Benjamin Gess

Xavier Lamy
email: xlamy@math.univ-toulouse.fr

REGULARITY OF SOLUTIONS TO SCALAR CONSERVATION LAWS WITH A FORCE

Keywords:

We prove regularity estimates for entropy solutions to scalar conservation laws with a force. Based on the kinetic form of a scalar conservation law, a new decomposition of entropy solutions is introduced, by means of a decomposition in the velocity variable, adapted to the non-degeneracy properties of the flux function. This allows a finer control of the degeneracy behavior of the flux. In addition, this decomposition allows to make use of the fact that the entropy dissipation measure has locally finite singular moments. Based on these observations, improved regularity estimates for entropy solutions to (forced) scalar conservation laws are obtained.

Introduction

We consider the regularity of solutions to scalar conservation laws ∂ t u + divA(u) = S on (0, T ) × R n (1.1) u(0) = u 0 on R n , for S ∈ L 1 ([0, T ] × R n ), u 0 ∈ L 1 (R n ) and A ∈ C 2 (R; R n ) satisfying a non-degeneracy condition to be specified below. In the special case, n = 1, S ≡ 0 and A convex, the one-sided Oleinik inequality for entropy solutions can be used to obtain optimal regularity estimates for (1.1). More precisely, assuming in addition that inf

(u,v)∈R 2 , u =v |A (u) -A (v)| |u -v| > 0
for some l > 0, Bourdarias, Gisclon and Junca have shown in [START_REF] Bourdarias | Fractional BV spaces and applications to scalar conservation laws[END_REF] that bounded entropy solutions for (1.1) satisfy u(t) ∈ W 1 -ε, loc (R) for all t, ε > 0. A typical example is A(u) = |u| +1 , ≥ 1. For a flux function A that fails to be convex, n = 1, S ≡ 0, the same regularity can be obtained under some restrictive assumptions on the zeroes of A , by combining results of Cheng [START_REF] Cheng | A regularity theorem for a nonconvex scalar conservation law[END_REF] and Jabin [START_REF] Jabin | Some regularizing methods for transport equations and the regularity of solutions to scalar conservation laws[END_REF]. In multiple dimensions, or for S non-smooth, these arguments do not apply anymore. In this case, the best known regularity estimates rely on the kinetic formulation of (1.1), as introduced by Lions, Perthame and Tadmor in [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related equations[END_REF]. In this work it was observed 1 that if u is an entropy solution to (1.1) then the kinetic function (1.2) f (t, x, v) := 1 0<v<u(t,x) -1 0>v>u(t,x) satisfies (1.3)

∂ t f + a(v) • ∇ x f = ∂ v m + δ v=u S,
for some Radon measure m ≥ 0 and a := A . Based on this and on averaging techniques, regularity estimates for bounded entropy solutions to (1.1) have been obtained in [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related equations[END_REF] assuming a non-degeneracy property for the flux A and S ≡ 0. For the special case of (1.1) with A(u) = u +1 this leads to

(1.4) u ∈ W s,p loc ((0, T ) × R n ) ∀s < 1 1 + 2 , p < 4 + 1 2 + 1 .
In this work, we provide improved regularity estimates in the case l > 1, based on a careful treatment of the degeneracy at u = 0.

Regularity estimates for scalar conservation laws. The improved regularity estimates for the special case A(u) = u +1 will be obtained as a consequence of a general regularity result for possibly higher dimensional fluxes A(u) with a finite number of degeneracy points. To state this result precisely, we need to introduce technical assumptions satisfied by the velocity field a(u) = A (u), which quantify the "degree of nonlinearity" at these isolated points and away from them. Loosely speaking, we ask that:

• The overall nonlinearity be "higher" than some threshold represented by a number α ∈ (0, 1], where α = 1 corresponds in one dimension to the least degenerate flux A(u) = u 2 /2 and α = 1/ to the degenerate flux A(u) = u +1 . See (1.6) below.

