Ë L Natha

Simone Alibaud

AND Espen R Cifani

Jakobsen

CONTINUOUS DEPENDENCE ESTIMATES FOR NONLINEAR FRACTIONAL CONVECTION-DIFFUSION EQUATIONS

Keywords: 2010 Mathematics Subject Classification. 35R09, 35K65, 35L65, 35D30, 35B30 Fractional/fractal conservation laws, nonlinear parabolic equations, pure jump Lévy processes, continuous dependence estimates

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Continuous dependence estimates for nonlinear fractional convection-diffusion equations

Nathael Alibaud, Simone Cifani, Espen

Introduction

This paper is concerned with the following Cauchy problem:

(1.1) ∂ t u(x, t) + div (f (u)) (x, t) = L µ [A(u(•, t))](x) in Q T := R d × (0, T ), u(x, 0) = u 0 (x), in R d ,
where u is the scalar unknown function, div denotes the divergence with respect to (w.r.t.) x, and the operator L µ is defined for all φ ∈ C ∞ c (R d ) by

L µ [φ](x) := ˆRd \{0} φ(x + z)φ(x)z • Dφ(x)1 |z|≤1 dµ(z), (1.2) where Dφ denotes the gradient of φ w.r.t. x and 1 |z|≤1 = 1 for |z| ≤ 1 and = 0 otherwise. Throughout the paper, the data (f, A, u 0 , µ) is assumed to satisfy the following assumptions: (1) Subtracting constants to f and A if necessary, there is no loss of generality in assuming that f (0) = 0 and A(0) = 0. (2) Our results also hold for locally Lipschitz-continuous nonlinearities f and A since solutions will be bounded; see Remark 2.3 for more details. (3) Assumption (1.6) and a Taylor expansion reveal that L µ [φ] is well-defined for e.g. bounded C 2 functions φ:

f ∈ W 1,∞ (R, R d ) with f (0) = 0, (1.3) A ∈ W 1,∞ (R) is nondecreasing with A(0) = 0, (1.4) u 0 ∈ L ∞ (R d ) ∩ L 1 (R d ) ∩ BV (R d ), (1.
|L µ [φ](x)| ≤ max |z|≤1 |D 2 φ(x + z)| ˆ0<|z|≤1 1 2 |z| 2 dµ(z) + 2 φ L ∞ ˆ|z|>1 dµ(z)
where D 2 φ is the Hessian of φ. If in addition D 2 φ is bounded on R d , then so is

L µ [φ].
Under (1.6), L µ is the generator of a pure jump Lévy process, and reversely, any pure jump Lévy process has a generator of like L µ (see e.g. [START_REF] Applebaum | Lévy processes and Stochastic Calculus[END_REF][START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF]). This class of diffusion processes contains e.g. the α-stable process whose generator is the fractional Laplacian -(-△) α 2 with α ∈ (0, 2). It can be defined for all φ ∈ C ∞ c (R d ) via the Fourier transform as (-△)

α 2 φ = F -1 (| • | α F φ) ,
or in the form (1.2) with the following Lévy measure (see e.g. [START_REF] Applebaum | Lévy processes and Stochastic Calculus[END_REF][START_REF] Droniou | Fractal first order partial differential equations[END_REF][START_REF] Imbert | A non-local regularization of first order Hamilton-Jacobi equations[END_REF]): dµ(z) = dz |z| d+α (up to a positive multiplicative constant). (1.7) Many other Lévy processes/operators of practical interest can be found in e.g. [START_REF] Applebaum | Lévy processes and Stochastic Calculus[END_REF][START_REF] Cont | Financial modelling with jump processes[END_REF]. Under assumption (1.4), L µ [A(•)] is an example of a nonlinear nonlocal diffusion operator. For recent studies of this and similar type of operators, we refer the reader to [START_REF] Biler | Fractal porous medium equation[END_REF][START_REF] Biler | A nonlinear diffusion of dislocation density and selfsimilar solutions[END_REF][START_REF] Caffarelli | Nonlinear porous medium flow with fractional potential pressure[END_REF][START_REF] Cifani | Entropy formulation for degenerate fractional order convectiondiffusion equations[END_REF][START_REF] De Pablo | A fractional porous medium equation[END_REF] and the references therein.

Equation (1.1) appears in many different contexts such as overdriven gas detonations [START_REF] Clavin | Instabilities and nonlinear patterns of overdriven detonations in gases[END_REF], mathematical finance [START_REF] Cont | Financial modelling with jump processes[END_REF], flow in porous media [START_REF] De Pablo | A fractional porous medium equation[END_REF], radiation hydrodynamics [START_REF] Rohde | The nonrelativistic limit in radiation hydrodynamics. I. Weak entropy solutions for a model problem[END_REF][START_REF] Rosenau | Extending hydrodynamics via the regularization of the Chapman-Enskog expansion[END_REF], and anomalous diffusion in semiconductor growth [START_REF] Woyczyński | Lévy processes in the physical sciences[END_REF]. Equations of the form (1.1) constitute a large class of nonlinear degenerate parabolic integrodifferential equations (integro-PDEs). Let us give some representative examples.

When A = 0 or µ = 0, (1.1) is the well-known scalar conservation law (see e.g. [START_REF] Dafermos | Hyperbolic Conservation Laws in Continuum Physics[END_REF] and references therein):

(1.8) ∂ t u + divf (u) = 0.

When A(u) = u and L µ is the fractional Laplacian, (1.1) is the so-called fractal/fractional conservation law:

(1.9)

∂ t u + divf (u) = -(-△) α 2 u.
Equation (1.9) has been extensively studied since the nineties [START_REF] Alibaud | Entropy formulation for fractal conservation laws[END_REF][START_REF] Alibaud | Non-uniqueness of weak solutions for the fractal Burgers equation[END_REF][START_REF] Alibaud | Occurence and non-appearance of shocks in fractal Burgers equation[END_REF][START_REF] Alibaud | Asymptotic properties of entropy solutions to fractal Burgers equation[END_REF][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] Chan | Regularity of solutions for the critical N -dimensional Burgers equation[END_REF][START_REF] Chan | Eventual regularization of the slightly supercritical fractional Burgers equation[END_REF][START_REF] Cifani | The discontinuous Galerkin method for fractal conservation laws[END_REF][START_REF] Cifani | The discontinuous Galerkin method for fractional degenerate convection-diffusion equations[END_REF][START_REF] Dong | Finite time singularities and global well-posedness for fractal Burgers equations[END_REF][START_REF] Droniou | Vanishing nonlocal regularization of a scalar conservation law[END_REF][START_REF] Droniou | A numerical method for fractal conservation laws[END_REF][START_REF] Droniou | Global solution and smoothing effect for a nonlocal regularization of a hyperbolic equation[END_REF][START_REF] Droniou | Fractal first order partial differential equations[END_REF][START_REF] Ivec | A generalization of H-measures and application on purely fractional scalar conservation laws[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] Jourdain | Convergence of stochastic particle approximation for fractional Scalar conservation laws[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] 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]. The case of more general Lévy diffusions, combined with nonlinear local diffusions, (1.10)

∂ t u + divf (u) = div(a(u)∇u) + L µ [u],
can be found in [START_REF] Karlsen | Stability of entropy solutions for Lévy mixed hyperbolic parabolic equations[END_REF].

When A is nonlinear, (1.1) can be seen as a generalization of the following classical convection-diffusion equation (possibly degenerate):

(1.11) ∂ t u + divf (u) = △A(u); see e.g. [START_REF] Bouchut | Kružkov's estimates for scalar conservation laws revisited[END_REF][START_REF] Carrillo | Entropy Solutions for nonlinear Degenerate Problems[END_REF][START_REF] Chen | L 1 -framework for continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations[END_REF][START_REF] Cockburn | Continuous Dependence on the nonlinearities of Solutions of Degenerate Parabolic Equations[END_REF][START_REF] Karlsen | On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients[END_REF] for precise references on (1.11). Nonlinear nonlocal diffusions have been invetigated in [START_REF] De Pablo | A fractional porous medium equation[END_REF] in the setting of nonlocal porous media equations and L 1 semi-group methods, and in [START_REF] Cifani | Entropy formulation for degenerate fractional order convectiondiffusion equations[END_REF] where an L ∞ ∩ L 1 entropy solution theory is developed for more general degenerate equations of the form (1.1) along with connections to Hamilton-Jacobi-Bellman equations of stochastic control theory. Other interesting examples concern the class of nonsingular Lévy measures satisfying ´Rd \{0} dµ(z) < +∞. In that case, (1.1) can also be seen as a generalization of Rosenau's models [START_REF] Kawashima | Weak solutions with a shock to a model system of the radiating gas[END_REF][START_REF] Kawashima | Shock waves for a model system of a radiating gas[END_REF][START_REF] Lattanzio | Global well-posedness and relaxation limits of a model for radiating gas[END_REF][START_REF] Liu | Critical thresholds in a convolution model for nonlinear conservation laws[END_REF][START_REF] Schochet | Regularized Chapman-Enskog expansion for scalar conservation laws[END_REF][START_REF] Serre | L 1 -Stability of Constants in a Model for Radiating Gases[END_REF] and nonlinear radiation hydrodynamics models [START_REF] Rohde | The nonrelativistic limit in radiation hydrodynamics. I. Weak entropy solutions for a model problem[END_REF] of the form ∂ t u + divf (u) = g * A(u) -A(u), (1.12) where * denotes the convolution product w.r.t. x and g ∈ L 1 (R d ) is nonnegative with ´Rd g(z) dz = 1.

Most of the results on these nonlocal equations concern Equation (1.9) with the α-stable linear diffusions, and convolution models (1.12) with nonlinear but nonsingular Lévy diffusions. It is known that shocks can occur in finite time [START_REF] Alibaud | Occurence and non-appearance of shocks in fractal Burgers equation[END_REF][START_REF] Dong | Finite time singularities and global well-posedness for fractal Burgers equations[END_REF][START_REF] Kawashima | Weak solutions with a shock to a model system of the radiating gas[END_REF][START_REF] Kawashima | Shock waves for a model system of a radiating gas[END_REF][START_REF] Kiselev | Blow up and regularity for fractal Burgers equation[END_REF][START_REF] Liu | Critical thresholds in a convolution model for nonlinear conservation laws[END_REF][START_REF] Schochet | Regularized Chapman-Enskog expansion for scalar conservation laws[END_REF], that weak solutions can be nonunique [START_REF] Alibaud | Non-uniqueness of weak solutions for the fractal Burgers equation[END_REF], and that the Cauchy problem is well-posed with the notion of entropy solutions in the sense of Kruzhkov [START_REF] Alibaud | Entropy formulation for fractal conservation laws[END_REF][START_REF] Lattanzio | Global well-posedness and relaxation limits of a model for radiating gas[END_REF][START_REF] Rohde | The nonrelativistic limit in radiation hydrodynamics. I. Weak entropy solutions for a model problem[END_REF]; see also the works [START_REF] Cockburn | On convergence of entropy solutions to a single conservation law[END_REF][START_REF] Jakubowski | On a nonlinear elliptic/parabolic integro-differential equation with L 1 -data[END_REF] for the related topic of time fractional derivatives. The entropy solution theory has been generalized in [START_REF] Karlsen | Stability of entropy solutions for Lévy mixed hyperbolic parabolic equations[END_REF] to singular but linear Lévy diffusions along with nonlinear local diffusions. Very recently, it has been extended in [START_REF] Cifani | Entropy formulation for degenerate fractional order convectiondiffusion equations[END_REF] to cover the full problem (1.1) for general singular Lévy measures and nonlinear A.

