I Chalendar
email: chalendar@math.univ-lyon1.fr.

J Esterle
email: jean.esterle@math.u-bordeaux1.fr.

J R Partington
email: j.r.partington@leeds.ac.uk.

Lower estimates near the origin for functional calculus on operator semigroups

Keywords: Mathematics Subject Classification (2000): Primary: 47D03, 46J40, 46H30 Secondary: 30A42, 47A60 strongly continuous semigroup, functional calculus, Laplace transform, maximum principle

This paper provides sharp lower estimates near the origin for the functional calculus F (-uA) of a generator A of an operator semigroup defined on the (strictly) positive real line; here F is given as the Laplace transform of a measure or distribution. The results are linked to the existence of an identity element or an exhaustive sequence of idempotents in the Banach algebra generated by the semigroup. Both the quasinilpotent and non-quasinilpotent cases are considered, and sharp results are proved extending many in the literature.

Introduction

This article is concerned with estimates for F (-uA)whereA is the generator of a strongly continuous semigroup (T (t)) t>0 on a Banach space. Here F is an entire function with F (0) = 0, given as the Laplace transform of a measure or distribution; the functional calculus defining F (-uA)i sg i v e nb ym e a n s of an integral.

This can be seen as providing a wide generalization of results in [START_REF] Bendaoud | Distances between elements of a semigroup and estimates for derivatives[END_REF][START_REF] Esterle | Distance entre éléments d'un semi-groupe dans une algèbre de Banach[END_REF][START_REF] Kalton | Power-bounded operators and related norm estimates[END_REF], for example, where quantities such as T (t) -T (2t) (or its spectral radius) are estimated near the origin. For example, if T (t) -T (2t) < 1/4o na n interval (0,t 0 ), then, roughly speaking, (T (t)) t>0 has a bounded infinitesimal generator (see [START_REF] Bendaoud | Distances between elements of a semigroup and estimates for derivatives[END_REF]).

There are two cases to consider, namely, the quasinilpotent and nonquasinilpotent cases, and the techniques used are based on strong maximum principles for analytic functions.

In Section 2, the case of quasinilpotent semigroups is considered. Then in Section 3 the non-quasinilpotent case is analysed, providing conditions to obtain either an identity in the closed algebra generated by the semigroup or else an exhaustive sequence (P n ) n≥1 of idempotents such that (P n T (t)) t has a bounded generator. Here, the sharpness of the estimates is shown by an example.

Notation:

We write C + = {z ∈ C :Rez>0}, and similarly for C -. Let D(a, R)d e n o t et h ec o m p l e xd i s c{|z -a| <R}.

For a Jordan curve Γ ⊂ C, we write int Γ (the interior of Γ) for the open set of points in C about which the winding number of Γ is non-zero. For S ⊂ C let M c (S)d e n o t et h es p a c eo fr e g u l a rB o r e lm e a s u r e sh a v i n g compact support contained in S.

Quasinilpotent semigroups

Suppose that (T (t)) t>0 is a nontrivial strongly continuous semigroup of quasinilpotent operators acting on a Banach space (X , . ). Then we write X 0 = t>0 T (t)X -. (closure in norm), and define a norm

x 1 =sup t≥0 T (t)x , where T (0)x = x,
on the subspace X 1 := {x ∈ X 0 : x 1 < ∞}, which is a Banach space under the norm . 1 . Further, we write

X 1 := t>0 T (t)X -. 1 ⊂ X 1 . (1)
The following result follows immediately from the main result of [START_REF] Feller | On the generation of unbounded semi-groups of bounded linear operators[END_REF]. It will be used to reduce the case of a quasinilpotent semigroup to that of a contractive quasinilpotent semigroup.

Theorem 2.1 Let (T (t)) t>0 be a nontrivial strongly continuous semigroup of quasinilpotent operators acting on a Banach space (X , . ). Then with ( X 1 , . 1 ) defined as in [START_REF] Bendaoud | Distances between elements of a semigroup and estimates for derivatives[END_REF] the semigroup (T (t) | X 1 ) t>0 is a strongly continuous semigroup of quasinilpotent contractions. Moreover for all operators R in the commutant {T (t):t>0} ′ we have R | X 1 1 ≤ R .

Some complex function theory

Theorem 2.2 Let f : C + → C be a continuous bounded nonconstant function, holomorphic on C + , such that f ([0, ∞)) ⊂ R, f (0) = 0, and with

lim x→∞,x∈R f (x)=0. Suppose that α > 0 is such that f (α) ≥ |f (x)| for all x ∈ [0, ∞).
Then there exist a 1 ,a 2 ∈ C + , a 0 ∈ (α,a 1 ) and a 3 ∈ iR with Im a j > 0 for j =1, 2, 3, and Im a 2 =Ima 3 , and a simple piecewise linear Jordan curve Γ 1 joining a 1 to a 2 in the upper right half-plane {z ∈ C :R ez>0, Im z>0} and δ > 0 such that

(i) |f (z)| ≥ f (α)+δ|z -α| m for all z ∈ [α,a 1 ], where m (even) is the smallest positive integer with f (m) (α) =0; (ii) |f (z)| > |f (a 0 )| for all z ∈ Γ 1 ∪ [a 2 ,a 3 ].

Proof :

Since f is holomorphic in C + , we have, by Taylor's theorem, constants M>0a n dη > 0s u c ht h a t

f (z) -f (α) - (z -α) m m! f (m) (α) ≤ M |z -α| m+1 ,
whenever |z -α| < η. By choosing a 1 with |a 1 -α| sufficiently small and with argument such that (a 1α) m < 0( e . g . a r g ( a 1α)=π/m), we have |f (a 1 )| > |f (α)| = f (α), and we may then choose a point a 0 ∈ [α,a 1 ] with to a 2 by a polygonal path in V . We may also guarantee that it is simple (does not cross itself): the only difficulty arises if it crosses itself on the arc (a 0 ,a 1 ), when we may replace a 1 by the crossing point closest to a 0 , or if it crosses itself on the line (a 2 ,a 3 ), when we may replace a 2 by the crossing point closest to a 3 .

|f (α)| < |f (a 0 )| < |f (a 1 )|. Let U = {z ∈ C + : |f (z)| > |f (a 0 )|}. Since a 1 ∈ U ,
The curve constructed in the proof of Theorem 2.2 may be seen as the upper part of Figure 1.

We shall also require the following easy result.

Lemma 2.3 Let f ∈ H(C) with f (0) = 0, f nonconstant.
Then there is an r>0 such that for all u ∈ C \{0}, with |u| <r, we have

sup |z|≤ r |u| |f (zu)| < sup x≥0 |f (xu)|. Proof : Choose R>0s u c ht h a tf (z) =0forallz with |z| = R. Let δ = inf{|f (z)| : |z| = R} so that δ > 0. Now take r>0suchthatsup |z|≤r |f (z)| < δ,

The main result

Recall that if (T (t)) t>0 is a uniformly bounded strongly continuous semigroup with generator A, then

(A + λI) -1 = - ∞ 0 e λt T (t) dt,
for all λ ∈ C with Re λ < 0. Here the integral is taken in the sense of Bochner with respect to the strong operator topology. If, in addition, (T (t)) t>0 is quasinilpotent, then

(A + λI) -1 = - ∞ 0 e λt T (t) dt, for all λ ∈ C. Similarly, if µ ∈ M c (0, ∞)w i t hL a p l a c et r a n s f o r m F (s):=Lµ(s)= ∞ 0 e -sξ dµ(ξ), (2)
and (T (t)) t>0 is a strongly continuous semigroup of bounded operators on X , then we have a functional calculus for its generator A, defined by

F (-A)= ∞ 0 T (ξ) dµ(ξ),
in the sense of the strong operator topology; i.e.,

F (-A)x = ∞ 0 T (ξ)xdµ(ξ), (x ∈ X ),
which exists as a Bochner integral.

Lemma 2.4 Let µ ∈ M c (0, ∞) and (T (t)) t>0 a strongly continuous quasinilpotent semigroup of contractions. Set F = Lµ. Then we have for Re λ ≥ 0,

(F (-A) -F (λ)I)(A + λI) -1 ≤ ∞ 0 td|µ|(t).
Proof : We have

F (-A)(A + λI) -1 = - ∞ 0 T (t) dµ(t) ∞ 0 e λs T (s) ds = - ∞ 0 e -λt ∞ 0 e λ(s+t) T (s + t) ds dµ(t) = - ∞ 0 e -λt ∞ t e λv T (v) dv dµ(t)
,

where v = s + t. This in turn equals - ∞ 0 e -λt ∞ 0 e λv T (v) dv dµ(t)+ ∞ 0 e -λt t 0 e λv T (v) dv dµ(t) = F (λ)(A + λI) -1 + ∞ 0 t 0 e λ(v-t) T (v) dv dµ(t).
For Re λ ≥ 0, we have

t 0 e λ(v-t) T (v) dv ≤ t,
and so the conclusion follows.

The following theorem applies to several examples studied recently in [START_REF] Bendaoud | Distances between elements of a semigroup and estimates for derivatives[END_REF][START_REF] Esterle | Distance near the origin between elements of a strongly continuous semigroup[END_REF][START_REF] Esterle | Distance entre éléments d'un semi-groupe dans une algèbre de Banach[END_REF][START_REF] Kalton | Power-bounded operators and related norm estimates[END_REF]; these include µ = δ 1δ 2 ,t h ed i fference of two Dirac measures, where F (s): =Lµ(s)=e -se -2s and F (-sA)=T (t) -T (2t). More importantly, the theorem applies to many other examples, such as

dµ(t)= (χ [1,2] -χ [2,3] )(t)dt and µ = δ 1 -3δ 2 + δ 3 + δ 4 ,
which are not accessible with the methods of [START_REF] Bendaoud | Distances between elements of a semigroup and estimates for derivatives[END_REF][START_REF] Esterle | Distance near the origin between elements of a strongly continuous semigroup[END_REF][START_REF] Esterle | Distance entre éléments d'un semi-groupe dans une algèbre de Banach[END_REF][START_REF] Kalton | Power-bounded operators and related norm estimates[END_REF].

Theorem 2.5 Let µ ∈ M c (0, ∞) be a real measure such that ∞ 0 dµ(t)=0,
and let (T (t)) t>0 be a strongly continuous quasinilpotent semigroup of bounded operators on a Banach space X . Set F = Lµ. Then there is an

η > 0 such that F (-sA) > max x≥0 |F (x)| for 0 <s≤ η.
Proof : It follows from Theorem 2.1 that we may assume without loss of generality that (T (t)) t>0 is a strongly continuous quasinilpotent semigroup of contractions. Let α > 0b es u c ht h a t|F (x)| ≤ |F (α)| for all x ≥ 0, and let s>0. By considering -µ instead of µ, if necessary, we may suppose that F (α) > 0.

By Lemma 2.4 applied to the semigroup (T (st)) t>0 , for Re λ ≥ 0w e obtain

F (-sA)(sA + λI) -1 ≥ F (λ)(sA + λI) -1 - ∞ 0 td|µ|(t).
It follows that

F (-sA) ≤|F (λ)| - 1 (sA + λI) -1 ∞ 0 td|µ|(t)
for s>0a n dR eλ ≥ 0.

Suppose that there exists s ∈ (0, 1) such that F (-sA) ≤F (α), and consider the simple Jordan curve

Γ := [α,a 1 ] ∪ Γ 1 ∪ [a 2 ,a 3 ] ∪ [a 3 , a 3 ] ∪ [a 3 , a 2 ] ∪ Γ 1 ∪ [a 1 , α],
where Γ 1 ,a 1 ,a 2 ,a 3 are defined as in Theorem 2.2, taking f = F (see Figure 1). We now make various estimates of (sA + λI) -1 for λ on three different parts of Γ.

α a 0 a 1 a 2 a 3 Γ 1
1) For λ ∈ [α,a 1 ] ∪ [a 1 , α]w eh a v e F (α) ≥ F (-sA) ≥|F (λ)| - 1 (sA + λI) -1 ∞ 0 td|µ|(t) ≥ F (α)+δ|λ -α| m - 1 (sA + λI) -1 ∞ 0 td|µ|(t).
Hence we obtain

(sA + λI) -1 ≤ 1 δ|λ -α| m ∞ 0 td|µ|(t). (3)
2

) For λ ∈ Γ 1 ∪ [a 2 ,a 3 ] ∪ [a 3 , a 2 ] ∪ Γ 1 we have F (α) ≥ F (-sA) ≥|F (λ)| - 1 (sA + λI) -1 ∞ 0 td|µ|(t) ≥ |F (a 0 )| - 1 (sA + λI) -1 ∞ 0 td|µ|(t).
It follows that

(sA + λI) -1 ≤ 1 |F (a 0 )| -F (α) ∞ 0 td|µ|(t). (4)
3) For x ∈ R,

(A + ixI) -1 = - ∞ 0 T (t)e ixt dt ≤ ∞ 0 T (t) dt < ∞,
since (T (t)) t>0 is quasinilpotent and contractive. Therefore

(sA + λI) -1 = 1 s A + λ s I -1 ≤ 1 s ∞ 0 T (t) dt (5)
for all λ ∈ [a 3 , a 3 ].

We can now provide estimates for the quantity (λ

-α) m A + λ s I -1
for λ on Γ.L e tR =max λ∈Γ |λ -α|.

By (3) (λ -α) m A + λ s I -1 ≤ s δ ∞ 0 td|µ|(t) for all λ ∈ [α,a 1 ] ∪ [a 1 , α]
.