• The flux at the degeneracy points be "flatter" than some threshold represented by an exponent κ > 0, where κ = -1 corresponds in one dimension to the degenerate flux A(u) = u +1 with > 1. See (1.7) below. • The nonlinearity away from the degeneracy points be "higher" than some threshhold β > α (that is, strictly higher than the overall nonlinearity), in a way quantified, as one approaches a degeneracy point, by an exponent τ > 0 (the higher τ is, the faster nonlinearity is "lost" as one approaches the point). In one dimension, the flux A(u) = u +1 corresponds to β = 1 and τ = -1. See (1.8) below.

More specifically, we consider a velocity field a ∈ C 1 (R; R n ) such that the set of degeneracy points (1.5) Z := {a = 0} is locally finite 1 In fact, [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related equations[END_REF] treated the case S ≡ 0 but the same applies to non-vanishing S.

and assume that there exist α < β ∈ (0, 1] and κ, τ ≥ 0 such that for any bounded interval I ⊂ R v and λ, δ > 0 it holds sup

τ 2 +|ξ| 2 =1 |{v ∈ I : |τ + a(v) • ξ| ≤ δ}| δ α , (1.6) sup v∈I, dist(v,Z)≤λ |a (v)| λ κ , (1.7) sup τ 2 +|ξ| 2 =1 |{v ∈ I : dist(v, Z) ≥ λ, |τ + a(v) • ξ| ≤ δ}| λ -τ δ β . (1.8)
Here the symbol denotes inequality up to a multiplicative constant that depends only on the interval I and the velocity field a. Note that since I is bounded, (1.6)-(1.8) are trivially satisfied for δ, λ large. We are now in a position to state our main general result on the regularity of entropy solutions to scalar conservation laws.

Theorem 1. Let A ∈ C 2 (R; R n ) satisfy (1.5)-(1.8), u 0 ∈ L 1 (R n x ), S ∈ L 1 ([0, T ] × R n
x ) and u(t, x) be an entropy solution of (1.1) with associated kinetic function f as in

(1.2). Then, for all φ ∈ C ∞ c (R), ˆf (t, x, v)φ(v) dv ∈ W s,r loc ((0, T ) × R n x ) ∀s < s * ,
where

s * = (1 -η)θ α + ηθ β , θ a = a a + 2 , (a = α, β), η = E 1 E 1 + E 2 , E 1 = min κ + 1, 1 α θ α , E 2 = max 2τ β -κ -1, τ -1 β , 0 θ β ,
and the order of integrability r is given by

1 r = 1 -η r α + η r β , 1 r a = 1 + θ a 2 , (a = α, β). In particular, if u 0 ∈ L ∞ (R n ) and S ∈ L ∞ ([0, T ] × R n ) then u ∈ W s,r loc ((0, T ) × R n x ) ∀s < s * .
We next provide several examples of fluxes A satisfying the assumptions

(1.5)-(1.8). Example 2. Let A ∈ C 2 (I; R n ) for some interval I ⊆ R. (1) Let A ∈ C ∞ (I; R), n = 1. The valuation of A at v ∈ I is defined as m A (v) = inf{k ≥ 1 : A (k+1) (v) = 0}, the degeneracy of A on I is m A := sup v∈I m A (v). If 0 < m A < ∞ we say that A is non-degenerate of order m A . In this case (1.6) is satisfied with α = 1/m A (cf. [3, Lemma 1]).
(2) Let a be κ-Hölder continuous, i.e. A ∈ C 2+κ (I; R n ). Then (1.7) is satisfied. (3) Assume n = 1, (1.5) and that for some τ ≥ 0 and all λ > 0

λ τ inf v∈I, dist(v,Z)≥λ |a (v)|.
Then a satisfies (1.8) with The regularity for entropy solutions in the special case A(u) = u +1 is an immediate consequence of Theorem 1.

