +✏] and UinU✏inon [1 ✏,1 +✏]. In that case
F1 1({0}) = [ 1 +✏,1 ✏].
Indeed, sinceU✏inUinon [ 1 ✏, 1 +✏], all the points on the graph ofU✏inwith abscissa in ( 1 ✏, 1 +✏) will reachx= 0 aftert = 1. The same is true of the points on the graph ofU✏inwith abscissa in (1 ✏,1 +✏). Thus the regularization does not a↵ect the dynamics of the points with abscissa in ( 1 +✏,1 ✏) for all t2[0,1], and in particular fort= 1.
5. On the structure ofµ(t)
Throughout this section, we assume thatUin2C(RN;RN) satisfies the sublin- earity condition (11) at infinity and the regularity condition (18).
Consider a monokinetic measureµinof the form (3) with⇢in2L1(RN), whose action on a test function 2Cb(RN⇥RN) is given by the formula
hµin, i:=
Z
RN
(y, Uin(y))⇢in(y)dy . In other words
µin=LxN ⌦⇢in(x) inUin(x).
LetH 2C2(RN⇥RN) satisfy (7), and let tbe the Hamiltonian flow generated byH.
For all t2R, let µ(t) = t#µin be the push-forward of µin under t, defined as follows: for each test function 2Cb(RN ⇥RN),
(30) hµ(t), i:=hµin, ti= Z
RN
( t(y, Uin(y)))⇢in(y)dy . Finally, let⇢(t) be the measure onRN defined as
(31) ⇢(t) :=⇧#µ(t)
where ⇧: T⇤RN 'RN⇥RN 3(x,⇠)7!x2RN is the canonical projection. In other words, for each test function 2Cb(RN)
(32) h⇢(t), i=
Z
RN
(Xt(y, Uin(y))⇢in(y)dy or, equivalently
(33) ⇢(t) =Ft#⇢in.
We shall also use the following definition
(34) Pt:={y2RN \E|Jt(y)>0}.
Our main result in this section bears on the structure of the measuresµ(t) and its projection⇢(t).
Theorem 5.1. Under the assumptions above, let⇢in2L1(RN)be such that⇢in 0 a.e. onRN.
a) for each t2R,
⇢in1PtLN ⌧JtLN, and⇢in1ZtLN ?JtLN b) for each t2R,
µ(t) =µa(t) +µs(t) withµa(t)?µs(t),
where 8
>>
<
>>
:
µa(t) :=LxN ⌦ X
y2Ft 1({x})
⇢in1Pt
Jt
(y) ⌅t(y,Uin(y)), µs(t) := t#(LxN ⌦⇢in(x)1Zt(x) Uin(x)). Moreover
c) for each t2R, one has
supp(µ(t))⇢⇤t:= t({(y, Uin(y))|y2RN}) ; d) for each t2R, the measure
⇢a(t) :=⇧#µa(t)⌧LN; with
d⇢a(t)
dLN (x) =1RN\C0
t(x) X
y2Ft 1({x})
d(⇢in1PtLN) d(JtLN) (y) whereCt0 is defined in (26);
e) for each t2R, the measure
⇢s(t) :=⇧#µs(t)is carried by Ct; in particular
⇢a(t)?⇢s(t).
A few remarks are in order before we give the proof of Theorem 5.1. In the smooth case — i.e. whenever Uin 2 C1(RN;RN) and satisfies (11), we recall that Ct is closed in RN for each t 2R, by statement b) of Proposition 3.3. For any given t 2 R, let ⌘ (x,⇠) be a test function in Cc(RN ⇥RN) such that
⇧supp( )\Ct=?. By (30) hµ(t), i=
Z
RN
( t(y, Uin(y)))⇢in(y)dy .