By (4) (λ -α) m A + λ s I -1 ≤ sR m |F (a 0 )| -F (α) ∞ 0 td|µ|(t) for all λ ∈ Γ 1 ∪ [a 2 ,a 3 ] ∪ [a 3 , a 2 ] ∪ Γ 1 . By (5) (λ -α) m A + λ s I -1 ≤ R m ∞ 0 T (t) dt for all λ ∈ [a 3 , a 3 ].
Since 0 <s≤ 1, for all z ∈ Γ ∪ int Γ we have

A + z s I -1 ≤ M |z -α| m ,
by the maximum modulus principle, where

M =max R m ∞ 0 T (t) dt, R m |F (a 0 )| -F (α) ∞ 0 td|µ|(t), 1 δ ∞ 0 td|µ|(t) .
Since by hypothesis F (0) = 0, there is an r ∈ (0, α)s u c ht h a t

sup |z|≤r |F (z)| <F(α). Since D(0,r) ∩ Γ ∩ C + = ∅,w eh a v eD(0,r) ∩ C + ⊂ Γ ∪ int Γ. Now if z ∈ D(0,r)w i t hR ez>0, we have |z -α| ≥ α -r,a n dt h u sw e have A + z s I -1 ≤ M |z -α| m ≤ M (α -r) m .

Also sup

Re z≤0

(A + zI) -1 ≤ ∞ 0 T (t) dt < ∞.
Now, since by Liouville's theorem the function z → (A + zI) -1 is unbounded on C, it follows that for all u>0s u fficiently small the inequality

A + z u I -1 ≤ M (α -r) m
fails to hold for some z ∈ D(0,r) ∩ C + , depending on u.

It follows that there is an η > 0s u c ht h a t F (-uA) >F(α)f o r a l l u ∈ (0, η].

If µ ∈ M c (0, ∞)i sn o wac o m p l e xm e a s u r e ,t h e nw ew r i t e F (z)=F (z), which is also an entire function, indeed, the Laplace transform of µ.

Corollary 2.6 Let µ ∈ M c (0, ∞) be a complex measure such that ∞ 0 dµ(t)= 0, and let (T (t)) t>0 be a strongly continuous quasinilpotent semigroup of bounded operators on a Banach space X . Set F = Lµ. Then there is an η > 0 such that

F (-sA) F (-sA) > max x≥0 |F (x)| 2
for 0 <s≤ η.

Proof : The result follows on applying Theorem 2.5 to the real measure ν := µ * µ, whose Laplace transform satisfies Lν(s)=F (s) F (s).

We now give similar results for smoother semigroups: let p>0b ea n integer, and write U (p) for the class of semigroups (T (t)) t>0 such that the mapping t → T (t)i sp times continuous differentiable with respect to the norm topology. Let E (p) denote the class of distributions of order p with compact support in (0, ∞). For ϕ ∈ E (p) its action on a C p function f may be specified in terms of measures µ 0 ,....µ p , namely,

f, ϕ = p j=0 ∞ 0 f (j) (t) dµ j (t).
The Laplace transform of ϕ is given by

F (z):=Lϕ(z)= p j-0 ∞ 0 (-z) j e -zt dµ j (t).
We write G j = Lµ j and F j (z)=(-z) j G j (z)f o re a c hj. Likewise

F (-A)= p j=0 A j G j (-A)= p j=0 ∞ 0 A j T (t) dµ j (t). ( 6
)
We begin with the counterpart of Lemma 2.4.

Lemma 2.7 Let p ≥ 1 and ϕ ∈ E (p) , and let (T (t)) t>0 be a quasinilpotent U (p) semigroup of contractions. Set F = Lϕ. Then we have for Re λ ≥ 0,

(F (-A) -F (λ)I)A -p (A + λI) -1 ≤ p m=0 c m A m-p + p m=0 d m m-1 k=0 |λ| k A m-1-k-p , where c m = ∞ 0 td|µ m |(t) and d m = ∞ 0 d|µ m |(t)
for m =0, 1,...,p.

noting the dependence on s and λ. With the notation of the proof of Theorem 2.5 we have three key estimates:

1) For λ ∈ [α,a 1 ] ∪ [a 1 , α]w eh a v e F (α) ≥ F (-sA) ≥|F (λ)| - K (sA + λI) -1 (sA) -p ≥ F (α)+δ|λ -α| m - K (sA + λI) -1 (sA) -p .
Hence we obtain

(sA + λI) -1 (sA) -p ≤ K δ|λ -α| m . (7) 2) For λ ∈ Γ 1 ∪ [a 2 ,a 3 ] ∪ [a 3 , a 2 ] ∪ Γ 1 we have F (α) ≥ F (-sA) ≥|F (λ)| - K (sA + λI) -1 (sA) -p ≥ |F (a 0 )| - K (sA + λI) -1 (sA) -p .

It follows that

(sA + λI) -1 (sA) -p ≤ K |F (a 0 )| -F (α) . ( 8
)
3) For x ∈ R,

(A + ixI) -1 A -p = - ∞ 0 T (t)e ixt dtA -p ≤ A -1 p ∞ 0 T (t) dt < ∞,
since (T (t)) t>0 is quasinilpotent and contractive. Therefore

(sA + λI) -1 (sA) -p ≤ 1 s p+1 A -1 p ∞ 0 T (t) dt (9)
for all λ ∈ [a 3 , a 3 ].

We estimate (λα) m A + λ s I -1 A -p for λ on Γ.

Let R =max λ∈Γ |λ -α|. By ( 7)

(λ -α) m A + λ s I -1 A -p ≤ Ks p+1 δ for all λ ∈ [α,a 1 ] ∪ [a 1 , α]
.

By (8) (λ -α) m A + λ s I -1 A -p ≤ Ks p+1 R m |F (a 0 )| -F (α) for all λ ∈ Γ 1 ∪ [a 2 ,a 3 ] ∪ [a 3 , a 2 ] ∪ Γ 1 . By (9) (λ -α) m A + λ s I -1 A -p ≤ R m A -1 p ∞ 0 T (t) dt for all λ ∈ [a 3 , a 3 ]. Since 0 <s≤ 1, for all z ∈ Γ ∪ int Γ with |z| ≤ r we have A + z s I -1 A -p ≤ M |z -α| m ≤ M (α -r) m , (10)
by the maximum modulus principle, where

M =s u p 0<s≤1 z∈Γ∪int Γ max K(s, z)s p+1 δ , K(s, z)s p+1 R m |F (a 0 )| -F (α) ,R m A -1 p ∞ 0 T (t) dt , which is finite.
With this new choice of M , the proof is now concluded as for the proof of Theorem 2.5, using the observation that (A + zI) -1 A -p is unbounded on C, and obtaining a contradiction from (10).

The non-quasinilpotent case

Let (T (t)) t>0 be a semigroup of non-quasinilpotent operators, and let A T denote the closed (commutative) algebra generated by the semigroup. We write A T for the maximal ideal space of A T . Recall that this is compact if and only if A T / Rad(A T )i su n i t a l ;o t h e r w i s ei ti sl o c a l l yc o m p a c t ,a n dt h e function a : χ → χ(a)i sc o n t i n u o u so n A T for every a ∈ A T .

Recall that A T is said to have an exhaustive sequence of idempotents (P n ) n≥1 if P 2 n = P n P n+1 = P n for all n and for every χ ∈ A T there is a p such that χ(P n )=1foralln ≥ p.

The following result is part of the folklore of the subject, and it is partly contained in [START_REF] Esterle | Distance entre éléments d'un semi-groupe dans une algèbre de Banach[END_REF]Lem. 3.1] and [START_REF] Bendaoud | Distances between elements of a semigroup and estimates for derivatives[END_REF]Lem. 3.1]. It enables us to regard A itself as an element of C( A T )b yd e fi n i n ga na p p r o p r i a t ev a l u eχ(A)=-a χ for each χ ∈ A T . Lemma 3.1 For a strongly continuous and eventually norm-continuous semigroup (T (t)) t>0 and a nontrivial character χ ∈ A T there is a unique a χ ∈ C such that χ(T (t)) = e -taχ for all t>0. Moreover, the mapping χ → a χ is continuous, and χ(F (-uA)) = F (ua χ ) in the case that F = Lµ, as in [START_REF] Dales | Banach algebras and automatic continuity,v o l u m e2 4o f London Mathematical Society Monographs. New Series[END_REF].

Proof : The existence of a χ is given in [START_REF] Esterle | Distance entre éléments d'un semi-groupe dans une algèbre de Banach[END_REF], and its uniqueness is clear since the values of e -taχ for t>0d e t e r m i n ea χ uniquely.

For the continuity, note that

χ(T (1)) = e -aχ and χ e λ ∞ 1 T (t)e -λt dt = 1 a χ + λ e -aχ
if λ is taken sufficiently large that the integral converges. Thus if we have an e tχ α → χ then e -aχ α → e -aχ and 1 aχ α +λ e -aχ α → 1 aχ+λ e -aχ , which easily implies that a χα → a χ .

The final observation follows from an easy argument using Bochner integrals.

The following result will also be required. If P is an idempotent of A T , then t>0 PT(t)A T is dense in the unital Banach algebra P A T . Hence P A T = t>0 PT(t)A T ; also PT(t)isin v ertible in P A T for some, and hence for all, t>0, and then lim t→0+ P -PT(t) =0, since the semigroup is eventually continuous.

For the last observation, it follows from Theorem 2.5 that π(T (t)) = 0 for every t>0, where π : A T → A T / n≥1 P n A T denotes the canonical surjection.

  using the continuity of f and the fact that f (0) = 0. Now sup x>0 |f (xu)| ≥ δ for all u ∈ C \{0}, and the conclusion follows.

Figure 1 :

 1 Figure 1: The curve constructed in the proof of Theorem 2.5

Lemma 3 . 2

 32 Let (T (t)) t>0 be a non-quasinilpotent and eventually norm-continuous semigroup in a Banach algebra, with infinitesimal generator A; let A T be the subalgebra generated by the semigroup and Λ = {a χ : χ ∈ A T }, as in Lemma 3.1. Then the following conditions are equivalent: (i) A T has an exhaustive sequence of idempotents.(ii) For each integer m ≥ 1 the set Λ m := {λ ∈ Λ :Reλ ≤ m} is contained in a compact relatively open subset of Λ.Proof :(i) ⇒ (ii):L e tθ : χ → a χ be the homeomorphism given by Lemma 3.1. If (i) holds, thenK := θ -1 (Λ m )i sac o m p a c ts u b s e to f A T . For each χ ∈ A T , let n χ be such that χ(P n )=1f o rn ≥ n χ .S o {χ ∈ A T : χ(P n )=1}n is an open cover of K. By compactness, there is an N such that χ(P N )=1f o ra l lχ ∈ K, so K ⊂ {χ ∈ A T : χ(P N )=1 }, and θ(K)i sc o m p a c t ,o pe ni nλ, and contains Λ m .(ii) ⇒ (i):C o n v e r s e l y ,i f( i i )h o l d s ,t h e nf o re a c hm ≥ 1, the set Λ m is contained in a compact relatively open set Ω m ⊂ Λ.S oθ -1 (Ω m ) ⊂ A T iscompact and open; hence by Shilov's idempotent theorem [2, Thm. 2.4.33] there is an idempotent P m in A T such that χ(P m )=1forχ ∈ θ -1 (Ω m )a n d 0 otherwise. Now (P m ) m≥1 is an exhaustive sequence of idempotents in A T .Theorem 3.3 Let (T (t)) t>0 be a strongly continuous and eventually normcontinuous non-quasinilpotent semigroup on a Banach space X , with generatorA. Let F = Lµ, where µ ∈ M c (0, ∞) is a real measure such that ∞ 0 dµ =0. If there exists (u k ) k ⊂ (0, ∞) with u k → 0 such that ρ(F (-u k A)) < sup x>0 |F (x)|,then the algebra A T possesses an exhaustive sequence of idempotents (P n ) n≥1 such that each semigroup (P n T (t)) t>0 has a bounded generator.If, further, F (-u k A) < sup x>0 |F (x)|, then n≥1 P n A T is dense in A T .Proof : For m ≥ 1l e tΛ m = {a χ :Rea χ ≤ m}, where χ(T (t)) = e -aχt as in Lemma 3.1. Now Re a χ ≤ m if and only if |χ(T (1))| ≥ e -m , which implies that Λ m is compact, by the fact that A T ∪ {0} is compact, and χ → a χ is continuous (Lemma 3.1). Therefore, there is an R m > 0s u c ht h a tΛ m ⊂ D(0,R m ). Note that, by the definition of Λ m ,w eh a v eΛ ∩ C -= Λ ∩ C -⊂ D(0,R m ).By hypothesis there exists a u k > 0s u c ht h a t|u k | < r R m ,where r>0i sg i v e nb yL e m m a2 . 1a n dρ(F (-u k A)) < sup x>0 |F (xu k )|. It follows that Λ m ⊂ D(0,R m ) ⊂ D(0,r/|u k |)a n dt h u s |F (u k a χ )| < sup x>0 |F (u k x)| for all a χ ∈ Λ m . Let α k be such that sup x>0 |F (xu k )| = |F (α k u k )|.By Theorem 2.2 there exists a curve Γ k,0 in {z :R ez ≥ 0, Im z>0} joining α k ∈ R + to v k ∈ iR + with |v k | >R m on which |F (u k z)| ≥ |F (α k u k )|. Let Γ k = Γ k,0 ∪ {z ∈ C :Rez<0, |z| = |v k |} ∪ Γ k,0 (see Figure 2).