The purpose of the present paper is to develop an abstract framework for finding error estimates for entropy solutions of (1.1). As applications, we focus in this paper on continuous dependence estimates and convergence rates for vanishing viscosity approximations. We refer the reader to [START_REF] Bouchut | Kružkov's estimates for scalar conservation laws revisited[END_REF][START_REF] Chen | L 1 -framework for continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations[END_REF][START_REF] Cockburn | Continuous Dependence on the nonlinearities of Solutions of Degenerate Parabolic Equations[END_REF][START_REF] Karlsen | On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients[END_REF][START_REF] Kuznetsov | Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation[END_REF] and the references therein for similar analysis on (1.11) and related local equations. As far as nonlocal equations are concerned, continuous dependence estimates for fully nonlinear integro-PDEs have already been derived in [START_REF] Jakobsen | Continuous dependence estimates for viscosity solutions of integro-PDEs[END_REF] in the context of viscosity solutions of Bellman-Isaacs equations; see also [START_REF] Droniou | Fractal first order partial differential equations[END_REF][START_REF] Imbert | A non-local regularization of first order Hamilton-Jacobi equations[END_REF][START_REF] Jakobsen | Continuous dependence estimates for viscosity solutions of integro-PDEs[END_REF] for error estimates on nonlocal vanishing viscosity approximations.

To the best of our knowledge, the first and up to now only continuous dependence estimate for nonlocal conservation laws can be found in [START_REF] Karlsen | Stability of entropy solutions for Lévy mixed hyperbolic parabolic equations[END_REF]; see also [START_REF] Alibaud | Entropy formulation for fractal conservation laws[END_REF][START_REF] Cockburn | On convergence of entropy solutions to a single conservation law[END_REF][START_REF] Droniou | Vanishing nonlocal regularization of a scalar conservation law[END_REF][START_REF] Droniou | Fractal first order partial differential equations[END_REF][START_REF] Lattanzio | Global well-posedness and relaxation limits of a model for radiating gas[END_REF][START_REF] Schochet | Regularized Chapman-Enskog expansion for scalar conservation laws[END_REF] for convergence rates for vanishing viscosity approximations of Equations (1.9) and (1.12). The general estimate in [START_REF] Karlsen | Stability of entropy solutions for Lévy mixed hyperbolic parabolic equations[END_REF] is established for Equation (1.10) for linear symmetric Lévy diffusions. Inspired by an early version of the present paper, a formal discussion on possible extensions to nonlinear nonlocal diffusions is also given. On the technical side, [START_REF] Karlsen | Stability of entropy solutions for Lévy mixed hyperbolic parabolic equations[END_REF] employs so-called entropy defect measures while we do not.

To finish with the bibliography, let us also refer the reader to [START_REF] Cifani | On numerical methods for degenerate fractional convectiondiffusion equations[END_REF][START_REF] Cifani | The discontinuous Galerkin method for fractal conservation laws[END_REF][START_REF] Cifani | The discontinuous Galerkin method for fractional degenerate convection-diffusion equations[END_REF][START_REF] Dedner | Numerical approximation of entropy solutions for hyperbolic integro-differential equations[END_REF][START_REF] Droniou | A numerical method for fractal conservation laws[END_REF][START_REF] Rohde | The nonrelativistic limit in radiation hydrodynamics. I. Weak entropy solutions for a model problem[END_REF] for the related topic of error estimates for numerical approximations.

Our main result is stated in Lemma 3.1, and it compares the entropy solution u of (1.1) with a general function v. Our main application consists in comparing u with the entropy solution v of (1.13)

∂ t v + divg(v) = L ν [B(v)], v(x, 0) = v 0 ,
where the data set (g, B, v 0 , ν) is assumed to satisfy (1.3)- (1.6). We obtain explicit continuous dependence estimates on the data stated in Theorems 3.3-3.4. Let us recall that when B = 0 or ν = 0, (1.13) is the pure scalar conservation law in (1.8). Equation (1.1) can thus be seen as a nonlinear nonlocal vanishing viscosity approximation of (1.8) if A or µ vanishes. The rate of convergence is then obtained as a consequence of Theorems 3.3-3.4, see Theorem 3.9.

It is natural to compare Theorems 3.3-3.4 and Theorem 3.9 with the known error estimates for the different equations above. One can see that a quite important part of them are particular cases of our general results. We discuss this point in Section 3 by giving precise examples. Let us mention that we also give an example of a simple Hamilton-Jacobi equation where we show that Theorems 3.3-3.4 are in some sense the "conservation law version" of the results in [START_REF] Jakobsen | Continuous dependence estimates for viscosity solutions of integro-PDEs[END_REF]; see Example 3.2.

To finish, let us mention that in the case of fractional Laplacians of order α ≥ 1, Theorems 3.3-3.4 can be improved by taking advantage of the homogeneity of the measures in (1.7). In order not to make this paper too long, this special case (including α < 1) is investigated in a second paper [START_REF] Alibaud | Optimal continuous dependence estimates for fractional degenerate parabolic equations[END_REF].

The rest of this paper is organized as follows. In Section 2 we recall the notion of entropy solution to (1.1). In Section 3, we state and discuss our main results. Sections 4-5 are devoted to the proofs of our main results; Section 4 states some preliminary results on the nonlocal operator. The symbol supp is used for the support. The superscripts ± are used for the positive and negative parts. The total variation of a Radon measure µ is denoted by |µ|. Its tensor product with the Lebesgue measure dw is denoted by dµ(z) dw.

Entropy formulation and well-posedness

Let us recall the formal computations leading to the entropy formulation of (1.1). First we split L µ into 3 parts:

(2.1) L µ [φ](x) = L µ r [φ](x) + div (b µ r φ) (x) + L µ,r [φ](x) for φ ∈ C ∞ c (R d ),
q f (u, k) := sgn (u -k) (f (u) -f (k)) ∈ R d ,
where we always use the following everywhere representative of the sign function:

(2.6) sgn (u) := ±1 if ±u > 0, 0 if u = 0.
By (1.4) it is readily seen that for all u, k ∈ R,

(2.7) sgn (u -k) (A(u) -A(k)) = |A(u) -A(k)|,
and we formally deduce from (2.1), (2.7), and the nonnegativity of µ that

sgn (u -k) L µ [A(u)] ≤ L µ r [|A(u) -A(k)|] + div (b µ r |A(u) -A(k)|) + sgn (u -k) L µ,r [A(u)].
Let u be a solution of (1.1), and multiply (1.1) by sgn (uk). Formal computations then reveal that

∂ t |u -k| + div (q f (u, k) -b µ r |A(u) -A(k)|) ≤ L µ r [|A(u) -A(k)|] + sgn (u -k) L µ,r [A(u)
]. The entropy formulation in Definition 2.1 below consists in asking that u satisfies this inequality for all entropy-flux pairs (i.e. for all k ∈ R) and all r > 0. Roughly speaking one can give a sense to sgn (uk) L µ,r [A(u)] for bounded discontinuous u thanks to (1.6). But since µ may be singular at z = 0, see Remark 1.1 (3), the other terms have to be interpreted in the sense of distributions: Multiply by test functions φ and integrate by parts to move singular operators onto test functions. For the nonlocal terms this can be done by change of variables: First take (z, x, t) → (-z, x, t) to see (formally) that

ˆQT φ div (b µ r |A(u) -A(k)|) dxdt = ˆQT Dφ • b µ * r |A(u) -A(k)| dxdt,
where µ * is the Lévy measure (i.e. it satisfies (1.6)) defined for all

φ ∈ C ∞ c (R d \{0}) by (2.8) ˆRd \{0} φ(z) dµ * (z) := ˆRd \{0} φ(-z) dµ(z).
In view of (2.2), we can take (z, x, t) → (-z, x + z, t) to find that

ˆQT φ L µ r [|A(u) -A(k)|] dxdt = ˆQT |A(u) -A(k)| L µ * r [φ] dxdt.
This leads to the following definition introduced in [START_REF] Cifani | Entropy formulation for degenerate fractional order convectiondiffusion equations[END_REF].

Definition 2.1. (Entropy solutions) Assume (1.3)-(1.6). We say that a func- tion u ∈ L ∞ (Q T ) ∩ C [0, T ]; L 1 is an entropy solution of (1.1) provided that for all k ∈ R, all r > 0, and all nonnegative φ ∈ C ∞ c (R d+1 ), (2.9)
ˆQT |u -k| ∂ t φ + q f (u, k) + b µ * r |A(u) -A(k)| • Dφ dxdt + ˆQT |A(u) -A(k)| L µ * r [φ] + sgn (u -k) L µ,r [A(u)] φ dx dt - ˆRd |u(x, T ) -k| φ(x, T ) dx + ˆRd |u 0 (x) -k| φ(x, 0) dx ≥ 0.
Remark 2.1.

(1) Under assumptions (1.3)-(1.6), the entropy inequality (2.9) is well-defined independently of the a.e. representative of u. To see this, note that since µ * satisfies (1.6), it easily follows that

L µ * r [φ] ∈ C ∞ c (R d+1
). Since sgn (uk), q f (u, k), and A(u) belong to L ∞ by (2.6) and (1.3)-(1.4), it is then clear that all terms in (2.9) are well-defined except possibly the L µ,r -term.

Here it may look like we are integrating Lebesgue measurable functions w.r.t. a Radon measure µ. However, the integrand does have the right measurability by a classical approximation procedure, see Remark 5.1 in [START_REF] Cifani | Entropy formulation for degenerate fractional order convectiondiffusion equations[END_REF]. We therefore find that since A(u) belongs to C([0, T ]; L 1 ), so does also L µ,r [A(u)] and we are done.

(2) Another way to understand the measurability issue in [START_REF] Alibaud | Entropy formulation for fractal conservation laws[END_REF], is simply to consider only Borel measurable a.e. representatives of the solutions. The reading of the paper would remain exactly the same, since our L 1 -continuous dependence estimate do not depend on the representatives. (3) In the definition of entropy solutions, it is possible to consider functions u only defined for a.e. t ∈ [0, T ] by taking test functions with compact support in Q T and adding an explicit initial condition, see e.g. [START_REF] Cifani | Entropy formulation for degenerate fractional order convectiondiffusion equations[END_REF]. (4) One can check that classical solutions are entropy solutions, thus justifying the formal computations leading to Definition 2.1. Moreover entropy solution are weak solutions and hence smooth entropy solutions are classical solutions. We refer the reader to [START_REF] Cifani | Entropy formulation for degenerate fractional order convectiondiffusion equations[END_REF] for the proofs.