β = 1. (4) Let A(v) = sin(v) or A(v) = cos(v). Then A satisfies (1.5)-(1.
Corollary 3. Let ≥ 1, u 0 ∈ L 1 (R), S ∈ L 1 ([0, T ] × R) and u(t, x)
be an entropy solution of (1.1)

with n = 1, A(v) = |v| +1 or A(v) = sgn(v)|v| +1 and associated kinetic function f as in (1.2). Then, for all φ ∈ C ∞ c (R), ˆf (t, x, v)φ(v) dv ∈ W s,1 1 3 , 1 + 1 .
In particular, if

u 0 ∈ L ∞ (R n ) and S ∈ L ∞ ([0, T ] × R n ) then u ∈ W s,1 loc ((0, T ) × R n ) ∀s < min 1 3 , 1 + 1 .
Motivated by some ideas going back to Tadmor and Tao [START_REF] Tadmor | Velocity averaging, kinetic formulations, and regularizing effects in quasi-linear PDEs[END_REF], the proof of Theorem 1 relies on introducing a new decomposition of entropy solutions u which allows to make use of the fact that apart from the degeneracy at u = 0, the flux A(u) = u +1 has non-vanishing second derivative. Using this aspect alone we show that it is possible to improve the regularity in (1.4) to s < 1 2+ . In the literature, a key draw-back of the methods to estimate the regularity of solutions to (1.4) based on averaging techniques is that these methods are not able to make use of the sign of the entropy dissipation measure m in (1.3). Indeed, these arguments could only use that m has locally finite mass. In contrast, we make use of the observation that for entropy solutions to (1.1) the entropy defect measure m has, thanks to its sign, locally finite singular moments, that is, |v| -γ m has locally finite mass for all γ ∈ [0, 1). This is, to our knowledge, the first time that a kinetic averaging lemma manages, when applied to scalar conservation laws, to take advantage of the sign of the entropy production (see also [START_REF] Gess | Sobolev regularity for the porous medium equation with a force[END_REF]). Specializing our results to the particular case A(u) = u l+1 we obtain the following result.

Remark 4. Solutions of (1.1) for which the entropy dissipation m is only assumed to be a locally finite signed measure are sometimes called quasi-solutions [START_REF] De Lellis | Structure of entropy solutions for multi-dimensional scalar conservation laws[END_REF]. For the model case in Corollary 3, the arguments in [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related equations[END_REF] still apply to this larger class of solutions and provide the regularity (1.4). However, when is an integer and S ≡ 0, Crippa, Otto and Westdickenberg obtain in [START_REF] Crippa | Regularizing effect of nonlinearity in multidimensional scalar conservation laws[END_REF]Proposition 4.4], without using averaging lemmata, a better order of differentiability s < 1/(2 + ) which has been shown to be optimal by De Lellis and Westdickenberg [START_REF] Lellis | On the optimality of velocity averaging lemmas[END_REF]. In the case where A is convex, Golse and Perthame [START_REF] Golse | Optimal regularizing effect for scalar conservation laws[END_REF] provide a proof of the same regularity that could be adapted to the presence of a forcing term S. Our arguments yield this optimal order of differentiability s < 1/(2 + ) for quasi-solutions and for all A as in Corollary 3 and in the presence of the forcing term S.

(v) = sin(v) or A(v) = cos(v), u 0 ∈ L ∞ (R) and S ∈ L ∞ ([0, T ] × R). Then u ∈ W s,r loc ((0, T ) × R) ∀s < 1 3 , r ≤ 3 2 ,
despite the existence of degeneracy points, i.e. {v ∈ R : A (v) = 0} = ∅. This improves the previously known regularity of s < 1 5 [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related equations[END_REF].

Our estimates are based on the strategy introduced by Lions, Perthame and Tadmor in [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related equations[END_REF], namely applying averaging lemmas to the kinetic formulation (1.3). Accordingly, a general regularity estimate for solutions to kinetic equations will be given in the following section.

Averaging lemmas for kinetic equations. It is a well-known phenomenon that under suitable nonlinearity assumptions on the velocity field a(v), velocity averages of f solving (1.3) are more regular than f . In [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related equations[END_REF], Lions, Perthame and Tadmor use the following assumption: there exists an α ∈ (0, 1] such that for every bounded interval I ⊂ R v and all δ > 0,

