Nathaël Alibaud
email: nathael.alibaud@ens2m.fr

Boris Andreianov
email: boris.andreianov@univ-fcomte.fr

Non-uniqueness of weak solutions for the fractal Burgers equation

Keywords: fractional laplacian, non-local diffusion, conservation law, Lévy-Khintchine's formula, entropy solution, admissibility of solutions, Oleȋnik's condition, non-uniqueness of weak solutions 35L65, 35L67, 35L82, 35S10, 35S30

The notion of Kruzhkov entropy solution was extended by the first author in 2007 to conservation laws with a fractional laplacian diffusion term; this notion led to well-posedness for the Cauchy problem in the L ∞ -framework. In the present paper, we further motivate the introduction of entropy solutions, showing that in the case of fractional diffusion of order strictly less than one, uniqueness of a weak solution may fail.

Introduction

This paper contributes to the study of the so-called fractal/fractional Burgers equation

∂ t u(t, x) + ∂ x u 2 2 (t, x) + L λ [u](t, x) = 0, (t, x) ∈ R + × R, (1.1)
u(0, x) = u 0 (x), x ∈ R, (1.2)
where L λ is the non-local operator defined for all Schwartz function ϕ ∈ S(R) through its Fourier transform by F(L λ [ϕ])(ξ) := |ξ| λ F(ϕ)(ξ) with λ ∈ (0, 1); (1.3) i.e. L λ denotes the fractional power of order λ/2 of the Laplacian operator -∆ with respect to (w.r.t. for short) the space variable. This equation is involved in many different physical problems, such as overdriven detonation in gas [START_REF] Clavin | Instabilities and nonlinear patterns of overdriven detonations in gases[END_REF] or anomalous diffusion in semiconductor growth [START_REF] Woyczyński | Lévy processes in the physical sciences[END_REF], and appeared in a number of papers, such as [START_REF] Biler | Fractal Burgers Equations[END_REF][START_REF] Biler | Asymptotics for multifractal conservation laws[END_REF][START_REF] Biler | Asymptotics for conservation laws involving Lévy diffusion generators[END_REF][START_REF] Biler | Critical nonliearity exponent and self-similar asymptotics for Lévy conservation laws[END_REF][START_REF] Droniou | Global solution and smoothing effect for a non-local regularization of an hyperbolic equation[END_REF][START_REF] Jourdain | A probabilistic approach for nonlinear equations involving the fractional Laplacian and singular operator[END_REF][START_REF] Jourdain | Probabilistic approximation and inviscid limits for one-dimensional fractional conservation laws[END_REF][START_REF] Droniou | Fractal first order partial differential equations[END_REF][START_REF] Alibaud | Entropy formulation for fractal conservation laws[END_REF][START_REF] Alibaud | Occurence and non-appearance of shocks in fractal Burgers equation[END_REF][START_REF] Karch | On convergence of solutions of fractal Burgers equation toward rarefaction waves[END_REF][START_REF] Kiselev | Blow up and regularity for fractal Burgers equation[END_REF][START_REF] Dong | Finite time singularities and global well-posedness for fractal Burgers equations[END_REF][START_REF] Miao | Well-posedness of the Cauchy problem for fractional power dissipative equations[END_REF][START_REF] Miao | Global well-posedness of the critical Burgers equation in critical Besov spaces[END_REF][START_REF] Chan | Regularity of solutions for the critical N -dimensional Burgers equation[END_REF][START_REF] Alibaud | Asymptotic properties of entropy solutions to fractal Burgers equation[END_REF][START_REF] Droniou | A numerical method for fractal conservation laws[END_REF][START_REF] Karlsen | Stability of entropy solutions for Lévy mixed hyperbolic parabolic equations[END_REF]11]. Recently, the notion of entropy solution has been introduced by Alibaud in [START_REF] Alibaud | Entropy formulation for fractal conservation laws[END_REF] to show the global-in-time well-posedness in the L ∞ -framework.

For λ > 1 the notion of weak solution (i.e. a solution in the sense of distributions; cf. Definition 2.2 below) is sufficient to ensure the uniqueness and stability result, see the work of Droniou, Gallouët and Vovelle in [START_REF] Droniou | Global solution and smoothing effect for a non-local regularization of an hyperbolic equation[END_REF]. Such a result has been generalized to the critical case λ = 1 by Kiselev, Nazarov and Shterenberg in [START_REF] Kiselev | Blow up and regularity for fractal Burgers equation[END_REF], Dong, Du and Li in [START_REF] Dong | Finite time singularities and global well-posedness for fractal Burgers equations[END_REF], Miao and Wu in [START_REF] Miao | Global well-posedness of the critical Burgers equation in critical Besov spaces[END_REF] and Chan and Czubak in [START_REF] Chan | Regularity of solutions for the critical N -dimensional Burgers equation[END_REF] for a large class of initial data (either periodic or L 2 or in critical Besov spaces).

In this paper, we focus on the range of exponent λ ∈ (0, 1). By analogy with the purely hyperbolic equation λ = 0 (cf. Oleȋnik [START_REF] Oleȋnik | Discontinuous solutions of non-linear differential equations. (Russian), Uspehi Mat. Nauk[END_REF] and Kruzhkov [START_REF] Kruzhkov | First order quasilinear equations with several independent variables[END_REF]), a natural conjecture was that in that case a weak solution to the Cauchy problem (1.1)-(1.2) need not be unique. Indeed, it has been shown by Alibaud, Droniou and Vovelle in [START_REF] Alibaud | Occurence and non-appearance of shocks in fractal Burgers equation[END_REF] that the assumption λ < 1 makes the diffusion term too weak to prevent the appearance of discontinuities in solutions of (1.1); see also Kiselev, Nazarov and Shterenberg [START_REF] Kiselev | Blow up and regularity for fractal Burgers equation[END_REF] and Dong, Du and Li [START_REF] Dong | Finite time singularities and global well-posedness for fractal Burgers equations[END_REF]. To the best of our knownledge, yet it was unclear whether such discontinuities in a weak solution can violate the entropy conditions of [START_REF] Alibaud | Entropy formulation for fractal conservation laws[END_REF].

Here we construct a stationary weak solution of (1.1)-(1.2), λ < 1, which does violate the entropy constraint (constraint which can be expressed under the form of Oleȋnik's inequality, cf. [START_REF] Oleȋnik | Discontinuous solutions of non-linear differential equations. (Russian), Uspehi Mat. Nauk[END_REF]). Thus the main result of this paper is the following.

Theorem 1.1. Let λ ∈ (0, 1). There exist initial data u 0 ∈ L ∞ (R) such that uniqueness of a weak solution to the Cauchy problem (1.1)-(1.2) fails.

The rest of this paper is organized as follows. The next section lists the main notations, definitions and basic results on fractal conservation laws. The Oleȋnik inequality for the fractal Burgers equation is stated and proved in Section 3. In Section 4, we present and solve a regularized problem in which we pass to the limit in Section 5 to construct a non-entropy stationary solution. Section 6 is devoted the proof of the main properties of the fractional Laplacian (see Lemma 4.1) that have been used in both preceding sections. Finally, technical proofs and results have been gathered in Appendices A-B.

Preliminaries

In this section, we fix some notations, recall the Lévy-Khintchine formula for the fractional Laplacian and the associated notions of generalized solutions to fractal conservation laws.

Notations

Sets. Throughout this paper, R ± denote the sets (-∞, 0) and (0, +∞), respectively; the set R * denotes R \ {0} and R denotes {-∞} ∪ R ∪ {+∞}.

Right-differentiability.

A function m : R + → R is said to be right-differentiable at t 0 > 0 if there exists the limit lim t > →t 0 m(t)-m(t 0 ) t-t 0 in R; in that case, this limit is denoted by m ′ r (t 0 ).

Function spaces. Further, C ∞ c = D denotes the space of infinitely differentiable compactly supported test functions, S is the Schwartz space, D ′ is the distribution space and S ′ is the tempered distribution space. The space of k times continuously differentiable functions is denoted by C k and C k b denotes the subspace of functions with bounded derivatives up to order k (if k = 0, the superscripts are omitted); C c denotes the subspace of C of functions with compact support; C 0 denotes the closure of C c for the norm of the uniform convergence; L p , L p loc and W k,p , W k,p loc (=: H k , H k loc if p = 2) denote the classical Lebesgue and Sobolev spaces, respectively; BV and BV loc denote the spaces of functions which are globally and locally of bounded variations, respectively.

When it comes to topology and if nothing else is precised, D ′ and S ′ are endowed by their usual weak-⋆ topologies and the other spaces by their usual strong topologies (of Banach spaces, Fréchet spaces, etc).