Here is a well-posedness result from [START_REF] Cifani | Entropy formulation for degenerate fractional order convectiondiffusion equations[END_REF].

Theorem 2.2. (Well-posedness) Assume (1.3)-(1.6). There exists a unique entropy solution u of (1.1). This entropy solution belongs to

L ∞ (Q T )∩C [0, T ]; L 1 ∩ L ∞ (0, T ; BV ) and
(2.10)

     u L ∞ (QT ) ≤ u 0 L ∞ (R d ) , u C([0,T ];L 1 ) ≤ u 0 L 1 (R d ) , |u| L ∞ (0,T ;BV ) ≤ |u 0 | BV (R d ) .
Moreover, if v is the entropy solution of (1.1) with v(0) = v 0 for another initial data v 0 satisfying (1.5), then

(2.11) u -v C([0,T ];L 1 ) ≤ u 0 -v 0 L 1 (R d ) .
Remark 2.3. By the L ∞ -estimate in (2.10), all the results of this paper also holds for locally Lipschitz-continuous nonlinearities (f, A). Simply replace the data (f,

A) by (f, A) ψ M , where ψ M ∈ C ∞ c (R) is such that ψ M = 1 in [-M, M ] for M = u 0 L ∞ (R d ) .

Main results

Our first main result is a Kuznetsov type of lemma that measures the distance between the entropy solution u of (1.1) and an arbitrary function v.

Let ǫ, δ > 0 and

φ ǫ,δ ∈ C ∞ (Q 2 T ) be the test function φ ǫ,δ (x, t, y, s) := θ δ (t -s) θǫ (x -y), (3.1
and v ∈ L ∞ (Q T )∩C [0, T ]; L 1 with v(0) = v 0 .
Then for all r > 0, ǫ > 0, and 0 < δ < T ,

u(T ) -v(T ) L 1 (R d ) ≤ u 0 -v 0 L 1 (R d ) + ǫ C θ |u 0 | BV (R d ) + 2(ω u (δ) ∨ ω v (δ)) - ¨Q2 T |v(x, t) -u(y, s)| ∂ t φ ǫ,δ (x, t, y, s) dw - ¨Q2 T q f (v(x, t), u(y, s)) + b µ * r |A(v(x, t)) -A(u(y, s))| • D x φ ǫ,δ (x, t, y, s) dw + ¨Q2 T |A(v(x, t)) -A(u(y, s))| L µ * r [φ ǫ,δ (x, t, •, s)](y) dw - ¨Q2 T sgn (v(x, t) -u(y, s)) L µ,r [A(u(•, s))](y) φ ǫ,δ (x, t, y, s) dw + ¨Rd ×QT |v(x, T ) -u(y, s)| φ ǫ,δ (x, T, y, s) dx dy ds - ¨Rd ×QT |v 0 (x) -u(y, s)| φ ǫ,δ (x, 0, y, s) dx dy ds (3.3)
where dw := dx dt dy ds, and C θ := 2 ´Rd |x| θd (x) dx.

Remark 3.2.

(1) The error in time only depends on the moduli of continuity of u and v at t = 0 and t = T . Here we simply take the global-in-time moduli of continuity ω u (δ) and ω v (δ), since this is sufficient in our settings. (2) When A = 0 or µ = 0 this lemma reduces to the well-known Kuznetsov lemma [START_REF] Kuznetsov | Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation[END_REF] for multidimensional scalar conservation laws. (3) Notice that the L µ * r -term vanishes when r → 0, see Lemma 4.5. (4) Lemma 3.1 has many applications. In this paper and in [START_REF] Alibaud | Optimal continuous dependence estimates for fractional degenerate parabolic equations[END_REF] we focus on continuous dependence results and error estimates for the vanishing viscosity method. Then in [START_REF] Cifani | On numerical methods for degenerate fractional convectiondiffusion equations[END_REF], we will use the lemma to obtain error estimates for numerical approximations of (1.1).

In this paper we apply Lemma 3.1 to compare the entropy solution u of (1.1) with the entropy solution v of (1.13). This is our second main result, and we present it in the two theorems below. The first focuses on the dependence on the nonlinearities (with µ = ν) and the second one on the Lévy measure (with A = B).

Theorem 3.3. (Continuous dependence on the nonlinearities) Let u and v be the entropy solutions of (1.1) and (1.13) respectively with data sets (f, A, u 0 , µ) and

(g, B, v 0 , ν = µ) satisfying (1.3)-(1.6). Then for all T, r > 0, u -v C([0,T ];L 1 ) ≤ u 0 -v 0 L 1 (R d ) + |u 0 | BV (R d ) T f ′ -g ′ L ∞ (R,R d ) + |u 0 | BV (R d ) c d T ˆ0<|z|≤r |z| 2 dµ(z) A ′ -B ′ L ∞ (R) + |u 0 | BV (R d ) T ˆr∧1<|z|≤r∨1 z dµ(z) A ′ -B ′ L ∞ (R) + T ˆ|z|>r u 0 (• + z) -u 0 L 1 (R d ) dµ(z) A ′ -B ′ L ∞ (R) , (3.4)
where c d = 4d 2 d+1 . Theorem 3.4. (Continuous dependence on the Lévy measure) Let u and v be the entropy solutions of (1.1) and (1.13) respectively with data sets (f, A, u 0 , µ) and (g, B = A, v 0 , ν) satisfying (1.3)- (1.6). Then for all T, r > 0,

u -v C([0,T ];L 1 ) ≤ u 0 -v 0 L 1 (R d ) + |u 0 | BV (R d ) T f ′ -g ′ L ∞ (R,R d ) + |u 0 | BV (R d ) c d T A ′ L ∞ (R) ˆ0<|z|≤r |z| 2 d|µ -ν|(z) + |u 0 | BV (R d ) T A ′ L ∞ (R) ˆr∧1<|z|≤r∨1 z d(µ -ν)(z) + T A ′ L ∞ (R) ˆ|z|>r u 0 (• + z) -u 0 L 1 (R d ) d|µ -ν|(z), (3.5)
where c d = 4d 2 d+1 . Remark 3.5. In the error estimates of Theorems 3.3 and 3.4, there are 3 terms accounting for the dependence on the fractional diffusion term in (1.1): One term accounts for the behavior near the singularity of µ at z = 0 (the integral over 0 < |z| ≤ r), another term accounts for the behavior near infinity (the integral over |z| ≥ r), and the last term (the integral over r ∧ 1 < |z| ≤ r ∨ 1) is a drift term that is only present for nonsymmetric measures µ. The square root estimate for the singular term is similar to estimates for 2nd derivative terms in the local case and for non-local equations with different structure, cf. e.g. [START_REF] Chen | L 1 -framework for continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations[END_REF][START_REF] Jakobsen | Continuous dependence estimates for viscosity solutions of integro-PDEs[END_REF][START_REF] Karlsen | Stability of entropy solutions for Lévy mixed hyperbolic parabolic equations[END_REF]. Remark 3.6. Since the initial data is L 1 ∩ BV , an application of Fubini's theorem shows that for any r > r > 0,

ˆ|z|>r u 0 (• + z) -u 0 L 1 (R d ) dµ(z) ≤ |u 0 | BV (R d ) ˆr<|z|≤r |z| dµ(z) + 2 u 0 L 1 (R d )
ˆ|z|>r dµ(z).

From Theorems 3.3 and 3.4 we can easily find a general continuous dependence estimate when both A and µ are different from B and ν, respectively. E.g. we can take an intermediate solution w of w t + div f (w) = L µ [B(w)] and w(0) = u 0 , and use the triangle inequality. Using this idea we can show that the following estimates always have to hold: Corollary 3.7. Let u and v be the entropy solutions of (1.1) and (1.13) respectively with data sets (f, A, u 0 , µ) and (g, B, v 0 , ν) satisfying

(1.3)-(1.6). Then for all T > 0 u -v C([0,T ];L 1 ) ≤ u 0 -v 0 L 1 (R d ) + |u 0 | BV (R d ) T f ′ -g ′ L ∞ (R,R d ) + C (T 1 2 ∨ T ) A ′ -B ′ L ∞ (R) + ˆRd \{0} (|z| 2 ∧ 1) d|µ -ν|(z) (3.6)
where C only depends on d and the data. Moreover, if in addition (1) All these estimates hold for arbitrary Lévy measures µ, ν and even for strongly degenerate diffusions where A, B may vanish on large sets. They are consistent (at least for the |µ-ν| term) with general results for nonlocal Hamilton-Jacobi-Bellman equations in [START_REF] Jakobsen | Continuous dependence estimates for viscosity solutions of integro-PDEs[END_REF]. When µ, ν have the special form (1.7) (with possibly different α's), then it is possible to use the extra symmetry and homogeneity properties to obtain better estimates, see [START_REF] Alibaud | Optimal continuous dependence estimates for fractional degenerate parabolic equations[END_REF]. (2) The optimal choice of the r, r in Remark 3.6 depends on the behavior of the Lévy measures at zero and infinity, see the discussion above and at the end of this section for more details.

ˆRd \{0} (|z| ∧ 1) dµ(z) + ˆRd \{0} (|z| ∧ 1) dν(z) < +∞, then we have the better estimate u -v C([0,T ];L 1 ) ≤ u 0 -v 0 L 1 (R d ) + |u 0 | BV (R d ) T f ′ -g ′ L ∞ (R,R d ) + CT A ′ -B ′ L ∞ (R) + ˆRd \{0} (|z| ∧ 1) d|µ -ν|(z) , (3.
Let us now consider the nonlocal vanishing viscosity problem

(3.8) ∂ t u ǫ + divf (u ǫ ) = ǫ L µ [A(u ǫ )], u ǫ (0) = u 0 ,
i.e. problem (1.8) with a perturbation term ǫ L µ [A(u ǫ )]. When ǫ > 0 tend to zero, u ǫ is expected to converge toward the solution u of (1.8). As an immediate application of Theorem 3.3 or 3.4, we have the following result: Theorem 3.9 (Vanishing viscosity). Assume (1.3)-(1.6). Let u and u ǫ be the entropy solutions of (1.8) and (3.8) respectively. Then for every T, ǫ > 0 and all r > r > 0,

u -u ǫ C([0,T ];L 1 ) ≤ C min r>r>0 d 1 2 T 1 2 ǫ 1 2 ˆ0<|z|≤r |z| 2 dµ(z) + T ǫ ˆr<|z|≤r |z| dµ(z) + ˆr∧1<|z|≤r∨1 z dµ(z) + ˆ|z|>r dµ(z) , (3.9)
where C only depends on

u 0 L 1 (R d )∩BV (R d ) and A ′ L ∞ (R) .
Outline of proof. Note that u can be seen as the entropy solution of (1.1) with A = 0 and µ as Lévy measure. Hence we can estimate u-u ǫ C([0,T ];L 1 ) from Theorem 3.3. The error coming from the difference of the derivatives of the nonlinearities is equal to ǫ A ′ L ∞ (R) . Inequality (3.9) then follows from (3.4) and Remark 3.6.