(1.9) sup

τ 2 +|ξ| 2 =1 |{v ∈ I : |τ + a(v) • ξ| < δ}| δ α .
Note that this is exactly our assumption (1.6). They prove that if (1.9) holds and f ∈ L p solves (1.3) with m ∈ L q for some p, q ∈ (1, 2], then for any bump function φ ∈ C ∞ c (I), the velocity averages

f (t, x) := ˆf (t, x, v)φ(v) dv, satisfy f ∈ W s,r loc ((0, T ) × R n ), ∀s < θ = α/p α(1/p -1/q ) + 2 , 1 r = 1 -θ p + θ q .
In [START_REF] Tadmor | Velocity averaging, kinetic formulations, and regularizing effects in quasi-linear PDEs[END_REF], Tadmor and Tao introduce the additional assumption

(1.10) sup |a (v) • ξ| : v ∈ I, τ 2 + ξ 2 = 1, |τ + a(v) • ξ| < δ δ µ .
They prove that if (1.9)-(1.10) hold, then the velocity averages satisfy

f ∈ W s,r loc , ∀s < θ = α/p α(1/p -1/q ) + 2 -µ , 1 r = 1 -θ p + θ q .
However, in many cases of interest this additional assumption is not satisfied. As an example let us consider the velocity field a(v) = v for some ≥ 1. Then (1.9) holds with α = 1 and this is used in [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related equations[END_REF] to obtain that entropy solutions of (1.1) enjoy differentiability of order s = 1/(1 + 2 ). On the other hand, choosing ξ = -τ = 1/ √ 2 and v = 1 in (1.10) shows that one cannot do better than µ = 0. Hence, for a(v) = v the result in [START_REF] Tadmor | Velocity averaging, kinetic formulations, and regularizing effects in quasi-linear PDEs[END_REF] can not provide any improvement on [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related equations[END_REF]. Theorem 1 will be obtained as a corollary of a general averaging lemma for the kinetic equation

(1.11) ∂ t f + a(v) • ∇ x f = ∂ v g + h on R t × R n x × R v .
As outlined above, our argument is based on the idea underlying the assumption (1.10) in [START_REF] Tadmor | Velocity averaging, kinetic formulations, and regularizing effects in quasi-linear PDEs[END_REF] but requires a finer decomposition, relying on our technical assumptions (1.5)-(1.8).

Theorem 6. Let a ∈ C 1 (R; R n ) satisfy (1.5)-(1.8). Let p, q ∈ [1, 2] with p ≥ q, γ ∈ [0, 1] and σ ∈ [0, 1). Assume that f ∈ L p loc (R t × R n x ; W σ,p loc (R v )) solves the kinetic equation (1.11) with (1.12) h, (1 + dist(v, Z) -γ )g ∈ L q loc (R t × R n x × R v ) if q ∈ (1, p], M loc (R t × R n x × R v ) if q = 1.
Then, for any

φ ∈ C ∞ c (R), the average f (t, x) = ´f (t, x, v)φ(v) dv satisfies f ∈ W s,r loc (R t × R n x ) ∀s ∈ [0, s * ),
where the order of differentiability s * is given by

s * = (1 -η)θ α + ηθ β , θ a = a/p a(1/p -1/q ) + 2 (a = α, β), η = E 1 E 1 + E 2 , p ∈ [ p 1 + σp , p ] ∩ (1, ∞), E 1 = min (κ + γ), 1 α -(1 -γ) θ α , E 2 = max 2τ β -κ -γ, τ -1 β + 1 -γ, 0 θ β ,
and the order of integrability r is given by

1 r = 1 -η r α + η r β , 1 r a = 1 -θ a p + θ a q (a = α, β).
The proof of Theorem 6 consists in splitting the velocity average into velocities which are close to the degeneracy set {v ∈ R : dist(v, Z) ≤ λ} and far away from it {v ∈ R : dist(v, Z) ≥ λ}. Close to Z, assumption (1.6) only allows us to obtain a differentiability of order θ α by arguing as in [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related equations[END_REF], but assumption (1.7) allows us (in the spirit of [START_REF] Tadmor | Velocity averaging, kinetic formulations, and regularizing effects in quasi-linear PDEs[END_REF]) to estimate the corresponding norms with λ E 1 . Away from Z, assumption (1.8) allows us to obtain differentiability of the better order θ β , with a corresponding estimate in λ -E 2 . Then optimizing the choice of λ yields the conclusion.