Weak-⋆ topology in BV. Let ∂ x : D ′ (R) → D ′ (R) denote the gradient (w.r.t x) operator in the distribution sense. We let L 1 (R) ∩ (BV (R)) w-⋆ denote the linear space L 1 (R) ∩ BV (R) endowed with the smallest topology letting the inclusion L 1 (R) ∩ BV (R) ⊂ L 1 (R) and the mapping ∂ x : L 1 (R) ∩ BV (R) → (C 0 (R)) ′ be
continuous, where L 1 (R) is endowed with its strong topology and (C 0 (R)) ′ with its weak-⋆ topology. Hence, one has:

v k → v in L 1 (R) ∩ (BV (R)) w-⋆ ⇐⇒ v k → v in L 1 (R), ∂ x v k w-⋆ ⇀ ∂ x v in (C 0 (R)) ′ .
We define in the same way the space (BV loc (R)) w-⋆ ∩ H 1 loc (R \ {0}), whose notion of convergence of sequences is the following one:

v k → v in (BV loc (R)) w-⋆ ∩ H 1 loc (R \ {0}) ⇐⇒ v k → v in H 1 (R \ [-R, R]), ∀R > 0, ∂ x v k w-⋆ ⇀ ∂ x v in (C c (R)) ′ ; from the Banach-Steinhaus theorem, one sees that (v k ) k is (strongly) bounded in BV loc (R) ∩ H 1 loc (R \ {0}), i.e.: ∀R > 0, sup k∈N * v k H 1 (R\[-R,R]) + |v k | BV ((-R,R)) < +∞,
where | • | BV denotes the BV semi-norm.

Spaces of odd functions. In our construction, a key role is played by the spaces of odd functions v which are in the Sobolev space H 1 :

H 1 odd := v ∈ H 1 v is odd ; notice that v ∈ H 1 odd (R * ) can be discontinuous at zero so that v(0 -) = -v(0 + ) in the sense of traces, whereas v(0 -) = v(0 + ) = 0 if v ∈ H 1 odd (R). The space H 1 odd (R * ) and more generally H 1 (R * ) can be considered as subspaces of L 2 (R); to avoid confusion, ∂ x v always denotes the gradient of v in D ′ (R), so that (∂ x v) | R * ∈ L 2 (R) is the gradient in D ′ (R * ). One has (∂ x v) | R * = ∂ x v
almost everywhere (a.e. for short) on R if and only if (iff for short) v is continuous at zero; in the other case, one has

∂ x v / ∈ L 1 loc (R). When the context is clear, the products R ϕ (∂ x v) | R * and R (∂ x v) | R * (∂ x ψ) | R * with ϕ ∈ L 2 (R) and ψ ∈ H 1 (R * ) are simply denoted by R * ϕ ∂ x v and R * ∂ x v ∂ x ψ, respectively.
Identity and Fourier operators. By Id we denote the identity function. The Fourier transform F on S ′ (R) is denoted by F; for explicit computations, we use the following definition on L 1 (R):

F(v)(ξ) := R e -2iπxξ v(x) dx.
Entropy-flux pairs. By η, we denote a convex function on R; following Kruzhkov [START_REF] Kruzhkov | First order quasilinear equations with several independent variables[END_REF], we call it an entropy and q : u → u 0 s dη(s) is the associated entropy flux. Truncature functions. The sign function is defined by:

u → sign u := ±1 if ±u > 0, 0 if u = 0.
During the proofs, we shall need to regularize the function u → min{|u|, n} sign u, where n ∈ N * will be fixed; T n denotes a regularization satisfying

     T n ∈ C ∞ b (R) is odd, T n = Id on [-n + 1, n -1], |T n | ≤ n. (2.1)

Lévy-Khintchine's formula

Let λ ∈ (0, 1). For all ϕ ∈ S(R) and x ∈ R, we have

L λ [ϕ](x) = -G λ R ϕ(x + z) -ϕ(x) |z| 1+λ dz, (2.2)
where

G λ = λΓ( 1+λ 2 ) 2π 1 2 +λ Γ(1-λ 2 )
> 0 and Γ is Euler's function, see e.g. [START_REF] Bony | Semi-groupe de Feller sur une variété à bord compacte et prblèmes aux limites intégro-différentiels du second-ordre donnant lieu au principe du maximum[END_REF][START_REF] Hoh | Pseudo differential operators with negative definite symbols and the martingale problem[END_REF] or [18, Theorem 2.1].

Entropy and weak solutions

Formula (2.2) motivates the following notion of entropy solution introduced in [START_REF] Alibaud | Entropy formulation for fractal conservation laws[END_REF].

Definition 2.1 (Entropy solutions). Let λ ∈ (0, 1) and

u 0 ∈ L ∞ (R). A function u ∈ L ∞ (R + × R) is said to be an entropy solution to (1.1)-(1.2) if for all non-negative test function ϕ ∈ C ∞ c ([0, +∞) × R), all entropy η ∈ C 1 (R) and all r > 0, R η(u 0 )ϕ(0) + R + R (η(u)∂ t ϕ + q(u)∂ x ϕ) + G λ R + R |z|>r η ′ (u(t, x)) u(t, x + z) -u(t, x) |z| 1+λ ϕ(t, x) dtdxdz + G λ R + R |z|≤r η(u(t, x)) ϕ(t, x + z) -ϕ(t, x) |z| 1+λ dtdxdz ≥ 0. (2.3)
Remark 2.1. In the above definition, r plays the role of a cut-off parameter; taking r > 0 in (2.3), one avoids the technical difficulty while treating the singularity in the Lévy-Khintchine formula (by doing this, one looses some information, recovered at the limit r → 0). Let us refer to the recent paper of Karlsen and Ulusoy [START_REF] Karlsen | Stability of entropy solutions for Lévy mixed hyperbolic parabolic equations[END_REF] for a different definition of the entropy solution, equivalent to the above one; note that the framework of [START_REF] Karlsen | Stability of entropy solutions for Lévy mixed hyperbolic parabolic equations[END_REF] encompasses Lévy mixed hyperbolic/parabolic equations.

The notion of entropy solutions provides a well-posedness theory for the Cauchy problem for the fractional conservation law (1.1); the results are very similar to the ones for the classical Burgers equation (cf. e.g. [START_REF] Oleȋnik | Discontinuous solutions of non-linear differential equations. (Russian), Uspehi Mat. Nauk[END_REF][START_REF] Kruzhkov | First order quasilinear equations with several independent variables[END_REF]).

Theorem 2.1 ( [START_REF] Alibaud | Entropy formulation for fractal conservation laws[END_REF]). For all u 0 ∈ L ∞ (R), there exists one and only one entropy solution u ∈ L ∞ (R + × R) to (1.1)-(1.2). Moreover, u ∈ C([0, +∞); L 1 loc (R)) (so that u(0) = u 0 ), and the solution depends continuously in C([0, +∞);

L 1 (R)) on the initial data in L 1 (R) ∩ L ∞ (R).
As explained in the introduction, the purpose of this paper is to prove that the weaker solution notion below would not ensure uniqueness.

Definition 2.2 (Weak solutions). Let u 0 ∈ L ∞ (R). A function u ∈ L ∞ (R + × R) is said to be a weak solution to (1.1)-(1.2) if for all ϕ ∈ C ∞ c ([0, +∞) × R), R + R u ∂ t ϕ + u 2 2 ∂ x ϕ -u L λ [ϕ] + R u 0 ϕ(0) = 0.
(2.4)

The Oleȋnik inequality

Notice that it can be easily shown that an entropy solution is also a weak one. The converse statement is false, which we will prove by constructing a weak non-entropy solution. A key fact here is the well-known Oleȋnik inequality (see [START_REF] Oleȋnik | Discontinuous solutions of non-linear differential equations. (Russian), Uspehi Mat. Nauk[END_REF]); in this section, we generalize it to entropy solutions of the fractal Burgers equation. Proposition 3.1 (Oleȋnik's inequality). Let u 0 ∈ L ∞ (R). Let u ∈ L ∞ (R + × R) be the entropy solution to (1.1)-(1.2). Then, we have for all t > 0

∂ x u(t) ≤ 1 t in D ′ (R). (3.1)
Remark 3.1. This result can be adapted to general uniformly convex fluxes. Moreover, we think that the Oleȋnik inequality gives a necessary and sufficient condition for a weak solution to be an entropy solution (as for pure scalar conservation laws, cf. [START_REF] Oleȋnik | Discontinuous solutions of non-linear differential equations. (Russian), Uspehi Mat. Nauk[END_REF][START_REF] Kruzhkov | First order quasilinear equations with several independent variables[END_REF]). Nevertheless, for the sake of simplicity, we only prove the above result, which is sufficient for our purpose.