Since ⇧(supp( )) is compact and included in the open set RN \Ct, it intersects at most finitely many connected components ofRN \Ct. Assume without loss of generality that⇧(supp( )) is connected, so that it intersect exactly one connected component ⌦of RN \Ct; on ⌦, the integer-valued function N is a constant de- noted byN⌦, by statement c) of Proposition 3.3. With the notation used in that proposition, for eachx2⇧(supp( ))⇢⌦
Ft 1({x}) ={yj(t, x)|j= 1, . . . ,N⌦}, and
yj(t,·)2C1(⌦) for all j= 1, . . . ,N⌦.
Thereforeyj(t,·) is aC1-di↵eomorphism from⌦on its imageOj, with inverseFt. Thus
Ft 1(⌦) =
N[⌦
j=1
Oj andOi\Oj=?ifi6=j , so that
ZZ
⌦⇥RN
(t, x)µ(t, dxd⇠) =
N⌦
X
j=1
Z
Oj
(Ft(y),⌅t(y, Uin(y)))⇢in(y)dy . In each of the integrals on the right hand side,Ftis aC1-di↵eomorphism mapping Oj on⌦, so that, changing variables, we see that
Z
Oj
(Ft(y),⌅t(y, Uin(y)))⇢in(y)dy
= Z
⌦
(x,⌅t(yj(t, x), Uin(yj(t, x))))⇢in(yj(t, x))|det(rxyj(t, x))|dx . Since
|det(Dxyj(t, x))|=Jt(yj(t, x)) 1,
we conclude that the restriction ofµ(t) to (R⇥RN)\Cis a measure-valued function of (t, x) given by the following formula:
(35) µ(t, x,·) :=
N(t,x)
X
j=1
⇢in(yj(t, x)
Jt(yj(t, x)) ⌅t(yj(t,x),Uin(yj(t,x)))
whenever (t, x)2/ C. This formula is strikingly similar to the one giving µa(t) in statement b) of Theorem 5.1. There are however subtle di↵erences, which we shall discuss in more detail in the next section. At this point, it suffices to say that Theorem 5.1 provides a formula forµ(t) that holds globally on RN ⇥RN instead of (RN\Ct)⇥RN, and that the argument above requires more regularity onUin than assumed in Theorem 5.1.
Proof of Theorem 5.1. LetA⇢RN; then the condition Z
A
Jt(y)dy= 0 implies that Jt(y) = 0 for a.e. y2A . ThereforeLN(Pt\A) = 0 so that
Z
A
(⇢in1Pt)(y)dy= Z
Pt\A
⇢in(y)dy= 0. Thus⇢in1PtLN ⌧JtLN.
On the other hand, for eacht2R
RN =Pt[Zt withPt\Zt=?. Since
Jt(y) = 0 fory2Zt\E i.e. LN-a.e. onZt
while
⇢in(y)1Zt(y) = 0 for ally2Pt, we conclude that⇢in1ZtLN ?JtLN, which proves a).
Define
µina :=LxN ⌦(⇢in1Pt)(x) Uinin(x), µins :=LxN⌦(⇢in1Zt)(x) Uinin(x), and, for eacht2R
µa(t) := t#µina , µs(t) := t#µins , so that one has indeed
µa(t) +µs(t) = t#(µina +µins ) = t#µin=µ(t). Then there exists a uniqueb2L1(RN;JtLN) such that
⇢in1PtLN =bJtLN
by the Radon-Nikodym theorem. Thus, for each 2Cc(RN ⇥RN), by the area formula (see Theorem 3.4 in [20] and Theorem A in [14])
hµa(t), i= Z
RN
(Ft(y),⌅t(y, Uin(y)))b(y)Jt(y)dy
= Z
RN
0
@ X
y2Ft 1({x})
b(y) (x,⌅t(y, Uin(y))) 1 Adx
= Z
RN
0
@ X
y2Ft 1({x})
b(y)h ⌅t(y,Uin(y)), (x,·)i 1 Adx . In the formula above
b:= d(⇢in1PtLN) d(JtLN)
is the Radon-Nikodym derivative of⇢in1Pt with respect toJtLN. SinceJt>0 on the setPt
(36) b=⇢in1Pt
Jt
a.e. onRN, which proves b).