Figure 2 :

 2 Figure 2: Diagram for the proof of Theorem 3.3

Remark 3 . 4 A

 34 similar result holds for complex measures µ ∈ M c (0, ∞); namely, we replaceρ(F (-u k A)) < sup x>0 |F (x)|, by the symmetrised version ρ(F (-u k A) F (-u k A)) < sup x>0 |F (x)| 2 ,as in Corollary 2.6.

  this is a nonempty open set; we let V denote the connected component of U containing a 1 .We claim that∂V ∩ iR = ∅. Indeed, note that if z ∈ ∂V ∩ C + , then |f (z)| ≤ |f (a 0 )|, as otherwise if |f (z)| > |f (a 0 )|, then by the continuity of |f | > |f (a 0 )| for all z ∈ C + with |za 3 | < β. It follows that there is a point a 2 ∈ C + with Im a 2 =Ima 3 and (a 3 ,a 2 ] ⊂ V .Since V is open and connected, it is path-connected, and so we may join a 1

	aneigh bourhoodofz is contained in V . But we cannot have |f (z)| ≤ |f (a 0 )|
	on the whole of ∂V , as then by the strong maximum principle (see, e.g. [7,
	Thm. 9.4]) this inequality would hold for all z ∈ V including a 1 .
	So there exists a 3 ∈ ∂V ∩ iR with Im a 3 > 0and|f (a 3 )| > |f (a 0 )|. By the
	continuity of |f |, there exists β > 0s u c ht h a t|f (z)|

Acknowledgements

This work was partially supported by the ANR project ANR-09-BLAN-0058-01, the London Mathematical Society (Scheme 2), and the Institut Camille Jordan.

Proof :

Write B := (F (-A) -F (λ)I)A -p (A + λI) -1 . Then by [START_REF] Kalton | Power-bounded operators and related norm estimates[END_REF] we have

This can be rewritten as

Now the first terms can be estimated using Lemma 2.4, and for the second we use the obvious estimate |G m (λ)| ≤ d m for Re λ ≥ 0.

Theorem 2.8 Let p>1 and ϕ ∈ E (p) be a real distribution given by measures µ 0 ,...,µ p such that ∞ 0 dµ 0 (t)=0, and let (T (t)) t>0 be a quasinilpotent U (p) semigroup of bounded operators on a Banach space X . Set F = Lϕ.

Then there is an η > 0 such that

for 0 <s≤ η.

Proof : The proof is very similar to the proof of Theorem 2.5, but using Lemma 2.7, so we indicate the changes necessary. It will be convenient to take 0 <s≤ 1a n dt ow r i t e K = K(s, λ)=


I Chalendar
email: chalendar@math.univ-lyon1.fr

J Esterle
email: jean.esterle@math.u-bordeaux1.fr.

J R Partington
email: j.r.partington@leeds.ac.uk

Dichotomy results for norm estimates in operator semigroups

here is the semigroup T (t):x 7 ! x t in the algebra C 0 ([0, 1]) of continuous functions on [0, 1] vanishing at the origin.

Later, we consider semigroups defined on a sector in the complex plane, in which case they will be assumed to be analytic: that is, complex-differentiable in the norm topology.

The results in this survey indicate that the quantitative behaviour of the semigroup at the origin provides additional qualitative information, such as uniform continuity or analyticity. Here are a few examples.

We recall the classical 0-1 law, asserting that if lim sup t!0 + kT (t)-Ik < 1, then in fact kT (t)-Ik!0andhencethesemigroupisuniformlycontinuous, and of the form e tA for some bounded operator A. To see this, set

 Thm. 10.3.6]). This result is usually stated for n =1 , but Hille's argument works for any positive integer.

Introduction

We recall that a one-parameter family (T (t)) 0<t<1 in a Banach algebra (often itself simply the algebra of bounded linear operators on a Banach space X) is a semigroup if T (s + t)=T (s)T (t)f o r a l l t, s > 0.

We shall be concerned here with semigroups that are strongly continuous on (0, 1), but not necessarily continuous at the origin. An important example Theorem 1.1 Let (T (t)) t>0 be a n-times continuously differentiable semigroup over the positive reals. If lim sup t!0 + kt n T (n) (t)k < ( n e ) n , then the generator of the semigroup is bounded.

In the direction of analyticity, a classical result of Beurling [START_REF] Beurling | On analytic extension of semigroups of operators[END_REF] (see also [START_REF] Fackler | Regularity of semigroups via the asymptotic behaviour at zero[END_REF]) is the following: Theorem 1.2 A C 0 -semigroup (T (t)) 0t<1 on a complex Banach space X is holomorphic if and only if there exists a polynomial p such that lim sup t!0 + kp(T (t))k < sup{|p(z):|z|  1}.

(1)

Kato [START_REF] Kato | A characterization of holomorphic semigroups[END_REF] and Neuberger [START_REF] Neuberger | Analyticity and quasi-analyticity for one-parameter semigroups[END_REF] proved the sufficiency of (1) with p(z)= z -1, and sup |p(z)| =2,pro vidinga0-2la wforanalyticit y . Ingeneralthe converse is not true with p(z)=z -1, although it holds if X is uniformly convex and the semigroup is contractive [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF].

The more recent results considered in this survey concern estimates of the norm or spectral radius of quantities such as T (t) -T ((n +1)t)ast tends to 0. These are often formulated as dichotomy results, such as the zero-quarter law (the case n =1inthefollo wingtheorem).

Theorem 1.3 ( [START_REF] Esterle | Distance entre éléments d'un semi-groupe dans une algèbre de Banach[END_REF][START_REF] Mokhtari | Distance entre éléments d'un semi-groupe continu dans une algèbre de Banach[END_REF]) Let n ≥ 1 be an integer, and let (T (t)) t>0 be a semigroup in a Banach algebra. If lim sup t!0 + kT (t) -T ((n +1)t)k < n (n +1) 1+1/n , then either T (t)=0for t>0 or else the closed subalgebra generated by (T (t)) t>0 is unital, and the semigroup has a bounded generator A: that is, T (t)=exp(tA) for t>0.

Another result that we mention here concerns the link between the norm and spectral radius ⇢, and was motivated also by the Esterle-Katznelson-Tzafriri results on estimates for kT n -T n+1 k, where T is a power-bounded operator (see [START_REF] Esterle | Quasimultipliers, representations of H 1 , and the closed ideal problem for commutative Banach algebras[END_REF][START_REF] Katznelson | On power bounded operators[END_REF]). We have rewritten it in the notation of differentiable groups (T (t)) = (e tA ), which may be defined for t 2 C if A is bounded; note that T 0 (t)=AT (t).

Theorem 1.4 ([14])

Let A be an operator on a Banach space, and let (T (t)) be the group given by T (t)=e tA . Then each of the following conditions implies that ⇢(A)=kAk.

(i) sup t>0 tkT 0 (t)k1/e;

(ii) sup t>0 kT (t) -T ((s +1)t)ks(s +1) -(1+1/s) for some s>0;

(iii) sup t>0 kT ((s + i)t) -T ((s -i)t)k2e -s arctan(1/s) / p 1+s 2 for some s ≥ 0.

The third condition of Theorem 1.4 is linked to the Bonsall-Crabb proof of Sinclair's spectral radius formula for Hermitian elements of a Banach algebra, given in [START_REF] Bonsall | The spectral radius of a Hermitian element of a Banach algebra[END_REF].

In Sections 2 and 3 we review the existing literature on dichotomy laws for semigroups, first for semigroups on R + and then for analytic semigroups defined on a sector in the complex plane; our main sources here are [START_REF] Bendaoud | Distances between elements of a semigroup and estimates for derivatives[END_REF] and [START_REF] Chalendar | Boundary values of analytic semigroups and associated norm estimates[END_REF]. Then in Section 4 we present some very recent generalizations of these results, formulated in the language of functional calculus: this discussion is based on [START_REF] Chalendar | [END_REF] and [START_REF] Chalendar | Lower estimates near the origin for functional calculus on operator semigroups[END_REF].

Semigroups on R +

We begin with a result on quasinilpotent semigroups.

Theorem 2.1 ( [START_REF] Esterle | Distance near the origin between elements of a strongly continuous semigroup[END_REF]) Let (T (t)) t>0 be a C 0 -semigroup of bounded quasinilpotent linear operators on a Banach space X. Then there exists δ > 0 such that kT (t) -T (s)k > ✓(s/t) for 0 <t<s<δ, where

✓(s/t)=(s -t)t t/(s-t) s s/(t-s) =m a x 0x1 (x t -x s ).
In particular, for all γ > 0, there exists δ > 0 such that

kT (t) -T ((γ +1)t)k > γ (γ +1) 1+1/γ
for all 0 <t<δ. This is a sharp result, in the sense that given ✏ :( 0 , 1) ! (0, 1) nondecreasing, there exists a nontrivial quasinilpotent semigroup (T ✏ (t)) t>0 on a Hilbert space such that:

kT ✏ (t) -T ✏ (s)k✓(s/t)+(s -t)✏(s).
It is a quantitative formulation of an intuitive fact: T (t)c a n n o tb eu n iformly too close to T (s)f o rs 6 = t, with s, t small when the generator is unbounded.

The following result is a strengthening of Theorem 2.3 of [START_REF] Esterle | Distance entre éléments d'un semi-groupe dans une algèbre de Banach[END_REF]. Note that Rad A denotes the radical of the algebra A.

Theorem 2.2 ([1]

) Let (T (t)) t>0 be a non-quasinilpotent semigroup in a Banach algebra, let A be the closed subalgebra generated by (T (t)) t>0 , and let γ > 0 be a real number. If

⇢(T (t) -T (t(γ +1)))< γ (γ +1) 1+ 1 γ
for 0 <t t 0 , for some t 0 > 0, then A/ Rad(A) is unital, and there exist an idempotent J in A, an element u of JA and a mapping r : R + ! Rad(JA), with the following properties:

(i) '(J)=1for all ' 2 b A;

(ii) r(s + t)=r(s)+r(t) for all s, t 2 R + ;

(iii) JT(t)=e tu+r(t) for t 2 R + , where e

v = J + X k≥1 v k k! for v 2 JA; (iv) (T (t) -JT(t)) t2R + is a quasinilpotent semigroup.
If A is semi-simple (that is, Rad(A)={0}), then the conclusion is much more straightforward.

Corollary 2.3 ([1]

) Let (T (t)) t2R + be a non-trivial semigroup in a commutative semi-simple Banach algebra, let A be the closed subalgebra generated by (T (t)) t2R + and let γ > 0.I f

⇢(T (t) -T ((γ +1)t)) < γ (γ +1) 1+ 1 γ ,
for 0 <t t 0 , for some t 0 > 0, then A is unital and there exists an element u 2 A such that T (t)=e tu for t 2 R + .

The following theorem needs no hypothesis on A, but requires stronger estimates, based on the norm rather than the spectral radius.

Theorem 2.4 ([1]

) Let (T (t)) t>0 be a non-trivial semigroup in a Banach algebra, let A be the closed subalgebra generated by (T (t)) t>0 and let n ≥ 1 be an integer. If there exists t 0 > 0 such that

k(T (t) -T (t(n +1)))k < n (n +1) 1+ 1 n
for 0 <t t 0 , then A possesses a unit J, lim t!0 + T (t)=J and there exists u 2 A such that T (t)=e tu for all t>0.

If (T (t) t>0 is a quasinilpotent semigroup, then the condition

kT (t) -T ((n +1)t)k < n (n +1) 1+1/n for 0 <t t 0 implies that T (t)=0.
The sharpness of the above result is shown by the following example.

Example 2.5 Let G be an additive subgroup of R, measurable, with G 6 = R. Then given (γ n ) n in R + such that tγ n 2 G for all t 2 G, t > 0, there exists a nontrivial semigroup (S(t)) t2G,t>0 in c 0 such that kS(t) -S(t(γ n +1))k < γ n (γ n +1) 1+1/γn , for all t 2 G, t>0.

Sectorial semigroups

In this section we discuss the behaviour of analytic semigroups defined on a sector

S ↵ = {z 2 C : Re(z) > 0a n d| arg(z)| < ↵}. with 0 < ↵  ⇡/2. We begin with the case ↵ = ⇡/2a n dS ↵ = C + . Theorem 3.1 ([1]
) Let (T (t) t2C + be an analytic non-quasinilpotent semigroup in a Banach algebra. Let A be the closed subalgebra generated by (T (t)) t2C + and let γ > 0. If there exists t 0 > 0 such that

sup t2C + ,|t|t 0 ⇢(T (t) -T (γ +1)t)) < 2
then A/ Rad A is unital, and the generator of (⇡(T (t)) t>0 is bounded, where

⇡ : A ! A/ Rad A denotes the canonical surjection.
A semigroup T (t)d e fi n e do v e rt h ep o s i t i v er e a l so ro nas e c t o ri ss a i d to be exponentially bounded if there exists c 1 > 0a n dc 2 2 R such that kT (t)kc 1 e c 2 |t| for every t. Beurling [START_REF] Beurling | On analytic extension of semigroups of operators[END_REF], in his work described in the introduction, showed that there exists a universal constant k such that every exponentially bounded weakly measurable semigroup (T (t)) t>0 of bounded operators satisfying lim sup

t!0 + kI -T (t)k = ⇢ < 2
admits an exponentially bounded analytic extension to a sector S ↵ with ↵ ≥ k(2 -⇢) 2 . From this one easily obtains the following result.