In order to prove this proposition, we need the following technical result: For a proof of this result, see e.g. the survey book of Danskyn [START_REF] Danskyn | The theory of Max Min[END_REF] on the min max theory; for the reader's convenience, a short proof is also given in Appendix A. We can now prove the Oleȋnik inequality.

Lemma 3.1. Let v ∈ C 1 (R + × R) be such that for all b > a > 0, lim |x|→+∞ sup t∈(a,b) v(t, x) = -∞. ( 3
Proof of Proposition 3.1. For ε > 0 consider the regularized problem

∂ t u ε + ∂ x u 2 ε 2 + L λ [u ε ] -ε∂ 2 xx u ε = 0 in R + × R, (3.3)
u ε (0) = u 0 on R. (3.4)
It was shown in [START_REF] Droniou | Global solution and smoothing effect for a non-local regularization of an hyperbolic equation[END_REF] that there exists a unique solution u ε ∈ L ∞ (R + × R) to (3.3)- (3.4) in the sense of the Duhamel formula, and that u ε ∈ C ∞ b ((a, +∞) × R)) for all a > 0. Furthermore, it has been proved in [START_REF] Alibaud | Asymptotic properties of entropy solutions to fractal Burgers equation[END_REF] that for all T > 0, u ε converges to u in C [0, T ]; L 1 loc (R) as ε → 0. Inequality (3.1) being stable by this convergence, it suffices to prove that u ε satisfies (3.1).

To do so, let us derivate (3.3) w.r.t. x. We get

∂ t v ε + v 2 ε + u ε ∂ x v ε + L λ [v ε ] -ε∂ 2 xx v ε = 0, (3.5)
with v ε := ∂ x u ε . Fix 0 < λ ′ < λ and introduce the "barrier function" Φ(x

) := (1 + |x| 2 ) λ ′ 2 . Then Φ is positive with lim |x|→+∞ Φ(x) = +∞; (3.6)
moreover Φ is smooth with

C Φ := ∂ x Φ ∞ + ∂ 2 xx Φ ∞ + L λ [Φ] ∞ < +∞, thanks to Lemma B.2 in Appendix B to ensure that L λ [Φ] ∈ C b (R) is well-defined by (2.2). For δ > 0 and t > 0, define m δ (t) := max x∈R {v ε (t, x) -δΦ(x)} .
Define K δ (t) := argmax x∈R {v ε (t, x) -δΦ(x)}. This set is non-empty and compact, thanks to the regularity of v ε and (3.6); moreover, by Lemma 3.1, m δ is right-differentiable w.r.t. t with:

(m δ ) ′ r (t) = max x∈K δ (t) ∂ t v ε (t, x) = ∂ t v ε (t, x δ (t))
for some x δ (t) ∈ K δ (t). This point is also a global maximum point of v ε (t) -δΦ, so that

∂ x v ε (t, x δ (t)) = δ∂ x Φ(x δ ), ∂ 2 xx v ε (t, x δ (t)) ≤ δ∂ 2 xx Φ(x δ ) and L λ [v ε ] ≥ δL λ [Φ] (t, x δ (t)) (the last inequality is easily derived from (2.2)). We deduce that |∂ x v ε (t, x δ (t))| ≤ δC Φ , ∂ 2 xx v ε (t, x δ (t)) ≤ δC Φ and L λ [v ε ] (t, x δ (t)) ≥ -δC Φ . By (3.5), we get (m δ ) ′ r (t)+v 2 ε (t, x δ (t)) ≤ Cδ, for some constant C that only depends on ε, u ε ∞ and C Φ . But, by construction m δ (t) = v ε (t, x δ (t)) -δΦ(x δ (t)) and Φ is non-negative, so that (m δ ) ′ r (t) + (m δ (t) + δΦ(x δ (t))) 2 ≤ Cδ and (m δ ) ′ r (t) -Cδ + (max{m δ (t), 0}) 2 ≤ 0. Now we set m δ (t) := m δ (t) -Cδt. Because the function r ∈ R → (max{r, 0}) 2 ∈ R is non- decreasing, we infer that m δ ∈ C(R + ) is right-differentiable with ( m δ ) ′ r (t) + (max{ m δ (t), 0}) 2 ≤ 0 for all t > 0. By Lemma B.1 in Appendix B, we can integrate this equation and conclude that m δ (t) ≤ 1 t for all t > 0. Finally, it is easy to prove that m δ (t) = m δ (t) -Cδt → sup x∈R v ε (t, x) as δ → 0, so that sup x∈R ∂ x u ε (t, x) ≤ 1
t (pointwise, for all t > 0). This proves (3.1) for u ε in the place of u, and thus completes the proof of the proposition.

A stationary regularized problem

The plan to show Theorem 1.1 consists in proving the existence of an odd weak stationary solution to (1.1) with a discontinuity at x = 0 not satisfying the Oleȋnik inequality. This nonentropy solution is constructed as limit of solutions to regularized problems, see Eqs. (4.2)-(4.3) below. This section focuses on the solvability of these problems. This is done in the second subsection; the first one lists some properties of L λ that will be needed.

Main properties of the non-local operator

In the sequel, L λ is always defined by the Lévy-Khintchine formula (2.2). Lemma 4.1. Let λ ∈ (0, 1). The operator L λ defined by the Lévy-Khintchine formula (2.2) enjoys the following properties:

(i) The operators L λ and L λ/2 are continuous as operators:

a) L λ : C b (R * ) ∩ C 1 (R * ) → C(R * ); b) L λ : H 1 (R * ) → L 1 loc (R) ∩ L 2 loc (R \ {0}); c) L λ/2 : H 1 (R * ) → L 2 (R).
Moreover, L λ is sequentially continuous as an operator:

d) L λ : L 1 (R) ∩ (BV (R)) w-⋆ → L 1 (R). (ii) If v ∈ H 1 (R * ), then the definition of L λ by Fourier transform (see (1.3)) makes sense; more precisely, L λ [v] = F -1 ξ → |ξ| λ F(v)(ξ) in S ′ (R). (iii) For all v, w ∈ H 1 (R * ), R L λ [v] w = R v L λ [w] = R L λ/2 [v] L λ/2 [w]. (iv) If v ∈ H 1 (R * ) is odd (resp. even), then L λ [v] is odd (resp. even). (v) Let 0 ≡ v ∈ C b (R * ) ∩ C 1 (R * ) be odd. Assume that x * > 0 is an extremum point of v such that v(x * ) = max R + v and v(x * ) ≥ 0 resp. v(x * ) = min R + v and v(x * ) ≤ 0 . Then, we have L λ [v](x * ) > 0 (resp. L λ [v](x * ) < 0).
Remark 4.1. Item (v) can be interpreted as a positive reverse maximum principle for the fractional Laplacian acting on the space of odd functions.

The proofs of these results are gathered in Section 6.

The regularized problem

Throughout this section, ε > 0 is a fixed parameter. Consider the space H 1 odd (R * ) with the scalar product

v, w := ε R * vw + ∂ x v ∂ x w + L λ/2 [v] L λ/2 [w] . (4.1)
By the item (i) (c) of Lemma 4.1, •, • is well-defined and its associated norm

• := •, • is equivalent to the usual H 1 (R * )-norm; in particular, H 1 odd (R * ) is an Hilbert space. Let us construct a solution v ∈ H 1 odd (R * ) to the problem ε(v ε -∂ 2 xx v ε ) + ∂ x v 2 ε 2 + L λ [v ε ] = 0 in R * , (4.2)
v ε (0 ± ) = ±1, (4.3)
where Eq. (4.2) is understood in the weak sense (e.g. in D ′ (R * )) and the constraint (4.3) is understood in the sense of traces. Setting

θ(x) := (1 -|x|) + sign x, (4.4)
we equivalently look for a weak solution of (4.2) living in the affine subspace of

H 1 odd (R * ) given by E := θ + H 1 odd (R) = v ∈ H 1 odd (R * ) v(0 ± ) = ±1
in the sense of traces . Here is the main result of this section. Proposition 4.1. Let λ ∈ (0, 1) and ε > 0. Eq. (4.2) admits a weak solution v ε ∈ E satisfying

0 ≤ v ε (x) sign x ≤ 1 for all x ∈ R * , (4.5)
sup

ε∈(0,1) R ε (∂ x v ε ) 2 | R * + L λ/2 [v ε ] 2 < +∞. (4.6)
Proof. The proof is divided into several steps.