Formula (3) obviously implies that
supp(µin)⇢⇤0:={(y, Uin)|y2RN}. Sinceµ(t) = t#µin, one has
supp(µ(t))⇢ t({(y, Uin)|y2RN}) =⇤t, which is precisely statement c).
In view of the first formula in b) and of the definition ⇢a(t) =⇧#µa(t) of the measure⇢a(t), one has
⇢a(t) = 0
@ X
y2Ft1({x})
b(y) 1 ALN,
with b as in (36). The set Ft 1({x}) can obviously be infinite, in which case the sum above can be undefined. However, this occurs only ifx2Ct0as defined in (26).
SinceCt0 isLN-negligible
⇢a(t) =ftLN
with
ft(x) :=1RN\C0 t(x)
0
@ X
y2Ft 1({x})
b(y) 1 A.
Thus,⇢a(t) is of the form⇢a(t) =ftLN withft 0 measurable on RN. Besides 0⇢a(t)⇢(t) and
Z
RN
⇢(t, dx) = Z
RN
⇢in(x)dx)<1
which implies in particular that ⇢a(t) ⌧ LN with the formula for the Radon- Nikodym derivative as in d).
Consider the measurable setA:=RN\Ct. By (33) applied to⇢in1Zt instead of
⇢in, one has
⇢s(t)(A) = Z
Ft 1(A)
⇢in(y)1Zt(y)dy= Z
Ft 1(A)\Zt
⇢in(y)dy= 0.
Indeed, by definition of Ct, one has Ft 1(A)\Zt = ?. In other words, ⇢s(t) is carried byCt.
Finally, since⇢a(t)⌧LN and⇢s(t) is carried byCtwhich isLN-negligible by (29), we conclude that⇢a(t)?⇢s(t) which is precisely statement e). ⇤
6. On the caustic and other exceptional sets
In the case of a smooth Uin profile — i.e. when Uin 2 C1(RN;RN) satisfies (11), the caustic C is the only exceptional set —Ct being equivalently defined as the image under the projection ⇧ of the set of points in the manifold ⇤t in (12) where the restriction⇧⇤
t is not di↵erentiable.
WhenUin is not everywhere di↵erentiable, this definition ofCt does not make sense in general since ⇤tis not a di↵erentiable manifold in the first place. In such cases, it is more natural to consider the measuresµ(t) and⇢(t) instead of the sets
⇤t andCt — all the more so sinceCt may not even be closed inRN. Thus, even though⇢s(t) is concentrated onCt, one cannot say thatCtis the support of⇢s(t) as Ctmay not be closed. On the other hand, the inclusion supp⇢s(t)⇢Ctis of little interest asCtmight be dense in some domain ofRN. Although⇢s(t) is concentrated on Ct, this obviously does not characterize Ct (if a measure is concentrated on a set, it is also concentrated on the complement in that set of any negligible set for that same measure).
There are analogous difficulties with the absolutely continuous part of the mea- sure⇢a(t). In formula (35), the restriction ofµa(t) to (RN\Ct)⇥RN is viewed as a function ofx2RN\Ctwith values in the set of Radon measures in the variable
⇠2RN. This viewpoint is obviously not appropriate ifUin is not at least of class C1 — for instance, if the setCtis dense in some domain of RN. In Theorem 5.1, the measure µ is a weakly continuous function of the time variablet with values in the space of Radon measures in the variables (x,⇠), and is therefore globally defined onRNx ⇥RN⇠ . Obviously, the ratio⇢in1Pt/Jtis just one possible choice of the Radon-Nikodym derivatived(⇢in1PtLN)/d(JtLN) and could be modified ar- bitrarily on any set ofJtLN-measure 0 — which could be of positiveLN-measure,