Theorem 3.2 ([1]

) Let (T (t)) t2C + be an analytic semigroup of bounded operators on a Banach space X. If the generator of the semigroup is unbounded, then we have, for

-⇡ 2 < ↵ < ⇡ 2 , lim sup t!0 + kI -T (t)k≥2 - r ⇡ 2 -|↵| k ,
where k is Beurling's universal constant.

We now consider similar results on smaller sectors than the half-plane, and in fact the result we prove will be stated in a far more general context.

Theorem 3.3 ([1]) Let 0 < ↵ < ⇡/2
and let f be an entire function with

f (0) = 0 and f (R) ✓ R, such that sup Re z>r |f (z)| ! 0 as r !1, (2)
and f is a linear combination of functions of the form z m exp(-zw) for m =0 , 1, 2,... and w>0. Let (T (t)) t2Sα =( e x p ( -tA)) t2Sα be an analytic non-quasinilpotent semigroup in a Banach algebra and let A be the subalgebra generated by (T (t)) t2Sα . If there exists t 0 > 0 such that P n k=1 a k = 0. This provides results analogous to those of [START_REF] Kalton | Power-bounded operators and related norm estimates[END_REF]Thm. 4.12], where the behaviour of expressions such as ktA exp(-tA)k and k exp(-tA) -exp(-stA)k was considered for all t>0.

sup t2Sα,|t|t 0 ⇢(f (tA)) <k(S ↵ ), with k(S ↵ )=s u p t2Sα |f (z)|, then A/ Rad A is
Remark 3.4 Another function considered in [START_REF] Kalton | Power-bounded operators and related norm estimates[END_REF] is f (z)=e -sz sin z, where we now require s>tan ↵ for f (tA) to be well-defined for t 2 S ↵ . This does not satisfy the condition (2), but we note that it holds for z 2 S ↵ , while for z 6 2 S ↵ there exists a constant C>0 such that for each z with Re z>Cthere exists λ 2 (0, 1) such that |f (λz)| ≥ sup z2Sα |f (z)|. Using this observation, it is not difficult to adapt the proof of Theorem 3.3 to this case.

As before, the sharpness of the constants can be shown by considering examples in C 0 ([0, 1]).

One particular case of the above is used in the estimates considered by Bendaoud, Esterle and Mokhtari [START_REF] Bendaoud | Distances entre exponentielles et puissances d'éléments de certaines algèbres de Banach[END_REF][START_REF] Esterle | Distance entre éléments d'un semi-groupe dans une algèbre de Banach[END_REF].

Corollary 3.5 Let γ > 0 and 0 < ↵ < ⇡/2. Let (T (t)) t2Sα be an analytic non-quasinilpotent semigroup in a Banach algebra and let A be the subalgebra generated by (T (t)) t2Sα . If there exists t 0 > 0 such that If A is the generator of a semigroup (T (t)) t2Sα analytic on S ↵ , we have

sup t2Sα,|t|t 0 ⇢(T (t) -T (t(γ +1)))<k(S ↵ ), with k(S ↵ )=sup t2Sα | exp(-t)-exp(-(γ +1)t)|, then A/ Rad A is
f n (tA)=t n T (n) (t). So the following result means that if sup t2Sα,0<|t|<δ kf n (tA)k <k n (↵)
for some δ > 0, then the generator of the semigroup is bounded.

The following result may be deduced from Hille's result, Theorem 1.1.

Theorem 3.6 ([1])

Let n ≥ 1 be an integer, let ↵ 2 (0, ⇡/2) and let (T (t)) t2Sα be a semigroup analytic on

S ↵ . If sup t2Sα,0<|t|<δ kt n T (n) (t)k < ⇣ n e cos(↵)
⌘ n for some δ > 0, then the closed algebra generated by the semigroup is unital, and the generator of the semigroup is bounded.

The remainder of this section is devoted to quasinilpotent semigroups.

We begin with the following observation. Remark 3.7 Analytic semigroups bounded near the origin can be extended to the closed sector. Assume that there exists r>0 such that sup t2D(0,r)\Sα kT (t)k < +1.

Then lim t→w t∈Sα T (t)x exists for every x 2 X and every w 2 @S ↵ . Moreover if we set T (w)=lim t→w t∈Sα

T (t)x,
then (T (t)) t2Sα is a semigroup of bounded operators which is continuous with respect to the strong operator topology. For we have lim t→w t∈Sα T (t)T (t 0 )x = T (t 0 )x for every t 0 > 0 and every x 2 X . Now the result follows immediately from the fact that S t>0 T (t)X is dense in X , given that sup z2D(0,r)\Sα kT (t)k < +1.

The next lemma demonstrates that nontrivial quasinilpotent analytic semigroups cannot be bounded on the right half-plane. In fact, more is true.

Lemma 3.8 ([5]) Let (T (t))

Re t>0 be a quasinilpotent analytic semigroup of bounded operators on a Banach space X . Suppose that there exists r>0 such that We are now in a position to give estimates for semigroups on the imaginary axis. Theorem 3.9 ( [START_REF] Chalendar | Boundary values of analytic semigroups and associated norm estimates[END_REF]) Let (T (t)) Re t>0 be a nonzero quasinilpotent analytic semigroup satisfying the conditions of Remark 3.7, and let s>0. Then

sup t2D + (0,r) kT (t)k < +1,
max(⇢(T (iy) -T (iy + is)), ⇢(T (-iy) -T (-iy -is))) ≥ 2,
for every y>0. for some δ > 0 and some µ>0, and let γ > 0. Then

sup t2D + (0,r) kT (t) -T ((1 + γ)t)k≥2,
for every r>0.

Lower estimates for functional calculus

In this section we summarise some very recent results from [START_REF] Chalendar | [END_REF] and [START_REF] Chalendar | Lower estimates near the origin for functional calculus on operator semigroups[END_REF], which provide a far-reaching generalization of earlier work.

Semigroups on R +

The quasinilpotent case

Recall that if (T (t)) t>0 is a uniformly bounded strongly continuous semigroup with generator A, then

(A + λI) -1 = - Z 1 0 e λt T (t) dt,
for all λ 2 C with Re λ < 0. Here the integral is taken in the sense of Bochner with respect to the strong operator topology.

If, in addition, (T (t)) t>0 is quasinilpotent, then

(A + λI) -1 = - Z 1 0 e λt T (t) dt, for all λ 2 C.
Similarly, if µ 2 M c (0, 1)w i t hL a p l a c et r a n s f o r m

F (s):=Lµ(s)= Z 1 0 e -s⇠ dµ(⇠), (3)
and (T (t)) t>0 is a strongly continuous semigroup of bounded operators on X , then we have a functional calculus for its generator A, defined by

F (-A)= Z 1 0 T (⇠) dµ(⇠),
in the sense of the strong operator topology; i.e.,

F (-A)x = Z 1 0 T (⇠)xdµ(⇠), (x 2 X ),
which exists as a Bochner integral.

The following theorem applies to several examples studied in [START_REF] Bendaoud | Distances between elements of a semigroup and estimates for derivatives[END_REF][START_REF] Esterle | Distance near the origin between elements of a strongly continuous semigroup[END_REF][START_REF] Esterle | Distance entre éléments d'un semi-groupe dans une algèbre de Banach[END_REF][START_REF] Kalton | Power-bounded operators and related norm estimates[END_REF]; these include µ = δ 1 -δ 2 ,thedifference of two Dirac measures, where F (s):= Lµ(s)=e -s -e -2s and F (-sA)=T (t) -T (2t). More importantly, the theorem applies to many other examples, such as dµ(t)=(χ [START_REF] Bendaoud | Distances between elements of a semigroup and estimates for derivatives[END_REF][START_REF] Bendaoud | Distances entre exponentielles et puissances d'éléments de certaines algèbres de Banach[END_REF] χ [START_REF] Bendaoud | Distances entre exponentielles et puissances d'éléments de certaines algèbres de Banach[END_REF][START_REF] Beurling | On analytic extension of semigroups of operators[END_REF] )(t)dt and µ = δ 1 -3δ 2 + δ 3 + δ 4 , which are not accessible with the methods of [START_REF] Bendaoud | Distances between elements of a semigroup and estimates for derivatives[END_REF][START_REF] Esterle | Distance near the origin between elements of a strongly continuous semigroup[END_REF][START_REF] Esterle | Distance entre éléments d'un semi-groupe dans une algèbre de Banach[END_REF][START_REF] Kalton | Power-bounded operators and related norm estimates[END_REF]. Theorem 4.1 Let µ 2 M c (0, 1) be a real measure such that

Z 1 0 dµ(t)=0,
and let (T (t)) t>0 be a strongly continuous quasinilpotent semigroup of bounded operators on a Banach space X . Set F = Lµ. Then there is an

η > 0 such that kF (-sA)k > max x≥0 |F (x)| for 0 <s η.
If µ 2 M c (0, 1)i sac o m p l e xm e a s u r e ,t h e nw ew r i t e e F = Lµ, so that e F (z)=F (z). By considering the real measure ν := µ ⇤ µ,w eo b t a i nt h e following result. Corollary 4.2 Let µ 2 M c (0, 1) be a complex measure such that Z 1 0 dµ(t)= 0, and let (T (t)) t>0 be a strongly continuous quasinilpotent semigroup of bounded operators on a Banach space X . Set F = Lµ. Then there is an

⌘ > 0 such that kF (-sA) e F (-sA)k > max x≥0 |F (x)| 2
for 0 <s ⌘.

The non-quasinilpotent case

Recall that A is said to have an exhaustive sequence of idempotents (P n ) n≥1 if P 2 n = P n P n+1 = P n for all n and for every χ 2 b A there is a p such that

χ(P n )=1foralln ≥ p.

Analytic semigroups

For 0 < α < π/2, let H(S ↵ ) denote the Fréchet space of holomorphic functions on S ↵ , endowed with the topology of local uniform convergence; thus, if (K n ) n≥1 is an increasing sequence of compact subsets of S ↵ with S n≥1 K n = S ↵ , we may specify the topology by the seminorms

kF k n := sup{|f (z)| : z 2 K n }.
Now let ' : H(S ↵ ) ! C be a continuous linear functional, in the sense that there is an index n and a constant M>0s u c ht h a t|hf, 'i|  M kf k n for all f 2 H(S ↵ ).

We define the Fourier-Borel transform of ' by FB(')(z)=he -z , 'i,

for z 2 C
, where e -z (⇠)=e -z⇠ for ⇠ 2 S ↵ .

If ' 2 H(S ↵ ) 0 , as above, then by the Hahn-Banach theorem, it can be extended to a functional on C(K n ), which we still write as ',a n di st h u s

given by a Borel measure µ supported on K n .

That is, we have

hf, 'i = Z Sα f (⇠) dµ(⇠),
where µ (which is not unique) is a compactly supported measure. For example, if hf, 'i = f 0 (1), then

hf, 'i = 1 2⇡i Z C f (z) dz (z -1) 2 ,
where C is any sufficiently small circle surrounding the point 1. Note that

  unital and the generator of ⇡(T (t)) t2Sα is bounded, where ⇡ : A ! A/ Rad(A) denotes the canonical surjection.Note that f (tA)i sw e l l -d e fi n e db ym e a n so fT (t)a n di t sd e r i v a t i v e s .Suitable choices of f (z)a r el i n e a rc o m b i n a t i o n so fz m exp(-z), m = 1, 2, 3,..., and exp(-z) -exp(-(γ +1)z); also real linear combinations of the form P n k=1 a k exp(-b k z)w ithb k > 0a n d

  unital and the generator of ⇡(T (t)) t2Sα is bounded, where ⇡ : A ! A/ Rad(A) denotes the canonical surjection. Now set f n (z)=z n e -z , and set k n (↵)=m a x z2Sα f n (z). An immediate computation shows that k n (↵)=

1 - 1 log

 11 and define T (iy) for y 2 R by Remark 3.7. If Z + kT (iy)k 1+y 2 dy < +1, then T (t)=0for Re t>0.

Corollary 3 .

 3 10 ([5]) Let (T (t)) Re t>0 be a quasinilpotent analytic semigroup such that sup y2R e -µ|y| kT (δ + iy)k < +1

Theorem 4 . 3

 43 Let (T (t)) t>0 be a strongly continuous and eventually normcontinuous non-quasinilpotent semigroup on a Banach space X , with gen-erator A. Let F = Lµ, where µ 2 M c (0, 1) is a real measure such that R 1 0 dµ =0. If there exists (u k ) k ⇢ (0, 1) with u k ! 0 such that ρ(F (-u k A)) < sup x>0 |F(x)|, then the algebra A generated by (T (t)) t>0 possesses an exhaustive sequence of idempotents (P n ) n≥1 such that each semigroup (P n T (t)) t>0 has a bounded generator. If, further, kF (-u k A)k < sup x>0 |F (x)|, then S n≥1 P n A is dense in A.