Step one. We first fix v ∈ E and introduce the auxiliary equation with modified convection term:

ε(v -∂ 2 xx v) + ρ n ∂ x ρ n T n (v) 2 2 + L λ [v] = 0, (4.7)
where for n ∈ N * , the truncation functions T n and ρ n are given, respectively, by (2.1) and by the formula

ρ n (x) := ρ x C(n,ε) with      ρ ∈ C ∞ c (R) even, 0 ≤ ρ ≤ ρ(0) = 1, -1 ≤ ρ ′ ≤ 0 on R + and with C(n, ε) := n 2 ε (4.8)
(this choice of the constant is explained in Step three). Note the property ρ n -→ n→+∞ 1 uniformly on compact subsets of R. (4.9)

It is straightforward to see that solving (4.7), (4.3) in the variational sense below,

find v ∈ E such that for all ϕ ∈ H 1 odd (R), R * ε (vϕ + ∂ x v∂ x ϕ) + L λ/2 [v]L λ/2 [ϕ] = R (ρn Tn(v)) 2 2 ∂ x (ρ n ϕ), (4.10)
is equivalent to finding a minimizer v ∈ E for the functional

J v,n : E -→ R u → 1 2 R * ε u 2 + (∂ x u) 2 + L λ/2 [u] 2 -(ρ n T n (v)) 2 ∂ x (ρ n u) . Notice that ρ n T n (v) ∈ L ∞ (R) and ρ n ∈ H 1 (R), so that (ρ n T n (v)) 2 (∂ x (ρ n u)) | R * ∈ L 1 (R) with R * (ρ n T n (v)) 2 ∂ x (ρ n u) ≤ C n u ;
let us precise that here and until the end of this proof, C n denotes a generic constant that depends only on n and eventually on the fixed parameter ε (and which can change from one expression to another). Then the functional J v,n is well-defined on E and coercive, because

J v,n (u) = 1 2 u 2 - 1 2 R * ρ n T n (v) 2 ∂ x (ρ n u) ≥ 1 2 u 2 -C n u (4.11)
tends to infinity as u → +∞. Finally, it is clear that J v,n is strictly convex and strongly continuous. Thus we conclude that there exists a unique minimizer of J v,n , which is the unique solution of (4.10). We denote this solution by F n (v), which defines a map F n : E -→ E.

Step two: apply the Schauder fixed-point theorem to the map F n . Note that

F n (E) is contained in the closed ball B Rn := B 0 H 1 odd (R * ) , R n of H 1 odd (R * )
for some radius R n > 0 (only depending on n and ε). Indeed, let v := F n (v); then by using (4.11), replacing the minimizer v with the function θ ∈ E in (4.4), and applying the Young inequality we get

v 2 ≤ 2J v,n (v) + C n v ≤ 2J v,n (θ) + 1 2 v 2 + C n .
We can restrict F n to the closed convex set C := E ∩ B Rn of the Banach space H 1 odd (R * ). It remains to show that F n : C -→ C is continuous and compact.

In order to justify the compactness of

F n (C), take a sequence (v k ) k ⊂ F n (C) and an associated sequence (v k ) k ⊂ C with v k = F n (v k ).
Because C is bounded, there exists a (not relabelled) subsequence of (v k ) k that converges weakly in H 1 (R * ) and strongly in L 2 loc (R) by standart embedding theorems; let v ∞ be its limit. One has v ∞ ∈ C because C is weakly closed in H 1 (R * ) as strongly closed convex subset. We can assume without loss of generality that the corresponding subsequence of (v k ) k converges weakly to some

v ∞ ∈ C in H 1 odd (R * ). Let us prove that v k converges strongly to v ∞ in H 1 odd (R * ) and that v ∞ = F n (v ∞ ).
By the above convergences, and the facts that

T n ∈ C ∞ b (R) and ρ n ∈ C ∞ c (R), one can see that R * ρ n T n (v k ) 2 ∂ x (ρ n v k ) → R * ρ n T n (v ∞ ) 2 ∂ x (ρ n v ∞ ) as k → +∞,
Moreover, using that v k is the minimizer of J vk ,n , we have

v k 2 - R * ρ n T n (v k ) 2 ∂ x (ρ n v k ) = 2J vk ,n (v k ) ≤ 2J vk ,n (v ∞ ) = v ∞ 2 - R * ρ n T n (v k ) 2 ∂ x (ρ n v ∞ ).
Thus passing to the limit as k → +∞ in this inequality yields:

v ∞ ≥ lim sup k→+∞ v k .
It follows that the convergence of v k to v ∞ is actually strong in H 1 odd (R * ). Passing to the limit as k → +∞ in the variational formulation (4.10) written for v k and vk , we deduce by the uniqueness of a solution to (4.10) that v ∞ = F n (v ∞ ); this completes the proof of the compactness of F n (C).

To prove the continuity of F n , one simply assumes that vk → v∞ strongly in H 1 odd (R * ) and repeats the above reasoning for each subsequence of (v k ) k . One gets that from all subsequence of (v k ) k one can extract a subsequence strongly converging to v ∞ = F n (v ∞ ); hence, the proof of the continuity of F n is complete.

We conclude that there exists a fixed point u n of F n in C. Then v := v = u n satisfies the formulation (4.10). In addition, (4.10) is trivially satisfied with a test function ϕ ∈ H 1 (R) which is even. Indeed, using the definitions of T n and ρ n and Lemma 4.1 (iv), we see that

ε vϕ + ∂ x v∂ x ϕ + L λ/2 [v]L λ/2 [ϕ] - ρ n T n (v) 2 2 ∂ x (ρ n ϕ)
is an odd function, so that its integral on R * is null. Since all function in H 1 (R) can be split into the sum of an odd function in H 1 odd (R) and an even function in H 1 (R), we have proved that the fixed point

u n ∈ E of F n satisfies for all ϕ ∈ H 1 (R), R ε u n ϕ + (∂ x u n ) | R * ∂ x ϕ + L λ/2 [u n ]L λ/2 [ϕ] = R (ρ n T n (u n )) 2 2 ∂ x (ρ n ϕ) (4.12)
(notice that the Rankhine-Hugoniot condition is contained in the fact that u 2 n 2 is even). In particular, using Lemma 4.1 item (iii), one has

ε(∂ 2 xx u n -u n ) = ρ n ∂ x (ρ n T n (u n )) 2 2 + L λ [u n ] in D ′ (R * ). ( 4

.13)

Step three: uniform estimates on the sequence (u n ) n . First, in order to prove a maximum principle for u n let us point out that u n is regular. Indeed, thanks to Lemma 4.1 (i) (b) and the facts that

T n ∈ C ∞ b (R) and ρ n ∈ C ∞ c (R)
, the right-hand side of (4.13) belongs to L 1 (I) for all compact interval I ⊂ R * . Eq. (4.13) then implies that

u n ∈ W 2,1 loc (R * ) ⊂ C 1 (R * ). Recall that u n ∈ H 1 (R * ) ⊂ C b (
R * ); thus using Lemma 4.1 (i) (a), we see that the right-hand side of (4.13) belongs to C(I). Exploiting once more Eq. (4.13), we infer that u n ∈ C 2 (R * ) and (4.13) holds pointwise on R * . Now, we are in a position to prove that for all x > 0 and n ∈ N * , 0 ≤ u n (x) ≤ 1. Indeed, because u n ∈ H 1 (R + ), we have lim x→+∞ u n (x) = 0; in addition, u n (0

+ ) = 1. Thus if u n (x) / ∈ [0, 1] for some x ∈ R + , there exists x * ∈ R + such that either u n (x * ) = max R + u > 1 or u n (x * ) = min R + u < 0. Consider the first case. Since u n ∈ C 2 (R * ), we have ∂ x u n (x * ) = 0 and ∂ 2 xx u n (x * ) ≤ 0.
In addition, by Lemma 4.1 (v) we have L[u n ](x * ) > 0. Therefore using (4.13) at the point x * , by the choice of ρ n and C(n, ε) in (4.8) we infer