Notation. All along the proofs of the main results Theorem 6 and Theorem 1 in Section 2 below, the functions f, g, h will be fixed, as well as the cut-off function φ, and we will systematically denote by C a constant that may depend on these functions and may change from line to line, but that will be independent of the interpolation parameters λ and δ to be introduced. Further, F = F t,x denotes the Fourier transform in the (t, x) variables and for (τ, ξ) ∈ R n+1 let (τ , ξ ) := 1

τ 2 + |ξ| 2 (τ, ξ), so that (τ ) 2 + |ξ | 2 = 1.
For p ≥ 1 we let p be the conjugate exponent, that is,

1 p + 1 p = 1. For Z ⊆ R, dist(v, Z) := inf z∈Z |v -z|.
Structure of the paper. The plan of the paper is as follows. In Section 2 we present the proof of the main results Theorem 6 and Theorem 1. Some background material on scalar conservation laws with an L 1 -force is presented in Appendix A. In Appendix B we recall a basic L p estimate for Fourier multipliers.

Proofs of the main results

Reduction to Z ∩ supp φ = {0} and localization. If Z ∩ supp φ = ∅ then Theorem 6 does not improve on [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related equations[END_REF], so we may assume that Z ∩ supp φ contains at least one element. If Z ∩ supp φ = {v 1 , . . . , v N }, we may choose a smooth partition of unity

φ 1 (v)+• • •+φ N (v) = 1 such that Z ∩ supp φ j = {v j } for all j ∈ {1, . . . , N }. Since f = f1 + • • • + fN with fj = ´f (t, x, v)φ(v)φ j (v)
dv, it suffices to prove Theorem 6 in the case where Z ∩ supp φ contains exactly one element. Translating v, we may moreover assume that this element is 0. Note that we may moreover assume that f, g, h have compact support: for φ(t, x, v) smooth and compactly supported, the function f = φf is compactly supported and satisfies (2.1)

∂ t f + a(v) • ∇ x f = ∂ v g + h, with h = f ∂ t φ + a(v)f ∇ x φ -(∂ v φ)g + hφ g = φg.
We note that h, g are compactly supported and satisfy (1.12) since q ≤ p. Hence, the assumptions (1.7)-(1.12) become sup

v∈I, |v|≤λ |a (v)| λ κ , (2.2) sup τ 2 +|ξ| 2 =1 |{v ∈ I : |v| ≥ λ, |τ + a(v) • ξ| ≤ δ}| λ -τ δ β , (2.3) h, (1 + |v| -γ )g ∈ L q (R t × R n x × R v ) if q ∈ (1, 2], M(R t × R n x × R v ) if q = 1.
(2.4)

Separating small and large velocities. We fix a bounded interval

I ⊂ [-Λ, Λ] ⊂ R and a bump function φ ∈ C ∞ c (I). We further fix a cut-off function η 1 ∈ C ∞ c (R) satisfying η 1 (v) ∈ [0, 1] for all v ∈ R, η 1 (v) ≡ 1 for |v| ≤ 1, η 1 (v) ≡ 0 for |v| ≥ 2.
Then we set η 2 := 1 -η 1 , so that for any λ > 0 it holds

f (t, x) = ˆf (t, x, v)φ(v)η 1 ( v λ ) dv + ˆf (t, x, v)φ(v)η 2 ( v λ ) dv =: A λ 1 f + A λ 2 f.
(2.5)

Note that for all λ ≥ Λ we have A λ 1 f = f and A λ 2 f = 0 so that in the sequel we will only need to consider λ ≤ Λ. Since A λ 2 f does not see small velocities, we could use assumption (2.3) and obtain from [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related equations[END_REF] that A λ 2 f has differentiability of order s = θ β . In contrast, for A λ 1 f we can only use (1.6) to see that it has differentiability of order s = θ α < θ β . But our assumptions allow us to take advantage of the fact that A λ 1 f only sees small velocities in two ways: first, by using that a (v) is small thanks to (2.2) -along the idea that led to introducing the assumption (1.10) in [START_REF] Tadmor | Velocity averaging, kinetic formulations, and regularizing effects in quasi-linear PDEs[END_REF]; and second, by using the finite singular moment assumption (2.4) on g. That way we find that the estimate for A λ 1 f comes with a constant that goes to zero when λ approaches zero (cf. Lemma 7 below). On the other hand, the estimate for A λ 2 f comes with a constant that blows up when λ approaches zero (cf. Lemma 8 below).