FBCorollary 4 . 5

 45 dµ(⇠). Now let T := (T (t)) t2Sα be an analytic semigroup on a Banach space X , with infinitesimal generator A. Let ' 2 H(S ↵ ) 0 and let F = FB(').We may thus define, formally to start with,F (-A)=hT,'i = Z Sα T (⇠) dµ(⇠),which is well-defined as a Bochner integral in A. It is easy to verify that the definition is independent of the choice of µ representing '. Indeed, if u 2 S ↵-β , where supp µ ⇢ S β and 0 < β < α, then we may also defineF (-uA)= Z S β T (uξ) dµ(ξ),since u⇠ lies in S ↵ .The following theorem extends[START_REF] Bendaoud | Distances between elements of a semigroup and estimates for derivatives[END_REF] Thm. 3.6]. In the following, a symmetric measure is a measure such that µ(S)=µ(S)f o rS ⇢ S ↵ . A symmetric measure will have a Fourier-Borel transform f satisfying f (z)= e f (z):=f (z) for all z 2 C.Theorem 4.4 Let 0 < ↵ < ⇡/2. Let ' 2 H(S ↵ ) 0 , induced by a symmetric measure µ 2 M c (S ↵ ) such that R Sα dµ(z)=0, and let f = FB('). Let (T (t)) t2Sα =(exp(tA)) t2Sα be an analytic non-quasinilpotent semigroup in a Banach algebra and let A be the subalgebra generated by (T (t)) t2Sα . If thereexists t 0 > 0 such that sup t2Sα,|t|t 0 ⇢(f (-tA)) < sup z2Sα |f (z)|,then A/ Rad A is unital and the generator of ⇡(T (t)) t2Sα is bounded, where ⇡ : A ! A/ Rad(A) denotes the canonical surjection. By considering the convolution of a functional ' 2 H(S ↵ ) 0 , with Fourier-Borel transform f , and the functional e ' with Fourier-Borel transform e f ,w e obtain the following result. Let 0 < ↵ < ⇡/2. Let ' 2 H(S ↵ ) 0 , induced by a measure µ 2 M c (S ↵ ) such that R Sα dµ(z)=0 , and let f = FB('). Let (T (t)) t2Sα = (exp(tA)) t2Sα be an analytic non-quasinilpotent semigroup in a Banach algebra and let A be the subalgebra generated by (T (t)) t2Sα . If there exists t 0 > 0 such that sup t2Sα,|t|t 0 ⇢(f (-tA) e f (-tA)) < sup z2Sα |f (z)|| e f (z)|, then A/ Rad A is unital and the generator of ⇡(T (t)) t2Sα is bounded, where ⇡ : A ! A/ Rad(A) denotes the canonical surjection. It is possible to obtain a similar conclusion, based only on estimates on the positive real line.

Theorem 4 . 6 Example 4 . 7

 4647 Let 0 < ↵ < ⇡/2. Let ' 2 H(S ↵ ) 0 , induced by a symmetric measure µ 2 M c (S ↵ ) such that R Sα dµ(z)=0 , and let f = FB('). Suppose that f (x) 2 R for x 2 R + . Let (T (t)) t2Sα =(exp(tA)) t2Sα be an analytic nonquasinilpotent semigroup in a Banach algebra and let A T be the subalgebra generated by (T (t)) t2Sα . If there exists t 0 > 0 such that⇢(f (-tA)) < sup x>0 |f (x)|,for all 0 <t t 0 , then A/ Rad A is unital and the generator of ⇡(T (t)) t2Sα is bounded, where ⇡ : A ! A/ Rad(A) denotes the canonical surjection.The following example shows that the hypotheses of Theorem 4.6 are sharp. Similar examples show the sharpness of other results given. In the Banach algebra A = C 0 ([0, 1]) consider the semigroup T (t):x 7 ! x t . Clearly there is no convergence as t approaches 0. For x 2 (0, 1] (which can be identified with the Gelfand space of A) let f = FB(µ) and f (-tA)(x)= Z Sα x -t⇠ dµ(⇠)= Z Sα e -t⇠ log x dµ(⇠), where µ 2 M c (S ↵ ), supposing that R Sα dµ(z)=0and that f is real for x>0. Thus f (-tA)(x)=f (-t log x) and ⇢(f (-tA)) = kf (-tA)k =sup x>0 |f (-t log x)| =sup r>0 |f (tr)|. Clearly sup t2Sα,|t|t 0 ⇢(f (-tA)) = sup t2Sα |f (z)| for all t 0 > 0.

Acknowledgements

This work was partially supported by the London Mathematical Society (Scheme 2).


Boris Andreianov

Carlotta Donadello

Ulrich Razafison

Massimiliano D Rosini

Riemann problems with non-local point constraints and capacity drop

Keywords: 35L65, 90B20 Riemann problem, non-local constrained hyperbolic PDE's, loss of self-similarity, loss of uniqueness, crowd dynamics, capacity drop

In the present note we discuss in details the Riemann problem for a onedimensional hyperbolic conservation law subject to a point constraint. We investigate how the regularity of the constraint operator impacts the wellposedness of the problem, namely in the case, relevant for numerical applications, of a discretized exit capacity. We devote particular attention to the case in which the constraint is given by a non-local operator depending on the solution itself. We provide several explicit examples.

We also give the detailed proof of some results announced in the paper [Andreainov, Donadello, Rosini, Crowd dynamics and conservation laws with non-local point constraints and capacity drop], which is devoted to existence and stability for a more general class of Cauchy problems subject to Lipschitz continuous non-local point constraints.

Introduction 1.Point constraints in traffic modeling

Traffic modeling is an exciting and fast-developing field of research with plentiful applications to real life. While this subject was initially limited to the description and the management of vehicular traffic, we see a growing interest nowadays on different applications as crowd dynamics and bio-mathematics. This note is related to an extensive on-going research project concerning the theoretical and the numerical study of macroscopic models for which the definition of solution involves an artificial limitation of the flux in a finite number of points. From the modeling point of view, this may correspond to a narrow exit in crowd modeling, a toll gate in vehicular traffic, a cell membrane in bio-medical modeling.

In the pioneering paper [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF], R. Colombo and P. Goatin introduced point constraints in the classical one-dimensional LWR road traffic model [START_REF] Lighthill | On kinematic waves. II. A theory of traffic flow on long crowded roads[END_REF][START_REF] Richards | Shock waves on the highway[END_REF], with the goal to model the presence of obstacles on the road as toll gates and road lights. This model reads as

∂ t ρ + ∂ x f (ρ) = 0 (t, x) ∈ R + × R (1a) f (ρ(t, 0±)) ≤ q(t) t ∈ R + (1b) ρ(0, x) = ρ 0 (x) x ∈ R, (1c)
where ρ = ρ(t, x) ∈ [0, R] is the unknown (mean) density at time t ∈ R + of vehicles moving along the road parameterized by the coordinate x ∈ R. Then, R ∈ R + is the maximal road density, f : [0, R] → R is the nonlinearity relating the flux in the direction of increasing x to the density, q : R + → R + is a given function prescribing the maximal flow allowed through the point x = 0, and ρ 0 : R → [0, R] is the initial (mean) density. Finally, ρ(t, 0-) denotes the left measure theoretic trace along the constraint implicitly defined by

lim ε↓0 1 ε +∞ 0 0 -ε |ρ(t, x) -ρ(t, 0-)| φ(t, x) dx dt = 0
for all φ ∈ C ∞ c (R 2 ; R). The right measure theoretic trace, ρ(t, 0+), is defined analogously.

In the above setting, the authors of [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF] were able to prove existence and wellposedness of solutions. Further theoretical considerations and numerics associated to this model have been developed in [START_REF] Andreianov | Finite volume schemes for locally constrained conservation laws[END_REF].

In view of the applications, however, we also need to consider the case in which the evolution of the constraint function q is not given beforehand, but depends on the solution ρ itself in a neighborhood of x = 0. In such situation we say that the point constraint is non-local. In this way we obtain crowd and cell membrane dynamics models described by coupled PDE-ODE systems for which the existence and well-posedness of solutions are not trivial matter. Nevertheless, this case has a practical interest. In crowd dynamics, as an example, the experimental observations by E. Cepolina in [START_REF] Elvezia | Phased evacuation: An optimisation model which takes into account the capacity drop phenomenon in pedestrian flows[END_REF] prove that the irrational behavior of pedestrians at bottlenecks ends up by reducing the maximal possible outflow. This phenomenon, called capacity drop, is also related to other effects observed in crowd dynamics, such Faster Is Slower and the Braess' paradox.

In full generality, we may consider the constraint function q as follows

q(t) = Q[ρ](t) ,
where Q : C 0 [0, T ]; L 1 (R; [0, R]) → L 1 ([0, T ]; [0, R]). The minimal regularity properties to impose on Q in order to achieve well-posedness of solutions are not known at the moment, and they are the object of one of our current research projects.

An example of non-local point constraint

In the paper [START_REF] Andreianov | Crowd dynamics and conservation laws with non-local constraints and capacity drop[END_REF], B. Andreianov, C. Donadello and M. D. Rosini proposed a model which generalizes the one in [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF] and consists of a Cauchy problem for a onedimensional hyperbolic conservation law as 1 subject to a non-local point constraint of the form

q(t) = p R- w(x) ρ(t, x) dx t ∈ R + . (2)
Here p : R + → R + prescribes the maximal flow allowed through an exit placed in x = 0 as a function of the weighted average density of pedestrians, ρ, in a left neighborhood of the exit and w : R -→ R + is the weight function used to average the density. The authors of [START_REF] Andreianov | Crowd dynamics and conservation laws with non-local constraints and capacity drop[END_REF] proved well-posedness of the Cauchy problem in L ∞ (R; [0, R]) for the model under the following assumptions on the regularity of f , w and p, see Figure 1,

(F) f ∈ Lip ([0, R]; [0, +∞[), f (0) = 0 = f (R) and there exists ρ ∈ ]0, R[ such that f (ρ) (ρ -ρ) > 0 for a.e. ρ ∈ [0, R] \ {ρ}. (W) w ∈ L ∞ (R -; R +
) is an increasing map, w L 1 (R-;R+) = 1 and there exists i w > 0 such that w(x) = 0 for any x ≤ -i w .

(P1) p belongs to Lip ([0, R] ; ]0, f (ρ)]) and it is a non-increasing map.

We recall that, in the previous literature, the only macroscopic model able to reproduce the capacity drop at bottlenecks is the CR model introduced by R. Colombo and M. D. Rosini in [START_REF] Colombo | Pedestrian flows and non-classical shocks[END_REF]. The Riemann solver for the model described in [START_REF] Colombo | Pedestrian flows and non-classical shocks[END_REF] is fairly intricate; in this note, our main goal is to describe in an exhaustive way the Riemann solver (or, rather, solvers) for the model of [START_REF] Andreianov | Crowd dynamics and conservation laws with non-local constraints and capacity drop[END_REF]. Notice that, in contrast to [START_REF] Colombo | Pedestrian flows and non-classical shocks[END_REF], a specific nonclassical Riemann solver has to be used only at the exit point x = 0 while the simple classical Riemann solver is used elsewhere.

The notion of solution we adopt is a natural extension of the one introduced in [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF], for a constrained Cauchy problem of the form 1.

Definition 1.1. Assume conditions (F), (W) and that p is a non-increasing, possibly multivalued, map with values in ]0,

f (ρ)]. A map ρ ∈ L ∞ (R + × R; [0, R]) ∩ C 0 (R + ; L 1 loc (R; [0, R])
) is an entropy weak solution to 1, 2 if the following conditions hold:

1. There exists q ∈ L ∞ (R + ; [0, f (ρ)]) such that for every test function φ ∈ C ∞ c (R 2 ; R + ) and for every k ∈ [0, R] R+ R [|ρ -k|∂ t φ + sign(ρ -k) (f (ρ) -f (k)) ∂ x φ] dx dt (3a) + 2 R+ 1 - q (t) f (ρ) f (k) φ(t, 0) dt (3b) + R |ρ 0 (x) -k| φ(0, x) dx ≥ 0 , (3c)
and

f (ρ(t, 0±)) ≤ q (t) for a.e. t ∈ R + . (3d)
2. In addition q is linked to ρ by the relation 2.

If q is given a priori, then 3 is the definition of entropy weak solution to problem 1. We refer to Proposition 2.6 in [START_REF] Andreianov | Finite volume schemes for locally constrained conservation laws[END_REF] for a series of equivalent formulations of conditions 3.

The next proposition lists the basic properties of a entropy weak solution of 1, 2, for the case of a single valued p, the proof is given in [START_REF] Andreianov | Crowd dynamics and conservation laws with non-local constraints and capacity drop[END_REF]. (2) Any discontinuity satisfies the Rankine-Hugoniot jump condition, see [START_REF] Dafermos | Hyperbolic conservation laws in continuum physics[END_REF].

(3) Any discontinuity away from the constraint is classical, i.e. satisfies the Lax entropy inequalities, see [START_REF] Dafermos | Hyperbolic conservation laws in continuum physics[END_REF].

(4) Nonclassical discontinuities, see [START_REF] Lefloch | Hyperbolic systems of conservation laws[END_REF], may occur only at the constraint location x = 0, and in this case the flow at x = 0 is the maximal flow allowed by the constraint. Namely, if the solution contains a nonclassical discontinuity for all times t ∈ I, I open in R + , then for a.e.

t in I f (ρ(t, 0-)) = f (ρ(t, 0+)) = p R- w(x) ρ(t, x) dx . (4)
If the constraint function p is multivalued the equalities in 2 and 4 should be interpreted as inclusions, and the result of the proposition remains true.