εu n (x * ) = ε∂ 2 xx u n (x * ) -L λ [u n ](x * ) -ρ n (x * ) ∂ x (ρ n (x * ) T n (u n (x * ))) 2 2 ≤ -(ρ n (x * ) T n (u(x * ))) 2 ∂ x ρ n (x * ) ≤ n 2 1 C(n, ε) sup R + (-∂ x ρ) ≤ ε.
Thus u(x * ) ≤ 1, which contradicts the definition of x * . The case u n (x * ) = min R + u < 0 is similar; we use in addition the fact that ∂ x ρ n ≤ 0 on R + . The function u n being even, from the maximum principle of Step three we have

|u n | ≤ 1 on R * . Since T n = Id on [-n + 1, n -1], we have T n (u n ) = u n in (4.13) for all n ≥ 2.
Let us finally derive the uniform H 1 odd (R * )-bound on (u n ) n . To do so, replace the minimum u n of the functional J un,n by the fixed function θ ∈ E in (4.4); we find

u n 2 = 2J un,n (u n ) + R * (ρ n u n ) 2 ∂ x (ρ n u n ) ≤ 2J un,n (θ) + R * ∂ x (ρ n u n ) 3 3 .
Since ρ n (0) = 1 = ±u n (0 ± ), we get

u n 2 ≤ 2J un,n (θ) - 2 3 = θ 2 - R * (ρ n u n ) 2 ∂ x (ρ n θ) - 2 3 .
To estimate the integral term, we use that θ is supported by [-1, 1] with |∂ x (ρ n θ)| ≤ 1 + ε n 2 , thanks again to the choice of ρ n in (4.8); Using finally the bound |u n | ≤ 1 derived above, we get

- R * (ρ n u n ) 2 ∂ x (ρ n θ) ≤ 2 + 2ε n 2 ;
hence, we obtain the following uniform estimate:

u n 2 ≤ θ 2 + 4 3 + 2ε n 2 . ( 4

.14)

Step four: passage to the limit as n → +∞. The H 1 odd (R * )-estimate of Step three permits to extract a (not relabelled) subsequence (u n ) n which converges weakly in H 1 (R * ) and strongly in L 2 loc (R), to a limit that we denote v ε . We have (u n ) n ⊂ E which is a closed affine subspace of H 1 (R * ), so that v ε ∈ E. The above convergences and the convergence of ρ n in (4.9) are enough to pass to the limit in (4.12); at the limit, we conclude that v ε is a weak solution of (4.2). Notice that v ε inherits the bounds on u n , namely the bound (4.14) and the maximum principle 0 ≤ u n (x) sign x ≤ 1. This yields (4.5) and (4.6), thanks to the definition of • via the scalar product (4.1). Remark 4.2. When passing to the limit as n → +∞ in (4.12) in the last step, one gets:

R ε v ε ϕ + (∂ x v ε ) | R * ∂ x ϕ + v ε L λ [ϕ] = R v ε 2 2 ∂ x ϕ for all ϕ ∈ H 1 (R). (4.15)

A non-entropy stationary solution

We are now able to construct a stationary non-entropy solution to (1.1) by passing to the limit in v ε as ε → 0. Let us explain our strategy. First, we have to use the uniform estimates of Proposition 4.1 to get compactness; this is done via the following lemma which is proved in Appendix A: Lemma 5.1. Assume that for all ε ∈ (0, 1), v ε ∈ H 1 (R * ) satisfies (4.5)-(4.6). Then the family {v ε | ε ∈ (0, 1)} is relatively compact in L 2 loc (R). With Lemma 5.1 in hands, we can prove the convergence of a subsequence of v ε , as ε → 0, to some stationary weak solution v of (1.1). Next, we need to control the traces of v at x = 0 ± . This is done by reformulating Definition 2.2 and by exploiting the Green-Gauss formula.

Let us begin with giving a characterization of odd weak stationary solutions of the fractional Burgers equation.

Proposition 5.1. An odd function v ∈ L ∞ (R) satisfies ∂ x v 2 2 + L λ [v] = 0 in D ′ (R), (5.1)

iff (i) and (ii) below hold true:

(i) there exists the trace γv 2 := lim h→0 + 1

h h 0 v 2 (x) dx; (ii) for all odd compactly supported in R test function ϕ ∈ C ∞ b (R * ), R * vL λ [ϕ] - v 2 2 ∂ x ϕ = ϕ(0 + ) γv 2 .
Proof. Assume (5.1). For all h > 0, let us set

ψ h (x) := 1 h (h -|x|) + sign x. Let us recall that θ(x) = (1 -|x|) + sign x. First consider θ h (x) := θ(x), x < 0 -ψ h (x), x ≥ 0 and θ 0 (x) = θ(x), x < 0 0, x ≥ 0.
By construction, θ h ∈ H 1 (R); therefore θ h can be approximated in H 1 (R) by functions in D(R) and thus taken as a test function in (5.1). This gives

- R + v 2 2 ∂ x ψ h = - R - v 2 2 ∂ x θ + R v L λ [θ h ].
But, it is obvious that θ h -→ θ 0 in L 1 (R)∩(BV (R)) w- * as h → 0 + ; thus using Lemma 4.1 (i) (d), we conclude that the limit in item (i) of Proposition 5.1 does exist, and

γv 2 := lim h→0 + 1 h h 0 v 2 = -lim h→0 + R + v 2 ∂ x ψ h = - R - v 2 ∂ x θ + 2 R v L λ [θ 0 ]. (5.2)
Further, take a function ϕ as in item (ii) of Proposition 5.1 and set ϕ h (x) := ϕ(x)ϕ(0 + )ψ h (x). One can take ϕ h ∈ H 1 (R) as a test function in (5.1). Taking into account the fact that v 2 2 ∂ x ϕ h and v L λ [ϕ h ] are even, thanks again to Lemma 4.1 (iv), we get 2

R + v L λ [ϕ] - v 2 2 ∂ x ϕ = 2ϕ(0 + ) R + v L[ψ h ] - v 2 2 ∂ x ψ h .
Now we pass to the limit as h → 0 + . As previously, because

ψ h -→ 0 in L 1 (R) ∩ (BV (R)) w- * , the term L λ [ψ h ] vanishes in L 1 (R). Using (5.
2), we get item (ii) of Proposition 5.1.

Conversely, assume that an odd function v satisfies items (i) and (ii) of Proposition 5.1. Take a test function ξ ∈ D(R) and write ξ = ϕ + ψ with ϕ ∈ D(R) odd (so that ϕ(0 + ) = 0) and ψ ∈ D(R) even. Then (ii) and the symmetry considerations, including Lemma 4.1 (iv), show that

R v L λ [ϕ] - v 2 2 ∂ x ϕ = ϕ(0 + ) γv 2 = 0, R v L λ [ψ] - v 2 2 ∂ x ψ = 0.
Hence we deduce that v satisfies (5.1).

Here is the existence result of a non-entropy stationary solution.

Proposition 5.2. Let λ ∈ (0, 1). There exists v ∈ L ∞ (R) that satisfies (5.1) and such that for all c > 0, v does not satisfy In order to conclude the proof, we will show that there exist the limits lim

∂ x v ≤ 1 c in D ′ (R).
h→0 + 1 h h 0 v = 1, lim h→0 + 1 h 0 -h v = -1.
(5.3) Indeed, (5.3) readily implies that for all c > 0, the function (v -1 c Id) does not admit a nonincreasing representative. Since ∂

x (v -1 c Id) = ∂ x v -1 c , the inequality ∂ x v -1 c
≤ 0 in the distribution sense fails to be true.

Thus it remains to show (5.3). To do so, we exploit the formulation (i)-(ii) of Proposition 5.1, the analogous formulation of the regularized problem (4.2), the fact that v ε k (0 ± ) = ±1, and (4.5).

Let us fix some odd compactly supported in

R function ϕ ∈ C ∞ b (R * ) such that ϕ(0 + ) = 1. Let us take the test function ϕ h (x) := ϕ(x) -ψ h (x) ∈ H 1 (R) in (4.15). We infer R * ε (v ε ϕ h + ∂ x v ε ∂ x ϕ h ) + v ε L λ [ϕ h ] - v 2 ε 2 ∂ x ϕ h = 0.
Each term in the above integrand is even; moreover, letting h → 0 + and using Lemma 4.

1 (i) (d) on L λ [ψ h ], we infer R * ε (v ε ϕ + ∂ x v ε ∂ x ϕ) + v ε L λ [ϕ] - v 2 ε 2 ∂ x ϕ = 2 lim h→0 + 1 h h 0 v 2 ε 2 -ε∂ x v ε = 1 - 2ε h [v ε ] h 0 = 1 - 2ε h (v ε (h) -1) ≥ 1; (5.4)
here in the last inequality, we have used 0

≤ v ε (x) ≤ 1 = v ε (0 + ) for x > 0.
Letting ε k → 0 in (5.4), using again (4.6) to vanish

R * ε ∂ x v ε ∂ x ϕ, we infer R * vL λ [ϕ] - v 2 2 ∂ x ϕ ≥ 1. (5.5)
Recall that v is odd and solves (5.1); thus it satisfies items (i) and (ii) of Proposition 5.1. From item (ii), we infer that lim h→0 + 1

h h 0 v 2 = γv 2 ≥ 1. But we also have 0 ≤ v ≤ 1 on [0, h]. Therefore lim h→0 + 1 h h 0 |1 -v| = lim h→0 + 1 h h 0 1 -v 2 1 + v ≤ lim h→0 + 1 h h 0 (1 -v 2 ) = 1 -γv 2 ≤ 0.
Whence the first equality in (5.3) follows. The second one is clear because v is an odd function. This concludes the proof.