Lemma 7. For all s ∈ [0, θ α ) there exists a constant C > 0 such that for any λ ≤ Λ it holds

A λ 1 f W s,rα ≤ Cλ E 1 ,
where E 1 is given by

E 1 = min (κ + γ), 1 α -(1 -γ) θ α .
Proof. The proof will follow the strategy of [19, Averaging Lemma 2.1], the main difference residing in the fact that we want to keep track of the dependence on λ of all the estimates. We fix ψ 0 (z) supported in |z| ≤ 2 and

ψ 1 (z) supported in 1/2 ≤ |z| ≤ 2 such that 1 ≡ ψ 0 (z) + j≥1 ψ 1 (2 -j z), ∀z ∈ C.
For any δ > 0 we decompose f as

(2.6) f = f 0 + f 1 ,
where

f 0 = F -1 ψ 0 iτ + ia(v) • ξ δ Ff, (2.7) f 1 = j≥1 F -1 ψ 1 iτ + ia(v) • ξ 2 j δ Ff = j≥1 f (j) .
Then we estimate the L p norm of A λ 1 f 0 and the Ẇ 1,q norm of A λ 1 f 1 and conclude using real interpolation. We treat first the case q > 1. Invoking Lemma 13 and using (1.6) we have

A λ 1 f 0 L p ≤ C sup τ 2 +|ξ| 2 =1 |{v ∈ I : |v| ≤ 2λ, |τ + a(v) • ξ| ≤ 2δ}| 1/p ≤ C min(δ α , λ) 1/p . (2.8) Using (1.3) in Fourier variables yields, for all (τ, ξ, v) ∈ R 2+n such that τ + iξ • a(v) = 0, Ff = 1 |(τ, ξ)| 1 iτ + iξ • a(v) F(∂ v g + h) = F(-∆ t,x ) -1/2 F -1 1 iτ + iξ • a(v) F(∂ v g + h).
Hence, setting ψ 1 (z) := ψ 1 (z)/z we find that for j ≥ 1 we have

F(-∆ t,x ) 1/2 A λ 1 f (j) = 1 2 j δ ˆ ψ 1 iτ + ia(v) • ξ 2 j δ F∂ v g φ(v)η 1 v λ dv + 1 2 j δ ˆ ψ 1 iτ + ia(v) • ξ 2 j δ Fh φ(v)η 1 v λ dv.
Integrating by parts thus yields

F(-∆ t,x ) 1/2 A λ 1 f (j) = - 1 (2 j δ) 2 ˆ ψ 1 iτ + ia(v) • ξ 2 j δ ia (v)|v| γ • ξ F|v| -γ g φ(v)η 1 v λ dv - 1 2 j δλ ˆ ψ 1 iτ + ia(v)ξ 2 j δ |v| γ F|v| -γ g φ(v)η 1 v λ dv - 1 2 j δ ˆ ψ 1 iτ + ia(v)ξ 2 j δ |v| γ F|v| -γ g φ (v)η 1 v λ dv + 1 2 j δ ˆ ψ 1 iτ + ia(v) • ξ 2 j δ Fh φ(v)η 1 v λ dv.
Invoking Lemma 13 with p = q, σ = 0, r = q , recalling that ξ is a bounded L q multiplier, that |v| -γ g ∈ L q and using (2.2), we deduce

A λ 1 f (j) Ẇ 1,q ≤ C 2 -2j δ -2 λ κ+γ (2 j δ) α q + 2 -j δ -1 λ γ-1 (2 j δ) α q + 2 -j δ -1 λ γ (2 j δ) α q + 2 -j δ -1 (2 j δ) α q ≤ C 2 -2j δ -2 λ κ+γ (2 j δ) α q + 2 -j δ -1 λ γ-1 (2 j δ) α q
.