Corollary 3.10. Assume (1.3)-(1.6). Let u and u ǫ be the entropy solutions of (1.8) and (3.8) respectively. Then for all T, ǫ > 0

u -u ǫ C([0,T ];L 1 ) ≤ C (T 1 2 ∨ T ) ǫ 1 2 ,
where C only depends on d and the data. Moreover, if in addition

(3.10) ˆRd \{0} (|z| ∧ 1) dµ(z) < +∞,
then we have the better estimate

u -u ǫ C([0,T ];L 1 ) ≤ CT ǫ,
where C depends on the data.

This corollary follows immediately from Theorem 3.9 or Corollary 3.7.

Remark 3.11.

(1) Our estimates are just as good or better than the standard O(ǫ

2 ) estimate for the classical vanishing viscosity method ((1.11) with A(u) = ǫ u).

(2) Our estimates hold for arbitrary Lévy measures µ and even for strongly degenerate diffusions where A may vanish on a large set! This is consistent with general results for nonlocal Hamilton-Jacobi-Bellman equations [START_REF] Jakobsen | Continuous dependence estimates for viscosity solutions of integro-PDEs[END_REF]. (3) As for the classical (local) vanishing viscosity method, better rates could be obtained if the solutions are more regular. E.g. if A(u ǫ ) is uniformly (in ǫ) bounded in W 2,1 , then the error estimate should be O(ǫ) even without assumption (3.10). Such a result can not be derived from (3.9), but should be proved directly. (4) Corollary 3.10 contains less information than Theorem 3.9; indeed, if µ is as in (1.7), the additional symmetry and homogeneity can be used to obtain better estimates which can be proved to be optimal. See Example 3.3 below. (5) The error estimates above trivially also holds for the more general vanishing viscosity equation

∂ t u ǫ + divf (u ǫ ) = L ν [B(u ǫ )] + ǫ L µ [A(u ǫ )], u ǫ (0) = u 0 .
Further discussion. We now make a more precise comparison of the results above with known estimates from the literature. We begin with continuous dependence estimates and finish with convergence rates for vanishing viscosity approximations. Let u and v denote the entropy solutions of (1.1) and (1.13), respectively. To simplify, we take the same data sets (f, A, u 0 ) = (g, B, v 0 ) and we only allow the Lévy measures µ and ν to be different. We also let C denote a constant only depending on T, d and the data.

´Rd \{0} (|z| 2 ∧ |z|) dµ(z) < +∞, µ = µ * .
For such kind of equations, the following continuous dependence estimate on the Lévy measure has been established in [START_REF] Karlsen | Stability of entropy solutions for Lévy mixed hyperbolic parabolic equations[END_REF]:

u -v C([0,T ];L 1 ) ≤ C ˆ0<|z|≤1 |z| 2 d|µ -ν|(z) + C ˆ|z|>1 |z| d|µ -ν|(z).
This estimate follows from Theorem 3.4 and Remark 3.6 by taking r = 1 and r = +∞ in (3.5).

Example 3.2. Consider the following one-dimensional Hamilton-Jacobi equation

U t + f (U x ) = L µ [U ]
with initial data U 0 (x) := ´x -∞ u 0 (y) dy. This particular equation is related to the nonlocal conservation law (1.8), since its solution is U (x, t) = ´x -∞ u(y, t) dy where u solves (1.8), see [START_REF] Cifani | Entropy formulation for degenerate fractional order convectiondiffusion equations[END_REF]. It is also an example of an integro-PDE for which the general theory of [START_REF] Jakobsen | Continuous dependence estimates for viscosity solutions of integro-PDEs[END_REF] applies, and this theory allows us to establish the following continuous dependence estimate on the Lévy measure:

sup

R×[0,T ] |U -V | ≤ C ˆR\{0} (|z| 2 ∧ 1) d|µ -ν|(z),
where V (x, t) := ´x -∞ v(y, t) dy. (This result is a version of Theorem 4.1 in [START_REF] Jakobsen | Continuous dependence estimates for viscosity solutions of integro-PDEs[END_REF] which follows from Theorem 3.1 by setting p 0 , . . . , p 4 , p s = 0 and ρ = |z| ∧ 1 in (A0)). Since sup

R×[0,T ] |U -V | ≤ u -v C([0,T ];L 1 ) ,
this estimate also follows from (3.6) in Corollary 3.7 when (A, f, u 0 ) = (B, g, v 0 ).

Let us now compare Theorem 3.9 with known convergence rates. We keep the same notation for u and u ǫ as in Theorem 3.9.

Example 3.3. Let us consider the case where A(u ǫ ) = u ǫ and L µ = -(-△) α 2 , α ∈ (0, 2). Then the following optimal rates have been derived in [START_REF] Alibaud | Entropy formulation for fractal conservation laws[END_REF][START_REF] Droniou | Vanishing nonlocal regularization of a scalar conservation law[END_REF]:

(3.11) u -u ǫ C([0,T ];L 1 ) =        O ǫ 1 α if α > 1, O (ǫ | ln ǫ|) if α = 1, O (ǫ) if α < 1.
Let us explain how these results can be deduced from (3.9). First we use (1.7) to explicitly compute the integrals in (3.9) and obtain

u -u ǫ C([0,T ];L 1 ) = O min r>r>0 ǫ r 2-α 2 -α + ǫ ˆr r dτ τ α + ǫ r-α .
We then deduce (3.11) by taking r = ǫ

∂ t u ǫ + divf (u ǫ ) = 1 ǫ (g ǫ * u ǫ -u ǫ ) ,
where g ǫ (z) := 1 ǫ d g z ǫ with an even and nonnegative kernel g

∈ L 1 (R d ) such that ˆRd |z| 2 g(z) dz < +∞.
This is the Rosenau's regularization of the Chapman-Enskog expansion for hydrodynamics [START_REF] Rosenau | Extending hydrodynamics via the regularization of the Chapman-Enskog expansion[END_REF]; see also Equations (1.1) and (2.3) of [START_REF] Schochet | Regularized Chapman-Enskog expansion for scalar conservation laws[END_REF]. Its convergence toward (1.8) has been established in [START_REF] Lattanzio | Global well-posedness and relaxation limits of a model for radiating gas[END_REF][START_REF] Schochet | Regularized Chapman-Enskog expansion for scalar conservation laws[END_REF]. In Corollary 5.2 of [START_REF] Schochet | Regularized Chapman-Enskog expansion for scalar conservation laws[END_REF] the following rate of convergence has been derived:

u -u ǫ C([0,T ];L 1 ) = O ǫ 1 2
.

This result can be recovered from Theorems 3.3 or 3.4. Indeed, we can choose e.g. A(u ǫ ) = u ǫ , dµ(z) = gǫ(z) ǫ dz and ν = 0, to get the desired equations. Next, we apply (3.5) with r = +∞ and rescale the z-variable to show that the error term is bounded above by

C ˆRd \{0} |z| 2 d|µ -ν|(z) = C ˆRd |z| 2 g z ǫ ǫ d+1 dz = C ǫ 1 2
ˆRd |z| 2 g(z) dz.

Auxiliary results

Before proving our main results in the next section, we state several technical lemmas.

Lemma 4.1. Assume (1.6) and r > 0. Then for all φ ∈ C ∞ c (R d ),

L µ r [φ] L 1 (R d ) ≤ ˆ0<|z|≤r |z| 2 dµ(z) φ W 2,1 (R d ) .
The proof easily follows from a Taylor expansion and Fubini's theorem.

Remark 4.2. Similarly L µ,r is (linear and) bounded from L 1 into itself, thanks to Remark 3.6 with r = r.

In the next result, we establish a Kato type inequality for L µ,r [A(u)].

≤ φ ∈ C ∞ c (R d ), ˆRd sgn (u -k) L µ,r [A(u)] φ dx ≤ ˆRd |A(u) -A(k)| L µ * ,r [φ] dx.
Proof. Note first that A(u) is L 1 by (1.4), and hence L µ,r [A(u)] is well-defined in L 1 by Remark 4.2. Easy computations then reveal that

ˆRd sgn (u -k) L µ,r [A(u)] φ dx, = ˆRd ˆ|z|>r sgn (u(x) -k) A(u(x + z)) -A(u(x)) φ(x) dµ(z) dx, = ˆRd ˆ|z|>r sgn (u(x) -k) (A(u(x + z)) -A(k)) -(A(u(x)) -A(k)) φ(x) dµ(z) dx, ≤ ˆRd ˆ|z|>r |A(u(x + z)) -A(k)| -|A(u(x)) -A(k)| φ(x) dµ(z) dx by (2.7), = ˆRd ˆ|z|>r |A(u(x + z)) -A(k)| φ(x) dµ(z) dx =:I - ˆRd ˆ|z|>r |A(u(x)) -A(k)| φ(x) dµ(z) dx =:J .
Note that all these integrals are well-defined, thanks to (1.6) ( 1 ). By the respective changes of variable (z, x) → (-z, x + z) and (z, x) → (-z, x), we find that

I = ˆRd ˆ|z|>r φ(x + z) |A(u(x)) -A(k)| dµ * (z) dx, J = ˆRd ˆ|z|>r φ(x) |A(u(x)) -A(k)| dµ * (z) dx.
Here the measure µ * in (2.8) appears because of the relabelling of z. This measure has the same properties as µ. Hence we can conclude that

ˆRd sgn (u -k) L µ,r [A(u)] φ dx ≤ I -J = ˆRd |A(u) -A(k)| L µ * ,r [φ] dx,
and the proposition follows.

The next lemma is a consequence of the Kato inequality, and it plays a key role in the doubling of variables arguments throughout this paper and in the uniqueness proof of [START_REF] Alibaud | Entropy formulation for fractal conservation laws[END_REF][START_REF] Cifani | Entropy formulation for degenerate fractional order convectiondiffusion equations[END_REF].

, v ∈ L ∞ (Q T ) ∩ C([0, T ]; L 1 ), 0 ≤ ψ ∈ L 1 (R d × (0, T ) 2 ), and r > 0. Then ¨Q2 T sgn (u(y, s) -v(x, t)) • L µ,r [A(u(•, s))](y) -L µ,r [A(v(•, t))](x) ψ(x -y, t, s) dw ≤ 0
(where dw = dx dt dy ds). 1 The measurability is immediate if the reader only consider Borel measurable representatives of u as suggested in Remark 2.1 (2).