From Propositions 3.1 and 5.2, Theorem 1.1 readily follows.

Proof of Theorem 1.1. Take u 0 := v. From (5.1) we derive that the function defined by u(t) := v for all t ≥ 0 is a weak solution to (1.1)-(1.2). But it is not the entropy solution, because it fails to satisfy (3.1).

6 Proof of Lemma 4.1

We end this paper by proving the main properties of the fractional Laplacian acting on spaces of odd functions. First, we have to state and prove some technical lemmata.

Here are embedding and density results that will be needed; for the reader's convenience, short proofs are given in Appendix A. Lemma 6.1. The inclusions

H 1 (R * ) ⊂ BV loc (R) ∩ H 1 loc (R \ {0}) ⊂ L ∞ (R) ∩ L 2 (R) (6.1)
and

(BV loc (R)) w-⋆ ∩ H 1 loc (R \ {0}) ⊂ L 2 (R). (6.2)
are continuous and sequentially continuous, respectively.

Lemma 6.2. The space D(R) is dense in H 1 (R * ) for the (BV loc (R)) w-⋆ ∩ H 1 loc (R \ {0})-topology.
The next lemma states weak continuity results for the fractional Laplacian. Until the end of this section, L λ denotes the operator defined by (2.2) and L F λ denotes the one defined by (1.3).

Lemma 6.3. Let λ ∈ (0, 1). Then the following operators are sequentially continuous:

L λ : (BV loc (R)) w-⋆ ∩ H 1 loc (R \ {0}) → L 1 loc (R) ∩ L 2 loc (R \ {0}), L F λ/2 : (BV loc (R)) w-⋆ ∩ H 1 loc (R \ {0}) → L 2 (R).
Proof. The proof is divided in several steps.

Step one: strong continuity of

L λ . Let v ∈ BV loc (R) ∩ H 1 loc (R \ {0}
) and let us derive some estimates on L λ [v]. For all r, R > 0, using the Fubini theorem one has

R -R R |v(x + z) -v(x)| |z| 1+λ dxdz = R -R |z|≤r |v(x + z) -v(x)| |z| 1+λ dxdz + R -R |z|>r |v(x + z) -v(x)| |z| 1+λ dxdz ≤ |v| BV ((-R-r,R+r)) |z|≤r |z| -λ dz + sup |z|>r v L 1 ((-R+z,R+z)) + v L 1 ((-R,R)) |z|>r |z| -1-λ dz = 2r 1-λ 1 -λ |v| BV ((-R-r,R+r)) + 2 λr λ sup |z|>r v L 1 ((-R+z,R+z)) + v L 1 ((-R,R)) . (6.3)
By (6.1) of Lemma 6.1, using the Cauchy-Schwarz inequality to control the L 1 -norms by the L 2norms, one sees that integral term in (2.2) makes sense a.e. with

L λ [v] L 1 ((-R,R)) ≤ 2G λ r 1-λ 1 -λ |v| BV ((-R-r,R+r)) + 4G λ λr λ √ 2R v L 2 (R) , (6.4)
for all r, R > 0. In the same way, by Minkowski's integral inequality one has for R > r > 0

R\[-R,R] R |v(x + z) -v(x)| |z| 1+λ dz 2 dx 1 2 ≤ R R\[-R,R] |v(x + z) -v(x)| 2 |z| 2+2λ dx 1 2 dz = |z|≤r |z| -1-λ R\[-R,R] |v(x + z) -v(x)| 2 dx 1 2 dz + |z|>r |z| -1-λ R\[-R,R] |v(x + z) -v(x)| 2 dx 1 2 dz ≤ 2r 1-λ 1 -λ ∂ x v L 2 (R\[-R+r,R-r]) + 4 λr λ v L 2 (R) ; therefore, one gets for all R > r > 0, L λ [v] L 2 (R\[-R,R]) ≤ 2G λ r 1-λ 1 -λ ∂ x v L 2 (R\[-R+r,R-r]) + 4G λ λr λ v L 2 (R) . (6.5)
Now (6.4)-(6.5) imply that L λ :

BV loc (R) ∩ H 1 (R \ {0}) → L 1 loc (R) ∩ L 2 loc (R \ {0}
) is well defined and continuous.

Step two: weak-⋆ sequential continuity of

L λ . Consider a sequence (v k ) k converging to zero in (BV loc (R)) w-⋆ ∩ H 1 loc (R \ {0}). For all R > 0, (v k ) k is bounded in the norm of H 1 (R \ [-R, R]
) and the semi-norm of BV ((-R, R)) by some constant C R . By (6.4), one deduces that lim sup

k→+∞ L λ [v k ] L 1 ((-R,R)) ≤ 2G λ r 1-λ 1 -λ C R+r . Letting r → 0, one concludes that L λ [v k ] converges to zero in L 1 ((-R, R)). In the same way, one can prove that L λ [v k ] converges to zero in L 2 (R \ [-R, R]
) by using (6.5). Since R is arbitrary, the proof of Lemma 6.3 is complete.

Step three: strong continuity of

L F λ/2 . Let us derive an L 2 -estimate on L F λ/2 [v]. Recall that by Lemma 6.1 (6.1), one has v ∈ L 2 (R) so that | • | F(v)(•) ∈ L 1 loc (R) and L F λ/2 [v] is well defined in S ′ (R).
Further, consider some fixed ρ ∈ C ∞ c (R) such that ρ = 1 on some neighborhood of the origin, say on [-1/2, 1/2], and

supp ρ ⊆ [-1, 1]. Then one has v = ρv+(1-ρ)v with supp (ρv) ⊆ [-1, 1], ρv ∈ L 1 (R) ∩ BV (R) (since v ∈ L 2 (R)) and (1 -ρ)v ∈ H 1 (R); moreover, one readily sees that ρv L 1 (R) ≤ C ρ v L 2 (R) , (6.6)
|ρv| BV (R) ≤ C ρ |v| BV ((-1/2,1/2)) + v H 1 (R\[-1/2,1/2]) , (6.7)
(1 -ρ)v L 2 (R) ≤ C ρ v L 2 (R) , (6.8)
∂ x ((1 -ρ)v) L 2 (R) ≤ C ρ v H 1 (R\[-1/2,1/2]) , (6.9)
where until the end of the proof C ρ denotes a generic constant only depending on ρ. By Plancherel's equality, we have

L F λ/2 [v] L 2 (R) = R |ξ| λ |F (v) (ξ)| 2 dξ = R |ξ| λ |F (ρv) (ξ)| 2 dξ + R |ξ| λ |F ((1 -ρ)v) (ξ)| 2 dξ =: I + J. (6.10)
Let us first bound J from above. For all r > 0, one has

J = |ξ|>r |ξ| λ |F ((1 -ρ)v) (ξ)| 2 dξ + |ξ|≤r |ξ| λ |F ((1 -ρ)v) (ξ)| 2 dξ ≤ |ξ|>r |ξ| λ-2 |ξ| 2 |F ((1 -ρ)v) (ξ)| 2 dξ + r λ F ((1 -ρ)v) 2 L 2 (R) ≤ 1 r 2-λ |ξ|>r |ξ| 2 |F ((1 -ρ)v) (ξ)| 2 dξ + r λ (1 -ρ)v 2 L 2 (R) .
Using the formula

F(∂ x w) = 2iπξ F(w) (6.11)
and again Plancherel's equality, one gets J ≤

1 4π 2 r 2-λ ∂ x ((1 -ρ)v) 2 L 2 (R) + r λ (1 -ρ)v 2 L 2 (R)
; so that by (6.8)-(6.9), one has

J ≤ C ρ r 2-λ v 2 H 1 (R\[-1/2,1/2]) + C ρ r λ v 2 L 2 (R) . (6.12)
To bound I from above, one uses the boundeness of F : L 1 (R) → L ∞ (R) and the pointwise estimate |ξ| |F(w)(ξ)| ≤ 1 2π |w| BV (R) that comes from (6.11). We get

I = |ξ|>r |ξ| λ |F (ρv) (ξ)| 2 dξ + |ξ|≤r |ξ| λ |F (ρv) (ξ)| 2 dξ ≤ 1 4π 2 |ρv| 2 BV (R) |ξ|>r |ξ| λ-2 dξ + ρv 2 L 1 (R) |ξ|≤r |ξ| λ dξ = 1 2π 2 (1 -λ)r 1-λ |ρv| 2 BV (R) + 2r 1+λ 1 + λ ρv 2 L 1 (R) ;
so that by (6.6)-(6.7), one has

I ≤ C ρ (1 -λ)r 1-λ |v| BV ((-1/2,1/2)) + v H 1 (R\[-1/2,1/2]) 2 + C ρ r 1+λ 1 + λ v 2 L 2 (R) . (6.13)
We deduce from (6.10), (6.12) and (6.13) the final estimate:

L F λ/2 [v] L 2 (R) ≤ C ρ r λ + r 1+λ 1 + λ v 2 L 2 (R) + C ρ 1 r 2-λ + 1 (1 -λ)r 1-λ |v| BV ((-1/2,1/2)) + v H 1 (R\[-1/2,1/2]) 2 . (6.14)
for all r > 0.