The existence result in [START_REF] Andreianov | Crowd dynamics and conservation laws with non-local constraints and capacity drop[END_REF] is achieved by a procedure which couples the operator splitting method [START_REF] Dafermos | Hyperbolic systems and balance laws with inhomogeneity and dissipation[END_REF], with the wave-front tracking algorithm, [START_REF] Dafermos | Polygonal approximations of solutions of the initial value problem for a conservation law[END_REF], see also [START_REF] Amadori | An Integro-Differential Conservation Law arising in a Model of Granular Flow[END_REF] for a similar technique. This approach allows us to approximate our problem by a problem with "frozen" constraint, as 1, at each time step.

The regularity of p plays a central role in the well-posedness result. While existence still holds for the Cauchy problem when p is merely continuous, it is difficult to justify uniqueness in this case. Further, in [START_REF] Andreianov | Crowd dynamics and conservation laws with non-local constraints and capacity drop[END_REF], the authors give some basic examples illustrating that solutions of a Riemann problem for the case of a non-decreasing piecewise constant p, see (P2) below, may fail to be unique, L 1 loccontinuous and consistent.

The case in which p is piecewise constant is extremely important both for the theoretical study of the problem and its numerical applications. First, it is related to the construction of the Riemann solver, which is the basic building block for the wave-front tracking algorithm, a precious tool in the study of existence and stability of the solutions for the general Cauchy problem. Moreover, the piecewise constant case is essentially the only case in which solutions can be computed explicitly, which is an undeniable atout when looking for examples and applications. To this aim, it is relevant to provide a detailed study of the different pathological behaviors one may encounter. Remarkably, we show that these behaviors can be easily forecast and avoided when looking for explicit examples of solutions.

In the following section we develop a detailed proof of the fact that, when p is piecewise constant, the Riemann solver for 1 with a constraint of the form 2, is not unique and does not satisfy the minimal requirements needed to develop the classical wave-front tracking approach. Additionally, we compare the two extreme Riemann solvers: the one that minimizes the capacity drop, and the one that maximizes it. In particular, for any time T > 0 we estimate the distance between the profiles of the solutions produced by the two Riemann solvers starting from the same initial condition.

The constrained Riemann problem

In this section we study constrained Riemann problems of the form

∂ t ρ + ∂ x f (ρ) = 0 (t, x) ∈ R + × R (5a) f (ρ(t, 0±)) ≤ p R- w(x) ρ(t, x) dx t ∈ R + (5b) ρ(0, x) = ρ L if x < 0 ρ R if x ≥ 0 x ∈ R , ( 5c
) with ρ L , ρ R ∈ [0, R].
The flux f and the weight function w satisfy (F) and (W), moreover, we adopt the following assumption on p (instead of (P1)) to allow an explicit construction of solutions to 5

(P2) p : [0, R] → ]0, f (ρ)
] is piecewise constant non-increasing map with a finite number of jumps. Unfortunately, the regularity of p required by (P2) is not enough to apply the well-posedness results in [START_REF] Andreianov | Crowd dynamics and conservation laws with non-local constraints and capacity drop[END_REF]. In particular, Example 2 in [START_REF] Andreianov | Crowd dynamics and conservation laws with non-local constraints and capacity drop[END_REF] illustrates the loss of uniqueness and stability of entropy weak solutions. In this section we present a systematic study of the possible pathological behaviors. We denote by R the classical Riemann solver. This means that the map [(t, x) → R[ρ L , ρ R ](x/t)] is the unique entropy weak solution for the unconstrained problem 5a, 5c, see for example [START_REF] Bressan | Hyperbolic systems of conservation laws[END_REF] for its construction. Whenever the classical weak solution given by R does not satisfy the constraint 5b, we replace it by a nonclassical weak solution, see [START_REF] Lefloch | Hyperbolic systems of conservation laws[END_REF] as a general reference,

ρ(t, x) = R[ρ L , ρ(q)](x/t) if x < 0 R[ρ(q), ρ R ](x/t) if x ≥ 0 , (6)
where the maps ρ, ρ : [0, f (ρ)] → [0, R] are implicitly defined by f (ρ(q)) = q = f (ρ(q)) and ρ(q) ≤ ρ ≤ ρ(q) . We stress that by Proposition 1 any nonclassical entropy weak solution is a classical entropy weak solution in the Kružkov sense, see [START_REF] Kružhkov | First order quasilinear equations with several independent variables[END_REF], [START_REF] Dafermos | Hyperbolic conservation laws in continuum physics[END_REF], in the half-planes R + ×R - and R + × R + . However, the jump at x = 0 is a nonclassical shock, in the sense that it does not satisfy the Lax entropy inequalities. First, we should notice that as soon as the constraint function t → q(t) is not constant, the solutions of the Riemann problem may not be self-similar.

Example 2.1. We use a Cauchy problem of the form 1 to model vehicular traffic in presence of a traffic light. Assume

f (ρ) = ρ(1-ρ) and q(t) = 0.25 k∈N χ [2k,2k+1[ (t).
When the traffic light is green, i.e. for t ∈ [2k, 2k + 1[, the flow at x = 0 is free from any constraint. Conversely, when the traffic light is red, i.e. for t ∈ [2k + 1, 2(k + 1)[, the admissible flow at x = 0 become zero. This means that the Riemann problem with initial condition at t = 0 given by ρ L = ρ R = 0 will not be self-similar, because the constant solution will not satisfy the constraint starting from t = 1.

The above example also shows that as soon as we use a nonclassical Riemann solver we lose the a priori BV bounds on the solution.

In the proof of Proposition 2 we show that any entropy weak solution of 5 is self-similar for sufficiently small times. Therefore, it makes sense to introduce nonclassical local Riemann solvers, see Definition 2.3. Then, the availability of a local Riemann solver allows us to construct a global solution to the Riemann problem 5 by a wave-front tracking algorithm in which the jumps in the map [t → q(t)] are interpreted as non-local interactions.

Aiming for a general construction of the solutions to 5, we allow p to be a multi-valued piecewise constant function, namely, see Fig. 1, right:

• there exist ξ 1 , . . . , ξ n ∈ ]0, R[ and p 0 , . . . , p n ∈ ]0, f (ρ)], with ξ i < ξ i+1 and p i > p i+1 , such that p(0) = p 0 , p(R) = p n , p χ ]ξ i , ξ i+1 [ = p i for i = 0, . . . , n-1, p(ξ i ) = [p i , p i-1 ] for i = 1, . . . , n, being ξ 0 = 0 and ξ n+1 = R.
In the following we will use the notations ρi = ρ(p i ) and ρi = ρ(p i ).

As it will become clear in Proposition 2, the possible loss of uniqueness and stability can be easily forecast once the piecewise constant constraint p and the flux f are given. In particular, for some respective configurations of p and f the solution of the Riemann problem exists and is unique, locally in time, for any initial data

(ρ L , ρ R ) in [0, R] 2 . Definition 2.2. Introduce the subset of [0, R] 2 C = (ρ L , ρ R ) ∈ [0, R] 2 : (ρ L , ρ R ) satisfies condition (C) ,
where we say that (ρ L , ρ R ) satisfies condition (C) if it satisfies one of the following conditions:

(C1) ρ L < ρ R , f (ρ R ) < f (ρ L ) and f (ρ R ) ≤ p(ρ L +); (C2) ρ L < ρ R , f (ρ L ) ≤ f (ρ R ) and f (ρ L ) ≤ p(ρ L +); (C3) ρ R ≤ ρ L ≤ ρ and f (ρ L ) ≤ p (ρ L +); (C4) ρ R ≤ ρ < ρ L and f (ρ) = p (ρ L +); (C5) ρ < ρ R ≤ ρ L , f (ρ R ) ≤ p (ρ L -) and f (ρ L ) < p (ρ L +). Analogously, introduce the subset of [0, R] 2 N = (ρ L , ρ R ) ∈ [0, R] 2 : (ρ L , ρ R ) satisfies condition (N) ,
where we say that (ρ L , ρ R ) satisfies condition (N) if it satisfies one of the following conditions:

(N1) ρ L < ρ R and f (ρ L ) > f (ρ R ) > p(ρ L +); (N2) ρ L < ρ R , f (ρ L ) ≤ f (ρ R ) and f (ρ L ) > p(ρ L -); (N3) ρ R ≤ ρ L ≤ ρ and f (ρ L ) > p (ρ L -); (N4a) ρ R ≤ ρ < ρ L , f (ρ) = p (ρ L -) and f (ρ L ) < p(ρ L +); (N4b) ρ R ≤ ρ < ρ L , f (ρ) = p (ρ L -) and f (ρ L ) > p(ρ L -); (N5a) ρ < ρ R ≤ ρ L , f (ρ R ) > p (ρ L -) and f (ρ L ) < p (ρ L +); (N5b) ρ < ρ R ≤ ρ L and f (ρ L ) > p (ρ L -).
Here, C stands for classical and N , for nonclassical, in relation with the nature of the shock appearing in the solution of 5 at x = 0. Observe that if the constraint function p is constant in a neighborhood of the state ρ L , then p(ρ L -) = p(ρ L +) and this simplifies the above conditions. Also a right or left continuity assumption on p would simplify the above definitions.

The next proposition says that uniqueness holds at least for small times if and only if the initial data are in C ∪ N . It is fundamental to remark that since nonuniqueness is possible only when p(ρ L -) = p(ρ L +) and p(ρ L -) ≥ f (ρ L ) ≥ p(ρ L +), non-uniqueness concerns at most a finite number of left states and therefore the region [0, R] 2 \ (C ∪ N ) is the union of a finite number of line segments. Proposition 2. Consider the constrained Riemann problem 5.

• If (ρ L , ρ R ) ∈ C, then the map [(t, x) → R[ρ L , ρ R ](x/t)]
is the unique entropy weak solution at least for t > 0 sufficiently small.

• If (ρ L , ρ R ) ∈ N , then there exists a unique p ∈ [p(ρ L +), p(ρ L -)] such that the map t → R[ρ L , ρ (p)](x/t) if x < 0 R[ρ (p) , ρ R ](x/t) if x ≥ 0
is the unique entropy weak solution at least for t > 0 sufficiently small.

• If (ρ L , ρ R ) ∈ [0, R] 2 \ (C ∪ N )
, then the corresponding constrained Riemann problem 5 admits more than one entropy weak solution.

The proof is deferred to Section 4.1

Remark 1. Once the function p is fixed, the time interval [0, τ ] on which the solution to the Riemann problem 5 is self-similar can be estimated from the initial datum (ρ L , ρ R ) provided it belongs to C ∪ N . On the contrary, when

(ρ L , ρ R ) ∈ [0, R] 2 \ (C ∪ N ),
we are not always able to forecast when the next "interaction with the constraint" will take place. In some situations, a whole one-parameter family of solutions exists, we refer to Example 2 in [START_REF] Andreianov | Crowd dynamics and conservation laws with non-local constraints and capacity drop[END_REF] for a detailed discussion of this case.

Riemann solvers

As the local solutions of the Riemann problem 5 are not unique in general, we are naturally led to question the existence of suitable selection criteria. All the solutions we introduce are solutions in the Kružkov sense in the open half-planes R + × R + and R + ×R -, so they satisfy the basic requirement of entropy dissipation. However, coming back to the real situations which our model aims to describe, we argue that the most interesting behaviors to track correspond to the extreme cases in which the flux at the exit is either the highest or the lowest possible from a given initial condition.

If, as an example, our model describes the evacuation of a narrow corridor, it is clear that the optimal solution corresponds to the highest admissible values of the flux at the exit. By opposition to the next case, we describe this situation as quiet behaviour. In analogy to the discussion in [START_REF] El-Khatib | On entropy weak solutions of Hughes' model for pedestrian motion[END_REF] we interpret all other possible solutions as consequences of an irrational behavior, which in literature is often described as panic. In particular, we can use the solution corresponding to the lowest admissible values of the flux at the exit to find an upper bound for the evacuation time.

From now on we restrict ourselves to the case in which p(ξ i ) can only take the values p i and p i+1 and not the intermediate values.

Definition 2.3. Two Riemann solvers R q and R p for 5 are defined as follows for t > 0 sufficiently small and x ∈ R:

(C) If (ρ L , ρ R ) ∈ C then R q [ρ L , ρ R ](t, x) = R p [ρ L , ρ R ](t, x) = R[ρ L , ρ R ](x/t) . (N) If (ρ L , ρ R ) ∈ N then R q [ρ L , ρ R ](t, x) = R p [ρ L , ρ R ](t, x) = R[ρ L , ρ (p)](x/t) if x < 0 R[ρ (p) , ρ R ](x/t) if x ≥ 0 , where p = p(ρ L -) if (ρ L , ρ R ) satisfies (N4a) or (N5a), otherwise p = p(ρ L +). (CN2), (CN3), (CNN5) If (ρ L , ρ R ) satisfies one of these sets of conditions then R q [ρ L , ρ R ](t, x) = R[ρ L , ρ R ](x/t) , R p [ρ L , ρ R ](t, x) = R[ρ L , ρ (p(ρ L +))](x/t) if x < 0 R[ρ (p(ρ L +)) , ρ R ](x/t) if x ≥ 0 . (NNN4), (NNN5) If (ρ L , ρ R ) satisfies one of these sets of conditions then R q [ρ L , ρ R ](t, x)
takes the form 11 with p = p(ρ L -) and R p [ρ L , ρ R ](t, x) takes the form 11 with p = p(ρ L +).

In the next proposition we collect the main properties of the Riemann solvers R q and R p . In particular (R6) means that the Riemann solver R q is the one which allows for the fastest evacuation, while R p is associated to the slowest one.