Proof. Note that sgn (u(y, s)v(x, t)) A(u(y + z, s)) -A(u(y, s))

-sgn (u(y, s) -v(x, t)) A(v(x + z, t)) -A(v(x, t)) = sgn (u(y, s) -v(x, t)) • A(u(y + z, s)) -A(v(x + z, t)) -A(u(y, s)) -A(v(x, t)) ≤ |A(u(y + z, s)) -A(v(x + z, t))| -|A(u(y, s)) -A(v(x, t))|
where these functions are both defined. By an integration w.r.t. 1 |z|>r dµ(z), we find that for all (t, s) ∈ (0, T ) 2 and a.e. (x, y) ∈ R 2d , sgn (u(y, s)

-v(x, t)) L µ,r [A(u(•, s))](y) -L µ,r [A(v(•, t))](x) ≤ ˆ|z|>r (|A(u(y + z, s)) -A(v(x + z, t))| -|A(u(y, s)) -A(v(x, t))|) dµ(z).
After another integration, this time w.r.t. ψ(xy, t, s) dw, we then get that

¨Q2 T sgn (u(y, s) -v(x, t)) L µ,r [A(u(•, s))](y) -L µ,r [A(v(•, t))](x) ψ dw ≤ ¨Q2 T ˆ|z|>r |A(u(y + z, s)) -A(v(x + z, t))| ψ(x -y, t, s) dµ(z) dw - ¨Q2 T ˆ|z|>r |A(u(y, s)) -A(v(x, t))| ψ(x -y, t, s) dµ(z) dw, =: I + J.
Note that these integrals are finite since A(u) C([0,T ];L 1 ) ≤ A ′ L ∞ u C([0,T ];L 1 ) (A is Lipschitz-continuous and 0 at 0) and by Fubini (note the convolution integrals in x and y),

I, J ≤ A(u) C([0,T ];L 1 ) + A(v) C([0,T ];L 1 ) ψ L 1 (R d ×(0,T ) 2 )
ˆ|z|>r dµ(z).

We then change variables (z, x, t, y, s) → (z, x + z, t, y + z, s) in I,

I = ¨QT ˆ|z|>r |A(u(y, s)) -A(v(x, t))| ψ(x -z -(y -z), t, s) dµ(z) dw,
to find that I + J = 0 and the proof is complete.

Lemma 4.5. Under the assumptions of Lemma 3.1,

I = ¨Q2 T |A(v(x, t)) -A(u(y, s))| L µ * r [φ ǫ,δ (x, t, •, s)](y) dw ≤ C ǫ ˆ0<|z|≤r |z| 2 dµ(z),
where C ǫ > 0 does not depend on r > 0.

Proof. Easy computations show that

L µ * r [φ ǫ,δ (x, t, •, s)](y) = θ δ (t -s) ˆ0<|z|≤r θǫ (x -y -z) -θǫ (x -y) + z • D θǫ (x -y) 1 |z|≤1 dµ * (z) = θ δ (t -s) ˆ0<|z|≤r θǫ (x -y + z) -θǫ (x -y) -z • D θǫ (x -y) 1 |z|≤1 dµ(z) = θ δ (t -s) L µ r [ θǫ ](x -y),
and by Fubini (there are again convolution integrals in I!),

I ≤ ¨Q2 T |A(u(y, s)) -A(v(x, t))| θ δ (t -s) L µ r [ θǫ ](x -y) dw ≤ A(u) L 1 (QT ) + A(v) L 1 (QT ) θ δ L µ r [ θǫ ] L 1 (R d+1 ) ≤ T A ′ L ∞ u C([0,T ];L 1 ) + v C([0,T ];L 1 ) θ δ L µ r [ θǫ ] L 1 (R d+1 ) .
By classical properties of approximate units and Lemma 4.1,

θ δ L µ r [ θǫ ] L 1 (R d+1 ) = θ δ L 1 (R) =1 L µ r [ θǫ ] L 1 (R d ) ≤ θǫ W 2,1 (R d ) ˆ0<|z|≤r |z| 2 dµ(z).
The proof is complete.

Proofs of the main results

The proofs of this section use the so-called doubling of variables technique of Kruzhkov [START_REF] Kruzhkov | First order quasilinear equations with several independent variables[END_REF] along with ideas from [START_REF] Jakobsen | Continuous dependence estimates for viscosity solutions of integro-PDEs[END_REF][START_REF] Kuznetsov | Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation[END_REF]; for other relevant references, see also e.g. [START_REF] Alibaud | Entropy formulation for fractal conservation laws[END_REF][START_REF] Cifani | Entropy formulation for degenerate fractional order convectiondiffusion equations[END_REF][START_REF] Karlsen | Stability of entropy solutions for Lévy mixed hyperbolic parabolic equations[END_REF] for nonlocal equations. It consists in considering u as a function of the new variables (y, s) and using the approximate units φ ǫ,δ in (3.1) as test functions. For brevity, we do not specify anymore the variables of u = u(y, s), v = v(x, t) and φ ǫ,δ = φ ǫ,δ (x, t, y, s) when the context is clear; recall also that dx dt dy ds is denoted by dw. 5.1. Proof of Lemma 3.1. Let (x, t) ∈ Q T be fixed and u = u(y, s), k = v(x, t), and φ(y, s) := φ ǫ,δ (x, t, y, s). The entropy inequality for u (see (2.9)) then takes the form

ˆQT |u -v| ∂ s φ ǫ,δ + q f (u, v) + |A(u) -A(v)| b µ * r • D y φ ǫ,δ dy ds + ˆQT |A(u) -A(v)| L µ * r [φ ǫ,δ (x, t, •, s)](y) dy ds + ˆQT sgn (u -v) L µ,r [A(u(•, s))](y) φ ǫ,δ dy ds - ˆRd |u(y, T ) -v(x, t)| φ ǫ,δ (x, t, y, T ) dy + ˆRd |u 0 (y) -v(x, t)| φ ǫ,δ (x, t, y, 0) dy ≥ 0.
We integrate this inequality w.r.t. (x, t) ∈ Q T , noting that q f in (2.5) is symmetric, and that ∂ s φ ǫ,δ = -∂ t φ ǫ,δ and D y φ ǫ,δ = -D x φ ǫ,δ by (3.1). Consequently we find that

I 1 + • • • + I 5 := - ¨Q2 T |u -v| ∂ t φ ǫ,δ + q f (v, u) + |A(u) -A(v)| b µ * r • D x φ ǫ,δ dw + ¨Q2 T |A(u) -A(v)| L µ * r [φ ǫ,δ (x, t, •, s)](y) dw + ¨Q2 T sgn (u -v) L µ,r [A(u(•, s))](y) φ ǫ,δ dw - ¨QT ×R d |u(y, T ) -v(x, t)| φ ǫ,δ (x, t, y, T ) dx dt dy + ¨QT ×R d |u 0 (y) -v(x, t)| φ ǫ,δ (x, t, y, 0) dx dt dy ≥ 0.
(5.1)

Note that the terms in the inequality above are well-defined since they are all essentially of the form of convolution integrals of L 1 -functions. See Lemma 4.1, Remark 4.2, and the discussions in the proofs of Lemmas 4.4 and 4.5 for more details.

A classical computation from [START_REF] Kuznetsov | Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation[END_REF] reveals that

I 4 + I 5 - ¨Rd ×QT |u(y, s) -v(x, T )| φ ǫ,δ (x, T, y, s) dx dy ds + ¨Rd ×QT |u(y, s) -v 0 (x)| φ ǫ,δ (x, 0, y, s) dx dy ds ≤ -u(T ) -v(T ) L 1 (R d ) + u 0 -v 0 L 1 (R d ) + ǫ C θ |u 0 | BV (R d ) + 2(ω u (δ) ∨ ω v (δ)),
where C θ is as in Lemma 3.1. Lemma 3.1 now follows from (5.1) and the above estimates on I 4 and I 5 .

5.2.

Proof of Theorem 3.3. The proof uses the Kuznetsov lemma, and morally speaking it amounts to subtracting the u(y, s) and v(x, t) equations, multiplying by sgn (uv), and then applying both new and classical tricks to arrive at an L 1 -estimate of |u -v|. We expect to see terms involving

sgn (u -v) L µ,r [A(u)] -L µ,r [B(v)] ,
and naively we can write this as

sgn (u -v)L µ,r [(A -B)(u)] + sgn (u -v)L µ,r [B(u) -B(v)].
These terms are estimated by Kato type inequalities (see Lemmas 4.3 and 4.4), the first term should give the dependence on A -B while the second term is a nonpositive term that also appears in the uniqueness proof. The problem with this approach is that we can not apply Kato for the first term because A -B then have to be monotone! There are different ways to overcome this monotonicity problem, and we have chosen to adapt ideas from [START_REF] Jakobsen | Continuous dependence estimates for viscosity solutions of integro-PDEs[END_REF] -a paper on continuous dependence estimates for fully nonlinear Bellman-Isaacs type of equations via viscosity solution techniques. We consider the region where A ′ ≥ B ′ and its complementary. Let E ± be sets satisfying:

(5.2)

     E ± ⊆ R are Borel sets; ∪ ± E ± = R and ∩ ± E ± = ∅; R \ supp(A ′ -B ′ ) ∓ ⊆ E ± .
For all u ∈ R, we define

A ± (u) := ˆu 0 A ′ (τ ) 1 E± (τ ) dτ, B ± (u) := ˆu 0 B ′ (τ ) 1 E± (τ ) dτ, C ± (u) := ±(A ± (u) -B ± (u)).
(5.3)

These functions satisfy the following properties:

Lemma 5.1. Under the assumptions of Theorem 3.3, [START_REF] Alibaud | Occurence and non-appearance of shocks in fractal Burgers equation[END_REF], in particular, they are monotone;

(i) A = A + + A -and B = B + + B -; (ii) A ± , B ± , C ± satisfy (1.
(iii) ± |C ± (u)| L 1 (0,T ;BV ) ≤ A ′ -B ′ L ∞ (R) |u| L 1 (0,T ;BV ) ; (iv) for all z ∈ R d \ {0}, ± C ± (u(• + z, •)) -C ± (u) L 1 (QT ) ≤ A ′ -B ′ L ∞ (R) u(• + z, •) -u L 1 (QT ) .
The proofs of (i) and (ii) are immediate, whereas (iii) and (iv) follow from standard arguments for Lipschitz-continuous and BV -functions (see e.g. [START_REF] Bouchut | Kružkov's estimates for scalar conservation laws revisited[END_REF][START_REF] Evans | Measure theory and fine properties of functions[END_REF][START_REF] Rudin | Real and complex Analysis[END_REF]); the details are left to the reader.

In the proof below, A ± -B ± will be the monotone functions replacing the nonmonotone function A -B of the formal argument above.

Proof of Theorem 3.3. Let us divide the proof into several steps.