One infers that

L F λ/2 : BV loc (R) ∩ H 1 loc (R \ {0}) → L 2 (R) is continuous.
Step four: weak-⋆ sequential continuity of L F λ/2 . By (6.2) of Lemma 6.1, one sees that if v k → 0 in the topological space (BV loc (R)) w-⋆ ∩ H 1 loc (R \ {0}), then v k → 0 in L 2 (R). One then argues exactly as in Step two by using (6.14) instead of (6.4)-(6.5); one deduces that L F λ/2 |v k ] → 0 in L 2 (R) and this completes the proof of the lemma.

We can now prove the main properties of L λ stated in Subsection 4.1.

Proof of Lemma 4.1. Let us prove the different items step by step.

Step one: item (i) (a) and (b). Item (i) (a) is an immediate consequence of the theorem of continuity under the integral sign; the details are left to the reader. Item (i) (b) is clear from Lemmata 6.1 and 6.3.

Step two: item (i) (d). Passing to the limit R → +∞ in (6.3), one gets

L λ [v] L 1 (R) ≤ 2G λ r 1-λ 1 -λ |v| BV (R) + 4G λ λr λ v L 1 (R) , (6.15)
for all v ∈ L 1 (R) ∩ BV (R) and r > 0. With this estimate in hands, we can argue as in the second step of the proof of Lemma 6.3 to show item (i) (d).

Step three: items (ii) and (i) (c). Let us prove item (ii) first. By Lemma 6.2, v ∈ H 1 (R * ) can be approximated by

v k ∈ S(R) in (BV loc (R)) w-⋆ ∩ H 1 loc (R \ {0}). One has L λ [v k ] = L F λ [v k ]
thanks to the classical Lévy-Khintchine formula. By Lemma 6.3, we infer that L λ [v k ] converges toward L λ [v] in S ′ (R) as k → +∞. But the embedding (6.2) of Lemma 6.1 implies

that v k → v in L 2 (R) so that F(v k ) → F(v) in L 2 (R). It follows that | • | λ F(v k )(•) → | • | λ F(v)(•)
in S ′ (R); hence, taking the inverse Fourier transform, one sees that

L F λ [v k ] → L F λ [v] in S ′ (R). By uniqueness of the limit, one has L λ [v] = L F λ [v]
and the proof of item (ii) is complete. As an immediate consequence, one deduces item (i) (c) by using in particular Lemmata 6.1 and 6.3.

Step four: item (iii). Take v k , w k ∈ S(R) converging in (BV loc (R)) w-⋆ ∩ H 1 loc (R \ {0}) to v, w ∈ H 1 (R * ). For such functions, it is immediate from the definition by Fourier transform

(1.3) that R L λ [v k ]w k = R v k L λ [w k ] = R L λ/2 [w k ]L λ/2 [v k ]. By Lemma 6.3, one has L λ [u k ] → L λ [u] in L 1 loc (R)∩L 2 loc (R\{0}) for u = v, w
. By Lemma 6.1 and Banach-Alaoglu-Bourbaki's theorem, one has the following convergence (up to a subsequence):

u k → u in L 2 (R) and in L ∞ (R) weak-⋆
for u = v, w; indeed, (6.2) implies the strong convergence in L 2 and (6.1) implies that (u

k ) k is bounded in L ∞ , since it is (strongly) bounded in BV loc (R)∩ H 1 loc (R\{0}) as converging sequence in (BV loc (R)) w-⋆ ∩ H 1 loc (R \ {0}
). Hence, one clearly can pass to the limit:

R L λ [v]w = lim k→+∞ R L λ [v k ]w k = lim k→+∞ R v k L λ [w k ] = R vL λ [w].
To pass to the limit in

R L λ/2 [w k ]L λ/2 [v k ],
one uses Lemma 6.3 and item (ii). The proof of item (iii) is complete.

Step five: item (iv). It suffices to change the variable by z → -z in (2.2).

Step six: item (v). We consider only the case where v(x * ) = max R + v ≥ 0, since the case v(x * ) = min R + v ≤ 0 is symmetric. Simple computations show that

L λ [v](x * ) = -G λ R v(x * + z) -v(x * ) |z| 1+λ dz = -G λ +∞ -x * v(x * + z) -v(x * ) |z| 1+λ dz -G λ -x * -∞ v(x * + z) -v(x * ) |z| 1+λ dz = -G λ +∞ -x * v(x * + z) -v(x * ) |z| 1+λ dz -G λ +∞ -x * v(-x * -z ′ ) -v(x * ) |z ′ + 2x * | 1+λ dz ′ ,
after having changed the variable by z ′ = -z -2x * . By the oddity of v, we get

L λ [v](x * ) = -G λ +∞ -x * v(x * + z) -v(x * ) |z| 1+λ - v(x * + z) + v(x * ) |z + 2x * | 1+λ dz.
Let f (z) denote the integrand above. Let us prove that for 0 = z > -x * , this integrand is non-positive. It is readily seen that for such z, one always has

1 |z| 1+λ - 1 |z+2x * | 1+λ > 0. Then, one has f (z) = v(x * + z) 1 |z| 1+λ - 1 |z + 2x * | 1+λ -v(x * ) 1 |z| 1+λ + 1 |z + 2x * | 1+λ ≤ v(x * ) 1 |z| 1+λ - 1 |z + 2x * | 1+λ -v(x * ) 1 |z| 1+λ + 1 |z + 2x * | 1+λ ; indeed, x * + z ∈ R + , so that v(x * + z) ≤ v(x * ). We infer that f (z) ≤ -v(x * ) 2 |z+2x * | 1+λ ≤ 0 and conclude that L λ [v](x * ) ≥ 0.
To finish, observe that f can not be identically equal to zero, whenever v is non-trivial. This proves that L λ [v](x * ) > 0 and completes the proof of the lemma.

A Proofs of Lemmata 3.1, 5.1, 6.1 and 6.2

m(t k ) -m(t 0 ) t k -t 0 = lim sup k→+∞ v(t k , x k ) -m(t 0 ) t k -t 0 ≤ lim sup k→+∞ v(t k , x k ) -v(t 0 , x k ) t k -t 0 = ∂ t v(t 0 , x 0 ),
thanks to the C 1 -regularity of v. But, one has x 0 ∈ K(t 0 ); indeed, for all x ∈ R, one has v(t k , x k ) ≥ v(t k , x) so that the limit as k → +∞ gives v(t 0 , x 0 ) ≥ v(t 0 , x). Hence, one has proved that lim sup k→+∞ m(t k )-m(t 0 ) t k -t 0 ≤ sup x∈K(t 0 ) ∂ t v(t 0 , x) . In the same way, for all x ∈ K(t 0 ) one has

lim inf k→+∞ m(t k ) -m(t 0 ) t n -t 0 ≥ lim inf k→+∞ v(t k , x) -v(t 0 , x) t k -t 0 = ∂ t v(t 0 , x). This shows that lim inf k→+∞ m(t k ) -m(t 0 ) t k -t 0 ≥ max x∈K(t 0 ) ∂ t v(t 0 , x) ≥ lim sup k→+∞ m(t k ) -m(t 0 ) t k -t 0 , for all t 0 > 0 and (t k ) k such that t k → t 0 , t k > t 0 . This means that m is right-differentiable with m ′ r (t 0 ) = max x∈K(t 0 ) ∂ t v(t 0 , x) on R + . Proof of Lemma 5.1. Let us estimate the translations of v ε . Fix h ∈ R and define T h v ε (x) := v ε (x -h). Classical formula gives F (T h v ε ) (ξ) = e -2iπξh F (v ε ) (ξ)
. By the Plancherel equality, we deduce that