In the second inequality we were able to discard the two last terms in the previous line because λ ≤ Λ and γ ≤ 1. Since α < q , summing over j ≥ 1 yields (2.9)

A λ 1 f 1 Ẇ 1,q ≤ C δ -2+ α q λ κ+γ + δ -1+ α q λ γ-1 .
From (2.8)-(2.9) we obtain for all t > 0 that

K(t, A λ 1 f ) := inf A λ 1 f = f 0 + f 1 f 0 L p + t f 1 Ẇ 1,q ≤ C δ α p + tδ α q -2 λ κ+γ + tδ α q -1 λ γ-1 .
Next we optimize in δ. We choose it of the form δ = t a λ b , where b will be chosen later and a is determined by balancing the powers of t in the first two terms:

a α p = 1 + a α q -2 i.e. a = p α θ α .
This gives

t -θα K(t, A λ 1 f ) ≤ C λ b α p + λ b α q -2 +κ+γ + t p α θα λ b α q -1 +γ-1 .
Note that the last term is small for small t. On the other hand for large t we can use the fact (obtained from (2.8) by sending δ → ∞) that

A λ 1 f L p ≤ Cλ 1 p ,
to deduce, for any µ > 0,

t -θα K(t, A λ 1 f ) ≤ C λ b α p + λ b α q -2 +κ+γ + µ p α θα λ b α q -1 +γ-1 for t ≤ µ, µ -θα λ 1 p for t ≥ µ.
Next we choose µ in order to balance the last terms of the above two lines, i.e.

µ = λ α α+ p 1 θα 1 p +b 1-α q +1-γ ,
and conclude that

t -θα K(t, A λ 1 f ) ≤ C λ b α p + λ b α q -2 +κ+γ + λ 1 α+ p 1-α(1-γ)-bα 1-α q .
Finally we want to choose b to optimize the above powers of λ : set

E 1 := sup b∈R min                α p b κ + γ -2 - α q b 1 α + p 1 -α(1 -γ) -α 1 - α q b               
.

We denote by L 1 (b), L 2 (b), L 3 (b) the three affine functions of b appearing in the definition of E 1 . Since L 1 is increasing and L 2 , L 3 are decreasing, the function min(L 1 , L 2 , L 3 ) is bounded from above, thus E 1 < +∞. Moreover E 1 is given by

E 1 = min(L 1 (L 1 = L 2 ), L 1 (L 1 = L 3 )) = min (κ + γ), 1 α -(1 -γ) θ α .
Then, denoting by • θ the norm in the real interpolation space [L p , Ẇ 1,q ] θ,∞ (see e.g. [START_REF] Bergh | Interpolation spaces. An introduction[END_REF] for definition and properties), we have

A λ 1 f θα ≤ Cλ E 1 ,
which implies the conclusion of Lemma 7.

In the case q = 1, we obtain the same estimates, but the space Ẇ 1,q = (-∆ t,x ) -1/2 L q has to be replaced with (-∆ t,x ) -1/2 M. Since this space contains Ẇ s,1 for all s < 1 we still obtain the conclusion.

Lemma 8. For all s ∈ [0, θ β ) there exists C > 0 such that for any

λ ≤ Λ it holds A λ 2 f W s,r β ≤ Cλ -E 2
, where E 2 is given by

E 2 = max 2τ β -κ -γ, τ -1 β + 1 -γ, 0 θ β .
Proof. As in the proof of Lemma 7 we consider the decomposition (2.6) and treat first the case q > 1.

Let η(v) = η 1 (v/2) -η 1 (v), so that η is supported inside {1 ≤ |v| ≤ 4} and η 2 (v) = k≥0 η(v/2 k ). Hence, it holds

A λ 2 f = k≥0 A (k) 2 f, A (k) 2 f = ˆf (t, x, v)φ(v) η v 2 k λ dv.
Next we estimate

A (k) 2 f θ β
. Fix k ≥ 0 and let µ := 2 k λ, so that

A (k) 2 f = ˆf (t, x, v)φ(v) η(v/µ) dv = j≥1 ˆf (j) (t, x, v)φ(v) η(v/µ) dv =: j≥1 A (k) 2 f (j) ,
with f (j) defined as in (2.7). Analogously,