1. We argue as in the beginning of the proof of Lemma 3.1 changing the roles of u and v. We fix (y, s) and take k = u(y, s) and φ ǫ,δ = φ ǫ,δ (x, t, y, s) in the entropy inequality for v = v(x, t) to find that

¨Q2 T |v -u| ∂ t φ ǫ,δ + q g (v, u) + |B(v) -B(u)| b µ * r • D x φ ǫ,δ dw + ¨Q2 T |B(v) -B(u)| L µ * r [φ ǫ,δ (•, t, y, s)](x) dw + ¨Q2 T sgn (v -u) L µ,r [B(v(•, t))](x) φ ǫ,δ dw - ¨Rd ×QT |v(x, T ) -u(y, s)| φ ǫ,δ (x, T, y, s) dx dy ds + ¨Rd ×QT |v 0 (x) -u(y, s)| φ ǫ,δ (x, 0, y, s) dx dy ds ≥ 0.
Then we add this inequality and inequality (3.3) in Lemma 3.1,

u(T ) -v(T ) L 1 (R d ) ≤ u 0 -v 0 L 1 (R d ) + ǫ C θ |u 0 | BV (R d ) + 2(ω u (δ) ∨ ω v (δ)) + ¨Q2 T (q g -q f )(v, u) • D x φ ǫ,δ dw =:I1 + ¨Q2 T |B(v) -B(u)| L µ * r [φ ǫ,δ (•, y, t, s)](x) dw =:I2 + ¨Q2 T |A(v) -A(u)| L µ * r [φ ǫ,δ (x, t, •, s)](y) dw =:I ′ 2 + ¨Q2 T |B(v) -B(u)| -|A(v) -A(u)| b µ * r • D x φ ǫ,δ dw =:I3 + ¨Q2 T sgn (v -u) L µ,r [B(v(•, t))](x) -L µ,r [A(u(•, s))](y) φ ǫ,δ dw =:I4 , (5.4)
where r, ǫ > 0, 0 < δ < T , and C θ > 0 only depends on the kernel θd from (3.2).

2.

It is standard to estimate I 1 (cf. e.g. [START_REF] Dafermos | Hyperbolic Conservation Laws in Continuum Physics[END_REF][START_REF] Kuznetsov | Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation[END_REF]), and I 2 + I ′ 2 can be estimated by Lemma 4.5,

I 1 ≤ |u 0 | BV (R d ) T f ′ -g ′ L ∞ (R,R d ) , (5.5)
I 2 + I ′ 2 ≤ C ǫ ˆ0<|z|≤r |z| 2 dµ(z), (5.6)
where C ǫ does not depend on r > 0. Now we focus on I 3 and I 4 .

3. Cutting w.r.t. E ± . We split I 3 and I 4 into four new terms using the sets E ± , see (5.2)-(5.3). By Lemma 5.1 (i), I 4 can be written as

I 4 = ± ¨Q2 T sgn (v -u) L µ,r [B ± (v(•, t))](x) -L µ,r [A ± (u(•, s))](y) φ ǫ,δ dw.
By Lemma 5.1 (ii), we can apply twice Lemma 4.4 with B + and A -instead of A, followed by the definitions of C ± , see (5.3), to show that

I 4 ≤ ¨Q2 T sgn (v -u) L µ,r B + (u(•, s)) -A + (u(•, s)) (y) φ ǫ,δ dw + ¨Q2 T sgn (v -u) L µ,r B -(v(•, t)) -A -(v(•, t)) (x) φ ǫ,δ dw = ¨Q2 T sgn (u -v) L µ,r C + (u(•, s)) (y) φ ǫ,δ dw + ¨Q2 T sgn (v -u) L µ,r C -(v(•, t)) (x) φ ǫ,δ dw =: I + 4 + I - 4 . (5.7)
Note that it is crucial to have u in the first term and v in the second -otherwise we will not be able to apply the Kato inequality later on! We now consider I 3 . By (2.7), Lemma 5.1 (i)-(ii), the formula D x φ ǫ,δ = -D y φ ǫ,δ , and the definitions D + = D y and D -= D x , it follows that

|B(v) -B(u)| -|A(v) -A(u)| D x φ ǫ,δ = sgn (u -v) (A(u) -B(u)) -(A(v) -B(v)) D y φ ǫ,δ = ± sgn (u -v) ± (A ± (u) -B ± (u)) ∓ (A ± (v) -B ± (v)) D ± φ ǫ,δ = ± |C ± (u) -C ± (v)| D ± φ ǫ,δ .
We can then rewrite I 3 as (5.8)

I 3 = ± ¨QT |C ± (u) -C ± (v)| b µ * r • D ± φ ǫ,δ dw =:I ± 3 .
4. Cutting w.r.t. z. We decompose L µ,r into two new terms using a new cutting parameter r 1 > r.

Let µ = µ 1 + µ | |z|>r 1 for µ 1 := µ | 0<|z|≤r 1 ,
and note that by (2.4), L µ,r = L µ1,r + L µ,r1 . Then

I + 4 = ¨Q2 T sgn (u -v) L µ1,r [C + (u(•, s))](y) φ ǫ,δ dw =:I + 5 + ¨Q2 T sgn (u -v) L µ,r1 [C + (u(•, s))](y) φ ǫ,δ dw.
(

Since C + satisfies (1.4) by Lemma 5.1 (ii) and µ 1 clearly satisfies (1.6), we can apply the Kato type inequality in Lemma 4.3 (with k = v(x, t) and A = C + ) to show that

I + 5 = ˆQT ˆQT sgn (u(y, s) -v(x, t)) L µ1,r [C + (u(•, s))](y) φ ǫ,δ dy ds dx dt ≤ ˆQT ˆQT |C + (u(y, s)) -C + (v(x, t))| L µ * 1 ,r [φ ǫ,δ (x, t,
•, s)](y) dy ds dx dt.

Adding I + 3 in the form (5.8) then gives (5.10)

I + 3 + I + 5 ≤ ¨Q2 T |C + (u) -C + (v)| b µ * r • D y φ ǫ,δ + L µ * 1 ,r [φ ǫ,δ (x, t, •, s)](y) dw.

Now easy computations show that

D y φ ǫ,δ = -θ δ (t-s) D θǫ (x-y), L µ * 1 ,r [φ ǫ,δ (x, t, •, s)](y) = θ δ (t-s) L µ1,r [ θǫ ](x-y).
Hence by adding and subtracting z • D θǫ (xy), we get that

b µ * r • D y φ ǫ,δ + L µ * 1 ,r [φ ǫ,δ (x, t, •, s)](y) = θ δ (t -s) ˆr<|z|≤r1 θǫ (x -y + z) -θǫ (x -y) -z • D θǫ (x -y) dµ(z) + θ δ (t -s) D θǫ (x -y) • -b µ * r + ˆr<|z|≤r1 z dµ(z) =sgn (r1-1) ´r1 ∧(1∨r)<|z|≤r 1 ∨1 z dµ(z) , (5.11)
where the last equality comes from (2.3) and the change of variable z → -z. We insert (5.11) into (5.10) and combine the resulting inequality with (5.9),

I + 3 + I + 4 ≤ ¨Q2 T |C + (u) -C + (v)| • θ δ (t -s) ˆr<|z|≤r1 θǫ (x -y + z) -θǫ (x -y) -z • D θǫ (x -y) dµ(z) dw + ¨Q2 T |C + (u) -C + (v)| • θ δ (t -s) D θǫ (x -y) • sgn (r 1 -1) ˆr1∧(1∨r)<|z|≤r1∨1 z dµ(z) dw + ¨Q2 T sgn (u -v) L µ,r1 [C + (u(•, s))](y) φ ǫ,δ dw =: J + 1 + J + 2 + J + 3 .
(5.12)

Similar arguments show that we can bound I - 3 + I - 4 (see (5.7)-(5.8)) as follows,

I - 3 + I - 4 ≤ ¨Q2 T |C -(v) -C -(u)| • θ δ (t -s) ˆr<|z|≤r1 θǫ (x -y -z) -θǫ (x -y) + z • D θǫ (x -y) dµ(z) dw - ¨Q2 T |C -(v) -C -(u)| • θ δ (t -s) D θǫ (x -y) • sgn (r 1 -1) ˆr1∧(1∨r)<|z|≤r1∨1 z dµ(z) dw + ¨Q2 T sgn (v -u) L µ,r1 [C -(v(•, t))](x) φ ǫ,δ dw ≤ ˜Q2 T sgn (v-u) L µ,r 1 [C-(u(•,s))](y) φ ǫ,δ dw by Lemma 4.4 =: J - 1 + J - 2 + J - 3 .
(5.13)

5. L 1 ∩ BV -regularity. It remains to estimate J ± i for i = 1, . . . , 3 in (5.12)-(5.13). For J ± 1 and J ± 2 , we integrate by parts to take advantage of the BV -regularity of u. After some technical computations detailed in Appendix A, we find that

|J ± 1 | ≤ 1 2ǫ ˆRd |D θd |dx ˆr<|z|≤r1 |z| 2 dµ(z) |C ± (u)| L 1 (0,T ;BV ) , (5.14) |J ± 2 | ≤ ˆr1∧(1∨r)<|z|≤r1∨1 z dµ(z) |C ± (u)| L 1 (0,T ;BV ) , (5.15)
and hence

± (J ± 1 + J ± 2 ) ≤ 1 2ǫ ˆRd |D θd |dx ˆr<|z|≤r1 |z| 2 dµ(z) ± |C ± (u)| L 1 (0,T ;BV ) + ˆr1∧(1∨r)<|z|≤r1∨1 z dµ(z) ± |C ± (u)| L 1 (0,T ;BV ) .
By Lemma 5.1 (iii) and a priori estimates for u, cf. (2.10), we see that

± (J ± 1 + J ± 2 ) ≤ 1 2ǫ ˆRd |D θd |dx |u| L 1 (0,T ;BV ) ≤|u0| BV (R d ) T ˆr<|z|≤r1 |z| 2 dµ(z) A ′ -B ′ L ∞ (R) + |u| L 1 (0,T ;BV ) ≤|u0| BV (R d ) T ˆr1∧(1∨r)<|z|≤r1∨1 z dµ(z) A ′ -B ′ L ∞ (R) . (5.16)
Let us now estimate J + 3 in (5.12). Easy computations (see the proofs of Lemmas 4.4-4.5) show that

J + 3 ≤ θ δ θǫ L 1 (R d+1 ) L µ,r1 [C + (u)] L 1 (QT ) . Let us recall that θ δ θǫ L 1 (R d+1 ) = θ δ L 1 (R) θǫ L 1 (R d ) = 1,
and then

J + 3 ≤ ˆT 0 ˆ|z|>r1 C + (u(• + z, s)) -C + (u(•, s)) L 1 (R d ) dµ(z)ds. Since C + (u) ∈ L ∞ ∩ C([0, T ]; L 1 ), (z, s) → C + (u(• + z, s)) -C + (u(•, s)) L 1 (R d
) is a continuous function, hence Borel and dµ(z) ds-measurable. Thus, we may change the order of the integration to find

J + 3 ≤ ˆ|z|>r C + (u(• + z, •)) -C + (u) L 1 (QT ) dµ(z).
We get a similar estimates for J - 3 and find by Lemma 5.1 (iii)-(iv) and (2.10) that

± J ± 3 ≤ ˆ|z|>r1 ± C ± (u(• + z, •)) -C ± (u) L 1 (QT ) dµ(z), ≤ ˆ|z|>r1 u(• + z, •)) -u L 1 (QT ) ≤T u0(•+z)-u0 L 1 (R d ) dµ(z) A ′ -B ′ L ∞ (R d ) .
(5.17)

The last inequality (under the bracket) comes from (2.11) applied to the solution u(• + z, •) of (1.1) with initial data u 0 (• + z).