Proposition 3. Let (ρ L , ρ R ) ∈ [0, R]
2 . Then, for = q, p:

(R1) [(t, x) → R [ρ L , ρ R ](t, x)
] is a weak solution to 5a, 5c.

(R2) R [ρ L , ρ R ] satisfies the constraint 5b in the sense that f (R [ρ L , ρ R ](t, 0±)) ≤ p R- w(x) R [ρ L , ρ R ] (t, x) dx . (R3) R [ρ L , ρ R ](t) ∈ BV (R; [0, R]). (R4) The map R : [0, R] 2 → L 1 loc (R + × R; [0, R]) is continuous in C ∪ N but not in all [0, R] 2 .
(R5) R is consistent, see [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF], [START_REF] Colombo | Characterization of Riemann solvers for the two phase p-system[END_REF] and the comment below.

(R6) R q [ρ L , ρ R ] maximizes the flux at the exit, in the sense that if E is the set of all entropy weak solutions of the Riemann problem 5, we have

max ρ∈E {f (ρ(t, 0±))} = f (R q [ρ L , ρ R ](0±)) .
Analogously, R p [ρ L , ρ R ] minimizes the flux at the exit, in the sense that

min ρ∈E {f (ρ(t, 0±))} = f (R p [ρ L , ρ R ](0±)) .
We recall that a Riemann solver is said to be consistent when the juxtaposition of the solutions of two Riemann problems with respective initial conditions (ρ L , ρ M ) and (ρ M , ρ R ) is the solution of the Riemann problem with datum (ρ L , ρ R ). Moreover the vice versa also holds true, in the sense that whenever the state ρ M is an intermediate state in the solution of the Riemann problem with initial condition (ρ L , ρ R ), then the solution consist of exactly the same states and waves which we would obtain by solving side by side the two Riemann problems with data (ρ L , ρ M ) and (ρ M , ρ R ).

The proof of Proposition 3 is deferred to Section 4.2.

Remark 2. It is important to observe that even if p(ξ i ) can only take the two values p i and p i+1 , this is not enough to rule out the existence of infinitely many different solutions as the ones described in Example 2 of [START_REF] Andreianov | Crowd dynamics and conservation laws with non-local constraints and capacity drop[END_REF], in the case

p i > f (ξ i ) = p i+1 .
However, each of the extremes Riemann solvers R , = p, q, selects one of them because it sticks to the constant level of constraint prescribed by Definition 2.3, the level p i for R q and the level p i+1 for R p , until a non-local interaction takes place.

On the use of Riemann solvers R , = q, p

Although the Riemann solvers R are not L 1 loc -continuous, an existence result for the Cauchy problem 1 can be obtained from a wave-front tracking algorithm based on R , see for instance [START_REF] Colombo | Existence of nonclassical solutions in a Pedestrian flow model[END_REF], [START_REF] Rosini | Nonclassical interactions portrait in a macroscopic pedestrian flow model[END_REF]. Such approach using R does not require the operator splitting method. However, the non-local nature of the approximating problems prevents us from a direct application of the Riemann solvers R . In fact, even in a arbitrary small neighborhood of x = 0, to prolong the approximating solution ρ n beyond a time t = t > 0 it is not sufficient to know the traces ρ n ( t, 0-), ρ n ( t, 0+), but also the value 0 -iw w(x) ρ n ( t, x) dx is needed. Roughly speaking, because of the non-local character of the constraint one cannot merely juxtapose the solution to the Riemann problem associated to the values of the traces at x = 0 with the solution to the Riemann problems away from the constraint. Finally, also jumps in [t → p (ξ(t))] have to be considered as (non-local) interactions. Therefore, the approach using R is considerably heavier and more technical than the one presented in [START_REF] Andreianov | Crowd dynamics and conservation laws with non-local constraints and capacity drop[END_REF].

On the comparison between the Riemann solvers R p and R q

In this section we aim to compare the solutions obtained by the two Riemann solvers introduced above, starting from the same initial condition (ρ L , ρ R ). It is clear from Definition 2.3 that as soon as (ρ L , ρ R ) belongs to C ∪ N the solutions obtained by the two Riemann solvers coincide.

As a preliminary remark we stress that adapting the proof of Theorem 3.1 of [3] to the case in which p is discontinuous, one can get a rough upper bound, exponential in time, for the L 1 distance between two solutions ρ 1 , ρ 2 obtained from the same initial condition (not necessarily of Riemann type). Indeed, instead of the bound

|p(ρ 1 ) -p(ρ 2 )| ≤ Lip(p) |ρ 1 -ρ 2 | , (7)
valid when p is Lipschitz continuous, in the discontinuous case one can use the bound

|p(ρ 1 ) -p(ρ 2 )| ≤ h + N |ρ 1 -ρ 2 | , ( 8
)
where h is the maximal size of jump in p(•) and N is a constant (observe that if p(•) is a discretization of some Lipschitz function, see Section 4.1 of [START_REF] Andreianov | Crowd dynamics and conservation laws with non-local constraints and capacity drop[END_REF], then N can be taken independent of h). From the fundamental stability estimate of Proposition 2.10 of [START_REF] Andreianov | Finite volume schemes for locally constrained conservation laws[END_REF], using 8 and the Gronwall inequality one easily gets the bound

ρ 1 (t) -ρ 2 (t) L 1 (R;R) ≤ h 2N [exp(2N t) -1] , (9)
whenever ρ 1 (0, •) = ρ 2 (0, •). This rough estimate is enough to show that as h goes to zero, the discrepancy between different solutions vanishes and this argument applies to any initial datum, not necessarily of the Riemann type. However, the exponential growth with respect to t of the upper bound 9 is clearly not optimal when we aim to compare the solutions of a Riemann problem, since it does not take into account the specific self-similar structure of solutions valid at least on a small interval of time [0, T ].

Let us demonstrate that in the case where different Riemann solvers co-exist, the L 1 distance of the associated solution grows at most linearly both in h and in t ∈ [0, T ] (see also the numerical experiment on Figure 6).

In order to keep our presentation as light as possible, we focus on only one of the possible cases in which R p and R q differ. All other cases can be handed in a similar way. Assume that p(ξ and ρ L = ξ, ρ R = ρ, see Figure 3. We only consider solutions in a small interval of time [0, T ] in which they are self-similar. Then we get

) = p 1 χ [0, ξ] (ξ) + p 2 χ ] ξ,R] (ξ), where f > p 1 > f ( ξ) > p 2 > 0,
R p [ρ L , ρ R ](t, x) = R[ρ L , ρ2 ](x/t) if x < 0 R[ρ 2 , ρ](x/t) if x ≥ 0 , R q [ρ L , ρ R ](t, x) = R[ρ L , ρ1 ](x/t) if x < 0 R[ρ 1 , ρ](x/t) if x ≥ 0 ,
where the values ρi and ρi , for i = 1, 2, are implicitly defined by the relations f (ρ i ) = f (ρ i ) = p i and ρi ≤ ρ ≤ ρi . More explicitly, we can say that the solution corresponding to R p consists of a shock of negative speed λ p between ρ L and ρ2 , a stationary nonclassical shock between ρ2 and ρ2 and a shock of positive speed µ p between ρ2 and ρ. The solution corresponding to R q consists of a rarefaction wave between ρ L and ρ1 , a stationary nonclassical shock between ρ1 and ρ1 and a shock of positive speed µ q between ρ1 and ρ, see Figure 4.

As the characteristics of this problem propagate with finite speed, we expect the solutions associated to the two solvers coincide outside a bounded interval. The geometry of the problem, see Figure 3, implies that µ p > µ q and that λ p is smaller than all the propagation speeds in the rarefaction wave between ρ L and ρ1 . Therefore, at time t ∈ [0, T ] fixed, the two solutions coincide outside the interval [λ p t, µ p t].

The value of the distance

R p [ρ L , ρ R ](t) -R q [ρ L , ρ R ](t) L 1 (R;R)
corresponds, loosely speaking, to the value of the area between the profiles of solutions, see Figure 4.

Following the same technique as in [START_REF] Bressan | Hyperbolic systems of conservation laws[END_REF], Chapter 7, we can estimate the distance between the solutions profiles at a fixed time t ∈ [0, T ]. For the reader convenience, we recall that the propagation speed of the shock discontinuity between states the ρ a and ρ b is given by the Rankine-Hugoniot condition

σ(ρ a , ρ b ) = f (ρ a ) -f (ρ b ) ρ a -ρ b , (10)
and that the propagation speed of the characteristics in a rarefaction wave joining the states ρ a and ρ b varies between the values f (ρ a ) and f (ρ b ). Also, by definition Figure 4: The solutions corresponding to R p , left, and to R q , center, and the comparison between their profiles at fixed time, right, as described in Section 2.3.

f (ρ i ) = f (ρ i ) = p i , for i = 1, 2. A direct calculation gives us R p [ρ L , ρ R ](t, •) -R q [ρ L , ρ R ](t, •) L 1 (R;R) ≤ R[ρ L , ρ2 ](•/t) -R[ρ L , ρ1 ](•/t) L 1 (R;R) + R[ρ 2 , ρ R ](•/t) -R[ρ 1 , ρ R ](•/t) L 1 (R;R) ≤ [λ p -f (ρ L )] (ρ 2 -ρ L ) t +f (ρ L ) (ρ 2 -ρ1 ) t + (ρ 1 -ρ2 ) µ q t + (ρ -ρ2 ) (µ q -µ p ) t ≤ 2 t [p 1 -p 2 + o(ρ 2 -ρ1 )] .
This means that whenever the piecewise constant function p we consider is the discretization of a smooth function we can bound a priori the size of the error due to the lack of uniqueness and we can make it smaller and smaller as h tends to 0.

Numerical results

We present here some numerical experiments in order to illustrate the results of the above section. The scheme used for the simulations combines the ideas of [START_REF] Andreianov | Finite volume schemes for locally constrained conservation laws[END_REF] with an explicitly updated constraint computed from weighted space averages of the discrete solution at previous time step. We will justify in the future work [START_REF] Andreianov | A finite volume scheme for conservation laws with non-local point constraints[END_REF] convergence of this scheme to an entropy solution of the nonlocally constrainted problem in the sense of Definition 1.1, where the constraint function p(•) must be taken multi-valued. While it is delicate or even impossible to identify a unique Riemann solver to which the scheme would converge, we can use the simulations on Figure 5 to illustrate the fact that non-uniqueness for the Riemann problem results as unstable behavior in a vicinity of some specific data.

For the examples, we consider the flux

f (ρ) = ρ (1 -ρ). The domain of computation is x ∈ [-5, 5], the constraint function is p(ξ) = p 1 χ [0,0.8] (ξ) + p 2 χ ]0.8,1] (ξ)
, where p 1 = 0.1875, p 2 = 0.05, the weight function is w(x) = 2(x + 1) χ ]-1,0] (x). The final time of computation is T = 1. In Figure 5 is shown the computed solutions ρ 1 and ρ 2 corresponding respectively to the following initial states

ρ 1 0 (x) = 0.8015 if x < 0 0.5 if x > 0 and ρ 2 0 (x) = 0.7984 if x < 0 0.5 if x > 0 .
Note that we have considered ρ 1 L and ρ 2 L such that ρ 1 L -ρ 2 L ∆x/8, where the space step ∆x = 0.025. Finally we assume the time step ∆t = ∆x/10. Nonetheless, as shown in Figure 6, in practice the instability is limited to a behavior of kind

ρ 1 -ρ 2 L 1 ([-5,5];R) ≤ ρ 1 0 -ρ 2 0 L 1 ([-5,5];R) + C h α ,
where h is the maximal size of jump in p(•), α is close to 1 and C > 0 is a constant. Indeed we reported in Figure 6 the computation of the L 1 -discrete norms of the difference ρ 1 -ρ 2 when we take, p 2 = 0.05, 0.075, 0.1, 0.125 and 0.15 in the definition of the constraint function. Using logarithmic scales, we deduce that the distance between the two solutions is approximatively proportional to |p 1 -p 2 | 0.9 .

Proofs

Proof of Proposition 2

First, we introduce the notation

ξ(t) = R- w(x) ρ(t, x) dx .
Therefore ξ(0) = ρ L and the map [t → ξ(t)] is continuous. We stress that any nonclassical entropy weak solution in the sense of Definition 1.1 is also a classical entropy weak solution in the Kružkov sense in the halfplanes R + × R -and R + × R + . Therefore, at least for t > 0 sufficiently small, by Proposition 1, assumption (P2) and the continuity of the map [t → ξ(t)], any nonclassical entropy weak solution of 5 must have the form, see Fig. 2,

ρ(t, x) = R[ρ L , ρ(p)](x/t) if x < 0 R[ρ(p), ρ R ](x/t) if x ≥ 0 . (11a) -2.6 -2.4 -2.2 -2 -1.8 -1.6 -1.4 -1.2
-3.4 -3.2 -3 -2.8 -2.6 -2.4 -2.2 -2 -1.8 L 1 -discrete norms of rho 1 -rho 2 in log scale ln|p 1 -p 2 | Figure 6: The norm ρ 1 -ρ 2 L 1 with respect to |p 1 -p 2 | in log/log scale.
Observe that 11a is uniquely identified once we know p which, by 4, satisfies

p = f (ρ(p)) = f (ρ(p)) . (11b)
We recall that 11b means in particular that the Rankine-Hugoniot jump condition is satisfied at x = 0 even when the solution to the Riemann problem is nonclassical. As a consequence of 11b, of assumption (P2) and of the continuity of [t → ξ(t)], we have that

p ∈ [p(ρ L +), p(ρ L -)] . (11c)
This implies that if p(ρ L +) = p(ρ L -), then p(ξ) is constant in a neighborhood of ρ L and, since the solution is in

C 0 R + ; L 1 loc (R; [0, R])
, uniqueness is ensured by the results in [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF]. However, the continuity of p at ρ L is not a necessary condition for uniqueness as we show in the following section.