R |T h v ε -v ε | 2 = R e -2iπξh -1 2 |F (v ε ) (ξ)| 2 dξ = R e -2iπξh -1 2 |ξ| λ |ξ| λ |F (v ε ) (ξ)| 2 dξ ≤ M h R |ξ| λ |F (v ε ) (ξ)| 2 dξ, where M h := max ξ∈R |e -2iπξh -1| 2 |ξ| λ
. Lemma 4.1 item (ii) and the Plancherel equality imply that

R |T h v ε -v ε | 2 ≤ M h R L λ/2 [v ε ] 2 .
By the assumptions of the lemma, we deduce that R |T h v εv ε | 2 ≤ C 0 M h for some constant C 0 (the constant comes from (4.6)). Using that e z -1 = O(|z|) in a neighborhood, it is easy to see that lim h→0 M h = 0, because λ ∈ (0, 2]. The family {v ε | ε ∈ (0, 1)} is bounded in L ∞ (R), and thus also in L 2 loc (R). By the Fréchet-Kolmogorov theorem, it is relatively compact in L 2 loc (R).

Proof of Lemma 6.1. For all v ∈ H 1 (R * ), there exist the traces v(0

± ) ∈ R; it is not difficult to show that |v(0 ± )| ≤ v H 1 (R * ) .
Further, for all ±x > 0, v(x) = v(0 ± ) + The sequential embedding (6.2) is clear from (6.1). Indeed, Helly's theorem and L q , L p interpolation inequalities imply that the inclusion L ∞ (R) ∩ BV loc (R) ⊂ L p loc (R) is continuous and compact for all p ∈ [1, +∞); since each converging sequence in (BV loc (R)) w-⋆ ∩ H 1 loc (R \ {0}) is (strongly) bounded in BV loc (R) ∩ H 1 loc (R \ {0}), the inclusions

(BV loc (R)) w-⋆ ∩ H 1 loc (R \ {0}) ⊂ L p loc (R) ∩ L 2 loc (R \ {0}) ⊂ L 2 (R)
are sequentially continuous.

Proof of Lemma 6.2. From (A.2), one deduces that if v ∈ H 1 (R * ) then Then, one has m(t) ≤ 1 t for all t > 0.

∂ x v = (∂ x v) | R * +(v(0 + )- v(0 -)) δ 0 ,
Proof. Let t 0 > 0 be such that m(t 0 ) is positive. The function m has to be positive on some neighborhood of t 0 ; since (B.1) implies that m is non-increasing, this neighborhood has to contain the interval (0, t 0 ]. Dividing (B.1) by m 2 = (max{m, 0}) 2 on this interval, we get: -1 m ′ r ≤ -1 in (0, t 0 ). Integrating this inequation, one deduces that for all t < t 0 , 1 m(t) -1 m(t 0 ) ≤ tt 0 , which implies that m(t 0 ) ≤ 1 m(t) + t 0t -1 ≤ (t 0t) -1 . Letting t → 0, we conclude that m(t 0 ) ≤ 1 The idea of the proof of this technical result comes from [START_REF] Alibaud | Fractional semi-linear parabolic equations with unbounded data[END_REF]; we give here a short proof for the reader's convenience.

Proof. In the sequel, C denotes a constant only depending on λ ′ , λ, M Φ and L Φ . For all x ∈ R and r > 0, one has (for some constant C not depending on x ∈ R and r > 0). This proves that L λ [Φ](x) is well-defined by (2.2) for all x ∈ R; moreover, we let the reader check that the continuity of L λ [Φ] can be easily deduced from the dominated convergence theorem. What is left to study is thus the behavior of L λ [Φ] at infinity; to do so, one takes r = |x| 2 (which is positive for large x) and gets from (B.2) the following estimate: |L λ [Φ](x)| ≤ C |x| -λ + |x| λ ′ -λ for large x. The proof is complete.

  Proof. First, by Proposition 4.1 and Lemma 5.1 there exists v ∈ L ∞ (R) and a sequence (ε k ) k , ε k ↓ 0 as k → +∞, such that the solution v ε k of (4.2) with ε = ε k tends to v in L 2loc (R) by being bounded by 1 in L ∞ -norm. Using in particular (4.6) to vanish the term √ ε (∂ x v ε ) | R * , we can pass to the limit in (4.15) and infer (5.1).

x 0 ( 1 ∂

 01 ∂ x v) | R * (y)dy. (A.2) It follows that for all R > 0, one has v ∈ BV ((-R, R)) with |v| BV ((-R,R)) ≤ |v(0 + )v(0 -)| + (∂ x v) | R * L 1 ((-R,R)) ≤ 2 + √ 2R v H 1 (R * ) .This shows that the inclusionH 1 (R * ) ⊂ BV loc (R) ∩ H 1 loc (R \ {0}) is continuous. Now take v ∈ BV loc (R) ∩ H 1 loc (R \ {0}). Then v is continuous on R * and v(x) = v(1) +x x v(y)dy, where ∂ x v can be a Radon measure with singular part supported by {0}. By the continuity of the inclusionH 1 (R \ [-1, 1]) ⊂ C b (R \ (-1, 1)), one deduces that v is bounded outside (-1, 1); since v is bounded by |v(1)| + |v| BV ((-1,1)) on [-1, 1], the inclusion BV loc (R) ∩ H 1 loc (R \ {0}) ⊂ L ∞ (R) is continuous.From this result, it is easy to show (6.1).

1 .

 1 where one has (∂ x v) | R * ∈ L 2 (R) and δ 0 is the Dirac delta at zero. Let (ρ k ) k ⊂ D(R) be an approximate unit and definev k := ρ k * v. Then it is easy to check that v k → v in L 2 (R) and that ∂ x v k = (∂ x v) | R * * ρ k + (v(0 + )v(0 -)) ρ k converges to ∂ x v in L 2 loc (R \ {0}) and in C c (R) Let m ∈ C(R + ) be right-differentiable with m ′ r (t) + (max{m, 0}) 2 ≤ 0 on R + . (B.1)

  t 0 whenever m(t 0 ) is positive. The proof is complete.Lemma B.2. Let λ ∈ (0, 1) and Φ : R → R be locally Lipschitz-continuous and such that there exist 0 < λ ′ < λ, M Φ and L Φ with|Φ(x)| ≤ M Φ (1 + |x| λ ′ ) and |∂ x Φ(x)| ≤ L Φ 1 + |x| 1-λ ′ for a.e. x ∈ R. Then L λ [Φ] is well-defined by (2.2) and belongs to C b (R).

R

  |Φ(x + z) -Φ(x)| |z| 1+λ dz ≤ ∂ x Φ L ∞ ((x-r,x+r)) |z|≤r |z| -λ dz + |z|>r |Φ(x + z) -Φ(x)| |z| 1+λ dz, ≤ C r 1-λ ∂ x Φ L ∞ ((x-r,x+r)) + |z|>r |Φ(x + z) -Φ(x)| |z| 1+λ dz.Since |x + z| λ ′ ≤ |x| λ ′ + |z| λ ′ for all x, z ∈ R, the last integral term is bounded above byC |z|>r 2 + 2|x| λ ′ + |z| λ ′ |z| 1+λ dz ≤ C r -λ 1 + |x| λ ′ + r λ ′ .We get finally:R |Φ(x + z) -Φ(x)| |z| 1+λ dz ≤ C r -λ 1 + |x| λ ′ + r λ ′ + r ∂ x Φ L ∞ ((x-r,x+r)) (B.2)

  Proof of Lemma 3.1. The supremum m(t) is achieved because of (3.2), so that K(t) = ∅; moreover, one has for all b > a > 0 supIt is quite easy to show that m is continuous and we only detail the proof of the derivability from the right. Let t 0 > 0 be fixed and (t k ) k , (x k ) k be such that lim k→+∞ t k = t 0 , t k > t 0 and x k ∈ K(t k ), m(t k ) = v(t k , x k ) for all k ≥ 1. By (A.1), (x k ) k is bounded; hence, taking a subsequence if necessary, one can assume that x k converges toward some x 0 . One has lim sup

	t∈(a,b), x∈K(t)	|x| < +∞.	(A.1)
	k→+∞

Acknowledgement

The first author would like to thank the Departement of Mathematics of Prince of Songkla University (Hat Yai campus) in Thailand, for having ensured a large part of his working facilities.