6. Conclusion. By (5.7)-(5.8) and (5.12)-(5.13), I 3 +I 4 ≤ ± 3 i=1 J ± i . Therefore we may estimate (5.4) by (5.5)-(5.6) and (5.16)- (5.17). For all r 1 > r > 0, ǫ > 0, and T > δ > 0, we find that

u(T ) -v(T ) L 1 (R d ) ≤ u 0 -v 0 L 1 (R d ) + |u 0 | BV (R d ) T f ′ -g ′ L ∞ (R,R d ) + ǫ C θ |u 0 | BV (R d ) + 2(ω u (δ) ∨ ω v (δ)) + C ǫ ˆ0<|z|≤r |z| 2 dµ(z) + 1 2ǫ ˆRd |D θd |dx |u 0 | BV (R d ) T ˆr<|z|≤r1 |z| 2 dµ(z) A ′ -B ′ L ∞ (R d ) + |u 0 | BV (R d ) T ˆr1∧(1∨r)<|z|≤r1∨1 z dµ(z) A ′ -B ′ L ∞ (R d ) + T ˆ|z|>r1 u 0 (• + z) -u 0 L 1 (R d ) dµ(z) A ′ -B ′ L ∞ (R d ) , (5.18)
where C ǫ > 0 does not depend on r > 0.

To finish, we first pass to the limit as r → 0 in (5.18). By the dominated convergence theorem, the result is equivalent to setting r = 0 in each term, and in particular the term C ǫ ´0<|z|≤r |z| 2 dµ(z) vanishes. Secondly, we pass to the limit as δ → 0 to get rid of the term 2(ω u (δ) ∨ ω v (δ)). Finally, we optimize the remaining terms w.r.t. ǫ > 0 by using the formula min ǫ>0 ǫ a + b ǫ = 2 √ ab (for a, b ≥ 0). This gives us the following continuous dependence estimate: For all r 1 > 0,

u -v C([0,T ];L 1 ) ≤ u 0 -v 0 L 1 (R d ) + |u 0 | BV (R d ) T g ′ -f ′ L ∞ (R,R d ) + 2 1 2 C θ ˆRd |D θd |dx |u 0 | 2 BV (R d ) T ˆ0<|z|≤r1 |z| 2 dµ(z) A ′ -B ′ L ∞ (R) + |u 0 | BV (R d ) T ˆr1∧1<|z|≤r1∨1 z dµ(z) A ′ -B ′ L ∞ (R) + T ˆ|z|≥r1 u 0 (• + z) -u 0 L 1 (R d ) dµ(z) A ′ -B ′ L ∞ (R) , (5.19
→ ω d -1 1 |•|<1 in L 1 , and ´Rd |Dθ n | dx → ω d -1 |1 |•|<1 | BV (R d ) .
Here ω d is the volume of the unit ball in R d . Note that the BV -semi-norm of the indicator function of the unit ball is equal to the surface area of the unit sphere, i.e.

|1 |•|<1 | BV (R d ) = dω d . Moreover, we have ˆRd |x||θ n (x)| dx → 1 ω d ˆ|x|<1 |x| dx = d d + 1
.

The proof of (3.4) is then complete after passing to the limit as n → +∞ in (5.19).

5.3. Proof of Theorem 3.4. We argue step by step as in the proof of Theorem 3.3. This time, E ± are taken such as (5.20)

     E ± ⊆ R d \ {0} are Borel sets; ∪ ± E ± = R d \ {0} and ∩ ± E ± = ∅; R d \ {0} \ supp(µ -ν) ∓ ⊆ E ± .
Let µ ± and ν ± denote the restrictions of µ and ν to E ± . It is clear that

(5.21)      µ = ± µ ± and ν = ± ν ± , ±(µ ± -ν ± ) = (µ -ν) ± , µ ± , ν ± , and ± (µ ± -ν ± ) all satisfy (1.6).
Proof of Theorem 3.4.

1. We apply Lemma 3.1 with A = B, but different Lévy measures µ and ν, along with the entropy inequality for v to show that for all r, ǫ > 0, 0

< δ < T u(T ) -v(T ) L 1 (R d ) ≤ u 0 -v 0 L 1 (R d ) + ǫ C θ |u 0 | BV (R d ) + 2(ω u (δ) ∨ ω v (δ)) + ¨Q2 T (q g -q f )(v, u) • D x φ ǫ,δ dw + ¨Q2 T |A(v) -A(u)| L ν * r [φ ǫ,δ (•, y, t, s)](x) dw + ¨Q2 T |A(v) -A(u)| L µ * r [φ ǫ,δ (x, t, •, s)](y) dw + ¨Q2 T |A(v) -A(u)| b ν * r -b µ * r • D x φ ǫ,δ dw =:I3 + ¨Q2 T sgn (v -u) L ν,r [A(v(•, t))](x) -L µ,r [A(u(•, s))](y) φ ǫ,δ dw =:I4 , (5.22)
where C ǫ > 0 does not depend on r > 0. Except for I 3 and I 4 , the other terms were estimated in the proof of Theorem 3.3.

2. Cutting w.r.t. E ± . We use the notation introduced in (5.20). We apply Lemma 4.4 twice with ν + and µ -instead of µ, along with linearity of L µ,r in µ, see (2.2), to see that

I 4 = ± ¨Q2 T sgn (v -u) L ν±,r [A(v(•, t))](x) -L µ±,r [A(u(•, s))](y) φ ǫ,δ dw ≤ ¨Q2 T sgn (v -u) L ν+,r [A(u(•, s))](y) -L µ+,r [A(u(•, s))](y) φ ǫ,δ dw + ¨Q2 T sgn (v -u) L ν-,r [A(v(•, t))](x) -L µ-,r [A(v(•, t))](x) φ ǫ,δ dw = ¨Q2 T sgn (u -v) L µ+-ν+,r [A(u(•, s))](y) φ ǫ,δ dw + ¨Q2 T sgn (v -u) L -(µ--ν-),r [A(v(•, t))](x) φ ǫ,δ dw =: I + 4 + I - 4 .
Again, it is crucial to have u in I + 4 and v in I - 4 in order to use Kato's inequality later on.

Let us now consider I 3 . By (2.3) and (2.8), b µ r and µ * are linear w.r.t µ. Easy computations using (5.21) then leads to

b ν * r -b µ * r • D x φ ǫ,δ = ± b ±(µ±-ν±) * r • D ± φ ǫ,δ
where D + = D y and D -= D x , and hence

I 3 = ± ¨QT |A(u) -A(v)| b ±(µ±-ν±) * r • D ± φ ǫ,δ dw =: I + 3 + I - 3 .
3. Cutting w.r.t. z. The computations of this step are similar to the ones in the proof of Theorem 3.3. For the reader's convenience, we estimate I - 3 + I - 4 , the terms that was left to the reader in the preceding proof.

For any measure μ we let μ1 = μ| 0<|z|≤r 1 and write μ = μ1 + μ| |z|>r 1 for r 1 > r. Then

I - 4 ≤ ¨QT sgn (v -u) L -(µ--ν-)1,r [A(v(•, t))](x) φ ǫ,δ dw =:I - 5 + ¨QT sgn (v -u) L -(µ--ν-),r1 [A(v(•, t))](x) φ ǫ,δ dw.
Recall that -(µ -ν -) 1 is a positive Lévy measure by (5.21), so we can apply Lemma 4.3 with -(µ -ν -) 1 instead of µ and k = u(y, s) to find that

I - 5 ≤ ¨Q2 T |A(v) -A(u)| L -(µ--ν-) * 1 ,r [φ ǫ,δ (•, t, y, s)](x) dw and
I - 3 + I - 5 ≤ ¨Q2 T |A(v) -A(u)| b -(µ--ν-) * r • D x φ ǫ,δ + L -(µ--ν-) * 1 ,r [φ ǫ,δ (•, t, y, s)](x) dw.
Easy computations then leads to

L -(µ--ν-) * 1 ,r [φ ǫ,δ (•, t, y, s)](x) = θ δ (t -s) ˆr<|z|≤r1 θǫ (x -y -z) -θǫ (x -y) d(ν --µ -)(z),
and we can rewrite the nonlocal operator as follows,

b -(µ--ν-) * r • D x φ ǫ,δ + L -(µ--ν-) * 1 ,r [φ ǫ,δ (•, t, y, s)](x) = θ δ (t -s) ˆr<|z|≤r1 θǫ (x -y -z) -θǫ (x -y) + z • D θǫ (x -y) d(ν --µ -)(z) -θ δ (t -s) D θǫ (x -y) • -b -(µ--ν-) * r + ˆr<|z|≤r1 z d(ν --µ -)(z) =sgn (r1-1) ´r1 ∧(1∨r)<|z|≤r 1 ∨1 z d(ν--µ-)(z)
.

Compare this expression with (5.11) that appear when I + 3 and I + 4 are considered.

We add the different estimates and find that for all r 1 > r,

I - 3 + I - 4 ≤ ¨Q2 T |A(u) -A(v)| θ δ (t -s) • ˆr<|z|≤r1 θǫ (x -y -z) -θǫ (x -y) + z • D θǫ (x -y) d(ν --µ -)(z) dw - ¨Q2 T |A(u) -A(v)| θ δ (t -s) D θǫ (x -y) • sgn (r 1 -1) ˆr1∧(1∨r)<|z|≤r1∨1 z d(ν --µ -)(z) dw + ¨Q2 T sgn (v -u) L -(µ--ν-),r1 [A(v(•, t))](x) φ ǫ,δ dw ≤ ˜Q2 T sgn (v-u) L -(µ --ν -),r 1 [A(u(•,s))](y) φ ǫ,δ dw by Lemma 4.4 = J - 1 + J - 2 + J - 3 .