Cases in which uniqueness holds

In this section we prove that: If (ρ L , ρ R ) ∈ C the corresponding classical solution satisfies 5 for all t > 0 sufficiently small and it is not possible to construct a different solution. If (ρ L , ρ R ) ∈ N the corresponding classical solution does not satisfy 5b, and there exists a unique nonclassical solution that satisfies 5. We list here two basic properties which will be of great help in the following case by case analysis.

By assumption (P2) and the continuity of the map [t → ξ(t)] we have that for any t > 0 sufficiently small bp1 if ξ(t) < ρ L , then p(ξ(t)) ≡ p(ρ L -); bp2 if ρ L < ξ(t), then p(ξ(t)) ≡ p(ρ L +). The case ξ(t) ≡ ρ L is somehow special and has to be studied separately for each specific case.

Second, when the solution is nonclassical, due to the finite speed of propagation of the waves, the assumption (P2) and properties bp1 and bp2, we have

np1 if R [ρ L , ρ(p)] (x) ≡ ρ L for x < 0, then p = f (ρ L ) ∈ [p(ρ L +), p(ρ L -)]; np2 if p = f (ρ L ) and ρ L < ρ(p), then p = p(ρ L +); np3 if p = f (ρ L ) and ρ(p) < ρ L , then p = p(ρ L -); np4 if p is continuous in ρ L , namely p(ρ L -) = p(ρ L +), then p = p(ρ L ).
Now we start the description of the possible cases and we proceed as follows. First, we show that for any initial datum satisfying (Ci), i = 1, . . . , 5, the problem actually has a unique solution and that the solution is classical. Second, we take into consideration the corresponding case (Ni), for which we prove that the classical solution is not suitable and that there exists a unique nonclassical solution.

In general the solutions to the constrained Riemann problem 5 are not selfsimilar. All the cases listed below describe self-similar solutions because we let the solutions evolve only on a small interval of time.

(C1) In this case [(t, x) → R[ρ L , ρ R ](x/t)]
N5a) In this case [(t, x) → R[ρ L , ρ R ](x/t)] does not satisfy 5b because f (ρ R ) > p(ρ L -), see case (C5) (
. Therefore, there does not exist any classical solution and we can consider only nonclassical solutions of the form 11.

(N5b) By assumption and 11c, f (ρ L ) < p(ρ L +) ≤ p and therefore ρ(p) < ρ L and by np3 we have p = p(ρ L -).

(N5b) By assumption and 11c, f (ρ L ) > p(ρ L -) ≥ p and therefore ρ(p) < ρ L and by np2 we have p = p(ρ L +).

Cases in which uniqueness is violated

Now we list the "pathological" cases, where we have more than one admissible solution. We stress once again that a necessary condition for non-uniqueness is then the nonclassical solutions of the form 11 which corresponds to p in the set {p(ρ L +), f (ρ L ), p(ρ L -)} satisfy 5. This is the situation considered in the Example 2 in [START_REF] Andreianov | Crowd dynamics and conservation laws with non-local constraints and capacity drop[END_REF]. Observe that such solutions are distinct as far as they correspond to distinct constraint levels p, and that in any case there exist at least two distinct nonclassical solutions. (CNN5) If ρ < ρ R ≤ ρ L , f (ρ R ) ≤ p(ρ L -), p(ρ L -) = p(ρ L +) and p(ρ L +) ≤ f (ρ L ), then the classical solution [(t, x) → R[ρ L , ρ R ](x/t)], which consists of a possible null rarefaction on the left of the constraint, as well as the nonclassical solutions of the form 11 corresponding to p ∈ {p(ρ L +), f (ρ L )} satisfy 5. Observe that the two nonclassical solutions are distinct as far as they correspond to distinct constraint levels p, and that in any case there exist at least two distinct solutions, one classical and one nonclassical. (NNN5) If ρ < ρ R < ρ L , f (ρ R ) > p(ρ L -) ≥ f (ρ L ) ≥ p(ρ L +) and p(ρ L -) = p(ρ L +), then the nonclassical solutions of the form 11 corresponding to p ∈ {p(ρ L +), f (ρ L ), p(ρ L -)} satisfy 5. Observe that such solutions are distinct as far as they correspond to distinct constraint levels p, and that in any case there exist at least two distinct nonclassical solutions.

p(ρ L -) = p(ρ L +) and p(ρ L -) ≥ f (ρ L ) ≥ p(ρ L +). (CN2) If ρ L < ρ R , f (ρ L ) ≤ f (ρ R )

Proof of Proposition 3

(R1) Any solution given by R coincides on each side of the constraint with a solution given by the classical Riemann solver R. Therefore it satisfies the Rankine-Hugoniot jump condition along any of its discontinuities away from the constraint. Finally, by definition of ρ and ρ, it satisfies the Rankine-Hugoniot jump condition also along the constraint. (R5) We first stress once again that we can discuss the consistency property of our Riemann solvers only locally in time because, in general, the solutions may be not even self-similar globally in time. However, locally in time, the efficiency of the exit can be assumed to be constant and it is thus sufficient to proceed as in [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF].

(R6) It is clear by the proof of Proposition 2.

Aknowledgements

All

Proposition 1 .

 1 Let [t → ρ(t)] be an entropy weak solution of 1, 2 in the sense of Definition 1.1. Then[START_REF] Amadori | An Integro-Differential Conservation Law arising in a Model of Granular Flow[END_REF] It is also a weak solution of the Cauchy problem 1a, 1c.

Figure 1 :

 1 Figure 1: Examples of functions satisfying conditions (F), (W), and (P2).

Figure 2 :

 2 Figure 2: The four possible configurations of nonclassical entropy weak solutions of the form 11.

Figure 3 :

 3 Figure 3: The flux f , the constraint function p| ] ρ1, ρ2[ and the values of the density ρ considered in Section 2.3.

Figure 5 :

 5 Figure 5: The computed densities ρ 1 and ρ 2

( N4a )

 N4a By assumption and 11c f (ρ L ) < p(ρ L +) ≤ p and therefore ρ(p) < ρ L and by np3 we have p = p(ρ L -).(N4b) By assumption and 11c f (ρ L ) > p(ρ L -) ≥ p and therefore ρ(p) > ρ L and by np2 we have p = p(ρ L +).(C5) In this case [(t, x) → R[ρ L , ρ R ](x/t)] performs a possible null rarefaction on the left of the constraint and it satisfies 5b because f (ρ R ) ≤ p(ρ L -) and p(ξ(t)) ≡ p(ρ L -) by bp1. Assume that there exists a nonclassical solution of the form 11. Since by assumption and 11c p ≥ p(ρ L +) > f (ρ L ), we have ρ(p) < ρ L and by np3 p = p(ρ L -), but in this case the nonclassical solution coincides with the classical one.

  and p(ρ L +) < f (ρ L ) ≤ p(ρ L -), then the classical solution [(t, x) → R[ρ L , ρ R ](x/t)], which consists of a shock with non negative speed, as well as the nonclassical solution 11, with p = p(ρ L +), are distinct solutions of 5.(CN3) If ρ R ≤ ρ L ≤ ρ and p(ρ L +) < f (ρ L ) ≤ p(ρ L -), then the classical solution [(t, x) → R[ρ L , ρ R ](x/t)],which consists of a possible null rarefaction on the right of the constraint, as well as the nonclassical solution 11, with p = p(ρ L +), are distinct solutions of 5.(NNN4) If ρ R ≤ ρ < ρ L , p(ρ L -) = p(ρ L +) and p(ρ L +) ≤ f (ρ L ) ≤ p(ρ L -),

( R2 )

 R2 It is clear by the proof of Proposition 2. (R3) It is proved as in (R1) since any classical solution is in BV.(R4) As was proved in[START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF], R is continuous onC ∪ N . If (ρ L , ρ R ) is not in C ∪ Nthen p experiences a jump at ξ = ρ L . Therefore, the local in time solutions of the Riemann problem for the initial conditions (ρ L + ε, ρ R ) and (ρ L -ε, ρ R ) are different and only one of the two converges to R [ρ L , ρ R ] as ε > 0 goes to zero.

  performs a shock with negative speed σ(ρ L , ρ R ) and satisfies 5b because f (ρ R ) ≤ p(ρ L +) and p(ξ(t)) ≡ p(ρ L +) by bp2. Assume that there exists a nonclassical solution of the form 11. Observe that the assumptions ρL < ρ R and f (ρ R ) < f (ρ L ) together imply that ρ < ρ R . Then ρ(p) ≤ ρ < ρ R and R[ρ(p), ρ R ]is given by a shock with non negative speed if and only if p ≤ f (ρ R ), or equivalently, ρ R ≤ ρ(p). As a consequence, p ≤ f (ρ R ) < f (ρ L ), ρ L < ρ R ≤ ρ(p) and by np2 p coincides with p(ρ L +). In conclusion we have p ≤ f (ρ R ) ≤ p(ρ L +) = p, namely f (ρ R ) = p and the nonclassical solution coincides with the classical one. L +), see case (C1). Therefore, there does not exist any classical solution and we can consider only nonclassical solutions of the form 11. If p is continuous in ρ L , then by np4 we have that p = p(ρ L ). If p experiences a jump at ρ L then, one may wonder which value in [p(ρ L +), p(ρ L -)] has to be chosen as p. As in the case (C1), the assumptions imply that ρ < ρ R and then that ρ(p) < ρ R and p ≤ f (ρ R ). Then p is strictly smaller than f (ρ L ) and ρ(p) > ρ L . As a consequence, property np2 forces us to choose the unique possible value of p, which is p(ρ L +). In this case [(t, x) → R[ρ L , ρ R ](x/t)] performs a shock with non negative speed σ(ρ L , ρ R ) and it satisfies 5b because f (ρ L ) ≤ p(ρ L +). Assume that there exists a nonclassical solution of the form 11. Observe that the as-sumptions ρ L < ρ R and f (ρ R ) ≥ f (ρ L ) together imply that ρ > ρ L . Then ρ(p) ≥ ρ > ρ L and R[ρ L , ρ(p)] is given by a shock with non positive speed if and only if p ≤ f (ρ L ). Thus p ≤ f (ρ L ) ≤ p(ρ L +) and this implies by 11c that p = f (ρ L ) = p(ρ L +) and that the nonclassical solution coincides with the classical one. L -) < f (ρ L ), and as a consequence, property np2 forces us to choose p = p(ρ L +). In this case [(t, x) → R[ρ L , ρ R ](x/t)] performs a possible null rarefaction on the right of the constraint and it satisfies 5b because f (ρ L ) ≤ p(ρ L +). Assume that there exists a nonclassical solution of the form 11. Since ρ L ≤ ρ ≤ ρ(p), R[ρ L , ρ(p)] is given by a shock that has non positive speed if and only if p ≤ f (ρ L ). Therefore p ≤ f (ρ L ) ≤ p(ρ L +) and this by 11c implies that p = f (ρ L ) = p(ρ L +) and that the nonclassical solution coincides with the classical one.(N3) In this case [(t, x) → R[ρ L , ρ R ](x/t)] does not satisfy 5b because f (ρ L ) > p(ρ L -), see case (C3). Therefore, there does not exist any classical solution and we can consider only nonclassical solutions of the form 11. By hypothesis and 11c we have f (ρ

	(N1) In this case [(t, x) → R[ρ L , ρ R ](x/t)] does not satisfy 5b because f (ρ R ) > p(ρ (C2) (N2) In this case [(t, x) → R[ρ L , ρ R ](x/t)] does not satisfy 5b because f (ρ L ) > p(ρ (C3)

L -), see case (C2). Therefore, there does not exist any classical solution and we can consider only nonclassical solutions of the form 11. As in the case (C2), the assumptions imply ρ(p) ≥ ρ > ρ L . Furthermore, by 11c we

have p ≤ p(ρ L ) > p(ρ L -) ≥ p. Therefore ρ L ≤ ρ < ρ(p)

and by np2 we have p = p(ρ L +).

(C4) In this case [(t, x) → R[ρ L , ρ R ](x/t)

] performs a rarefaction with speeds between λ(ρ L ) < 0 and λ(ρ R ) ≥ 0 and it satisfies 5b because f (ρ) = p(ρ L +) implies that p(ρ) = f (ρ) for all ρ ≤ ρ L . Moreover, it implies also that p is continuous in ρ L and therefore, by np4, any nonclassical solution of the form 11 must have p = p(ρ L ) = f (ρ), but in this case the nonclassical solution coincides with the classical one.

(N4) In this case [(t, x) → R[ρ L , ρ R ](x/t)]

does not satisfy 5b because f (ρ) > p(ρ L -), see case (C4). Therefore, there does not exist any classical solution and we can consider only nonclassical solutions of the form 11.

  the authors are supported by French ANR JCJC grant CoToCoLa and Polonium 2014 (French-Polish cooperation program) No.331460NC. The second author is also supported by the Université de Frache-Comté, soutien aux EC 2014. The last author is also supported by ICM, University of Warsaw, and by Narodowe Centrum Nauki, grant 4140.