Similar arguments also lead to

I + 3 + I + 4 ≤ ¨Q2 T |A(u) -A(v)|θ δ (t -s) • ˆr<|z|≤r1 θǫ (x -y + z) -θǫ (x -y) -z • D θǫ (x -y) d(µ + -ν + )(z) dw + ¨Q2 T |A(u) -A(v)| θ δ (t -s) D θǫ (x -y) • sgn (r 1 -1) ˆr1∧(1∨r)<|z|≤r1∨1 z d(µ + -ν + )(z) dw + ¨Q2 T sgn (u -v) L (µ+-ν+),r1 [A(u(•, s))](y) φ ǫ,δ dw, =: J + 1 + J + 2 + J + 3 .
4. L 1 ∩BV -regularity. We estimate J ± i (i = 1, . . . , 3). Almost all the computations have already been done in the preceding proof. Indeed, J ± 1 are of the same form as in (5.12)-(5.13), with the new nonlinearity A and the new measures ±(µ ±ν ± ). Arguing as for (5.14) thus gives This term is of the same form than J + 2 in (5.12) (or J - 2 in (5.13)) with the new nonlinearity A and the new flux ´r1∧(1∨r)<|z|≤r1∨1 z d(µν)(z). Arguing as for (5.15) and using (2.10) thus give

± J ± 1 ≤ 1 2ǫ ˆRd |D θd |dx • |A(u)| L 1 (0,T ;BV ) ≤|u0| BV (R d ) T A ′ L ∞ (R) by (2.10) ˆr<|z|≤r1 |z| 2 d ± ±(µ ± -ν ± )
± J ± 2 ≤ |u 0 | BV (R d ) T A ′ L ∞ (R)
ˆr1∧(1∨r)<|z|≤r1∨1 z d(µν)(z) .

Finally, Hence, we change the order of integration (using Fubini) to see that

± J ± 3 ≤ T A ′ L ∞ (R) ˆ|z|≥r1 u 0 (• + z) -u 0 L 1 (R d ) d|µ -ν|(z).
|div y q(C + (v(x, t)), C + (u(•, s)))| ≤ |DC + (u(•, s))| q u L ∞ (R,R d ) ,
where q u L ∞ (R,R d ) denotes the Lipschitz constant of q w.r.t. its second variable. We thus may integrate by parts in y to see that

|J| ≤ q u L ∞ (R,R d )
ˆRd ˆRd θǫ (xy) d|DC + (u(•, s))|(y) dx.

Changing the order of integration, we find that

J ≤ |C + (u(s))| BV (R d ) q u L ∞ (R,R d ) ,
and hence by (A.3) and integrating first w.r.t. t, we get that

|J + 2 | ≤ q u L ∞ (R,R d ) ˆT 0 |C + (u(s))| BV (R d ) ds.
The proof of (5.15) is now complete since by (A.2),

q u L ∞ (R,R d ) = ˆr1∧ (

  [START_REF] Alibaud | Asymptotic properties of entropy solutions to fractal Burgers equation[END_REF] andµ is a nonnegative Radon measure on R d \ {0} satisfying (1.6) ˆRd \{0} (|z| 2 ∧ 1) dµ(z) < +∞,where we use the notation a ∧ b = min{a, b}. The measure µ is a Lévy measure.Remark 1.1.

  Notation. Hereafter, a ∨ b := max{a, b}, while • and | • | denote the Euclidean inner product and norm. For A ∈ R d×d , |A| := max{A w : w ∈ R d , |w| ≤ 1}. The symbols • and | • | are used for norms and semi-norms of functions respectively.

  r > 0, and x ∈ R d , whereL µ r [φ](x) := ˆ0<|z|≤r φ(x + z)φ(x)z • Dφ(x) 1 |z|≤1 dµ(r [φ](x) := ˆ|z|>r (φ(x + z)φ(x)) dµ(z). (2.4)Consider then the Kruzhkov[START_REF] Kruzhkov | First order quasilinear equations with several independent variables[END_REF] entropies | • -k|, k ∈ R, and entropy fluxes(2.5)

1 . 3 . 1 (

 131 , time and space approximate units with kernel θn with n = 1 and n = d satisfying (3.2) θn ∈ C ∞ c (R n ), θn ≥ 0, supp θn ⊆ {|x| < 1}, and ˆRn θn (x) dx = 1. We also let ω u (δ) be the modulus of continuity of u ∈ C [0, T ]; L Lemma Kuznetsov type Lemma). Assume (1.3)-(1.6). Let u be the entropy solution of (1.1)

7 )

 7 where C only depends on the data. Outline of proof. To prove (3.6), we use Theorems 3.3 and 3.4 with r = 1 and the triangle inequality. We also use estimates like |a -b| ≤ |a| + |b| |a -b|, |µ -ν| ≤ |µ| + |ν| etc. To prove (3.7), we also use Remark 3.6 and set r = 0 and r = 1. Remark 3.8.

Example 3 . 1 .

 31 Let us consider Equation (1.10) with a = 0. Let us also consider the class of Lévy operators satisfying

1 αExample 3 . 4 .

 134 and r = +∞ if α > 1, r = ǫ and r = 1 if α = 1, and r = 0 and r = 1 if α < 1. Let us finally consider the vanishing approximation (3.8) with the viscous term

Lemma 4 . 3 .

 43 Assume (1.4) and(1.6). Then for all u ∈ L 1 (R d ), k ∈ R, r > 0, and all 0

Lemma 4 . 4 .

 44 Assume (1.4) and (1.6), and let u

  (µ ±ν ± ) = µν and hence ± J ± 2 = ¨Q2 T |A(u) -A(v)| θ δ (ts) D θǫ (xy) • sgn (r 1 -1)ˆr1∧(1∨r)<|z|≤r1∨1 z d(µν)(z) dw.

5 .

 5 Conclusion. The rest of the proof is the same as for Theorem 3.3; i.e. we use the estimates on J ± i to estimateI 3 + I 4 ≤ 3 i=1 ± J ± i in(5.22) and pass to limit and/or optimizes w.r.t. the parameters r, ǫ, δ > 0. The proof is complete.Appendix A. Technical computations Proof of (5.14) and (5.15). We start by proving(5.14) in the + case. Similar arguments give the proof also in thecase. From Taylor's formulaθǫ (xy + z) -θǫ (xy)z • D θǫ (xy) = ˆ1 0 (1τ ) D 2 θǫ (xy + τ z) z • z dτ,and hence by Fubini's theorem, J + 1 in (5.12) can be written as(A.1) J + 1 = ¨(0,T ) 2 ˆr<|z|≤r1 ˆ1 0 θ δ (ts) (1τ ) • ˆRd ˆRd |C + (v(x, t)) -C + (u(y, s))| D 2 θǫ (xy + τ z) z • z dy dx =:J dτ dµ(z) dt ds. For any k ∈ R, it is classical that η k (C + (u(•, s))) = |k -C + (u(•, s))| ∈ BV with |Dη k (C + (u(•, s))| ≤ |DC + (u(•, s))|,since it is the composition of a Lipschitz-continuous function with a BV -function; see e.g.[START_REF] Bouchut | Kružkov's estimates for scalar conservation laws revisited[END_REF][START_REF] Evans | Measure theory and fine properties of functions[END_REF][START_REF] Rudin | Real and complex Analysis[END_REF]. Integration by parts w.r.t. y (for fixed z, x, t, s), then leads to|J| = ˆRd ˆRd D θǫ (xy + τ z) • z z • dDη C+(v(x,t)) (C + (u(•, s)))(y) dx , ≤ |z| 2 ˆRd ˆRd |D θǫ (xy + τ z)| d|DC + (u(•,s))|(y) dx. Note that by the definition of θǫ (just below (3.1)), we have ˆRd |D θǫ (x)| dx = 1 ǫ ˆRd |D θd |dx.

  )where θd is an arbitrary approximate unit (3.2) and C θ = 2 ´Rd |x| θd (x) dx by Lemma 3.1.Let θd = θ n where {θ n } n∈N is a sequence of kernels s.t. θ n satisfies (3.2), θ n

  |J| ≤ |z| 2 |C + (u(s))| BV (R d ) ˆRd |D θǫ (x)| dx ≤ |z| 2 |C + (u(s))| BV (R d )Let us recall that the integrand above is dτ dµ(z)dt ds-measurable since s → |u(s)| BV (R d ) is lower semi-continuous. By Fubini we then integrate first w.r.t. t and use that ´T 0 θ δ (ts) dt ≤ 1 to see that For fixed (x, t, s), q(C + (v(x, t), •) is Lipschitz-continuous and C + (u(•, s)) is BV ; hence, the composition q(C + (v(x, t)), C + (u(•, s))) is in BV (R d , R d ) with

	1 ǫ ˆRd |C + (u(s))| BV (R d ) ds, |D θd |dx, z dµ(z), and note that it is Lipschitz-continuous. Again by Fubini's theorem, J + and then from (A.1) that |J + 1 | ≤ 1 ǫ ˆRd |D θd |dx • ¨(0,T ) 2 ˆr<|z|≤r1 ˆ1 0 1 | ≤ 1 ǫ ˆRd |D θd |dx ˆ1 0 (1 -τ ) dτ ˆr<|z|≤r1 |z| 2 dµ(z) ˆT 0 and the proof of (5.14) is complete. We prove (5.15) by similar arguments. Define (A.2) q(v, u) := |v -u| sgn (r 1 -1) ˆr1∧(1∨r)<|z|≤r1∨1 2 in (5.12) can be written as (A.3) θ |J + J + 2 = ¨(0,T ) 2 θ

δ (ts) (1τ ) |z| 2 |C + (u(s))| BV (R d ) dτ dµ(z) dt ds. δ (ts) ˆRd ˆRd D θǫ (xy) • q(C + (v(x, t), C + (u(y, s))) dy dx =:J dt ds.

  Nathaël Alibaud) UMR CNRS 6623, Université de Franche-Comté, 16 route de Gray, 25 030 Besanc ¸on cedex, France, and, École Nationale Supérieure de Mécanique et des Microtechniques, 26 chemin de l' Épitaphe, 25030 Besanc ¸on cedex, France, and, Department of Mathematics, Faculty of Science, Prince of Songkla University, Hat Yai, Songkhla, Thailand, 90112 E-mail address: nathael.alibaud@ens2m.fr (Simone Cifani) Department of Mathematics, Norwegian University of Science and Technology (NTNU), N-7491 Trondheim, Norway E-mail address: simone.cifani@math.ntnu.no URL: http://www.math.ntnu.no/~cifani/ (Espen R. Jakobsen) Department of Mathematics, Norwegian University of Science and Technology (NTNU), N-7491 Trondheim, Norway E-mail address: erj@math.ntnu.no URL: http://www.math.ntnu.no/~erj/

1∨r)<|z|≤r1∨1 z dµ(z) .

(

Acknowledgement

We would like to thank the two anonymous referees for their careful reports. They helped us improve the paper a lot.

This research was supported by the Research Council of Norway (NFR), through the project "Integro-PDEs: Numerical methods, Analysis, and Applications to Finance", and by the "French ANR project CoToCoLa". 1