A (k) 2 f 0 = ˆf 0 (t, x, v)φ(v) η(v/µ) dv. Note that A (k) 2 f is nonzero only for k such that µ = 2 k λ ≤ Λ, since φ is supported in [-Λ, Λ]
and η(v/µ) vanishes for |v| ≤ µ. By Lemma 13 and assumption (2.3) it holds

A (k) 2 f 0 L p ≤ C sup τ 2 +ξ 2 =1 |{v ∈ I : 4µ ≥ |v| ≥ µ, |τ + a(v)ξ| ≤ 2δ}| 1/p ≤ C min(µ -τ δ β , µ) 1 p . (2.10)
As in the proof of Lemma 7 we have

F(-∆ t,x ) 1/2 A (k) 2 f (j) = - 1 (2 j δ) 2 ˆ ψ 1 iτ + ia(v) • ξ 2 j δ ia (v)|v| γ • ξ F|v| -γ g φ(v) η v µ dv - 1 2 j δµ ˆ ψ 1 iτ + ia(v) • ξ 2 j δ |v| γ F|v| -γ g φ(v) η v µ dv - 1 2 j δ ˆ ψ 1 iτ + ia(v) • ξ 2 j δ |v| γ F|v| -γ g φ (v) η v µ dv + 1 2 j δ ˆ ψ 1 iτ + ia(v) • ξ 2 j δ Fh φ(v) η v µ dv
which yields, using assumptions (2.2)-(2.4),

(2.11)

A (k) 2 f 1 Ẇ 1,q ≤ C δ -2+ β q µ -τ q +κ+γ + δ -1+ β q µ -1-τ q +γ .
Here as in the proof of Lemma 7 we estimated the third and fourth term by the second term on the right hand side since they come with higher powers of µ 1. The estimates (2.10)-(2.11) then imply

K(t, A (k) 2 f ) := inf A (k) 2 f = f 0 + f 1 f 0 L p + t f 1 Ẇ 1,q ≤ C µ -τ p δ β p + tδ β q -2 µ -τ q +κ+γ + tδ β q -1 µ -1-τ q +γ .
Equilibrating the first and the second term yields the choice

δ = t p β θ β µ b . Since A (k) 2 f L p ≤ µ 1 p we also have K(t, A (k) 2 f ) 1 for all t ≥ 0. We thus obtain t -θ β K(t, A (k) 2 f ) ≤ C µ -τ p +b β p + µ -τ q +κ+γ-b 2-β q + ν p β θ β µ -1-τ q +γ-b 1-β q for t ≤ ν, ν -θ β µ 1 p for t ≥ ν. We choose ν = µ 1 θ β β β+ p 1-γ+ τ q +b 1-β q + 1 p to deduce t -θ β K(t, A (k) 2 f ) ≤ C µ -τ p +b β p + µ -τ q +κ+γ-b 2-β q + µ -1 β+ p -1+β 1-γ+ τ q +β 1-β q b .
Then optimizing in b we set

E = inf b∈R max                  τ p - β p b τ q -κ -γ + 2 - β q b 1 β + p -1 + β 1 -γ + τ q + β 1 - β q b                 
, and obtain (recall µ ≤ Λ) that (2.12)

A (k) 2 f θ β ≤ Cµ -E = 2 -kE λ -E . We denote by L 1 (b), L 2 (b), L 3 (b) the three affine functions of b appear- ing in the definition of E. Since L 1 is increasing while L 2 , L 3 are in- creasing, the function max(L 1 , L 2 , L 3 ) is bounded from below, thus E > -∞. Moreover it holds E = max(L 1 (L 1 = L 2 ), L 1 (L 1 = L 3 )) = max 2τ β -κ -γ, τ -1 β + 1 -γ θ β .
If E > 0, then summing (2.12) over k ≥ 0 yields

A λ 2 f θ β ≤ Cλ -E .
If E ≤ 0, then summing (2.12) over those

k satisfying µ = 2 k λ ≤ Λ yields A λ β f θ 2 ≤ Cλ -E 0≤k≤log(Λ/λ) (2 -E ) k ≤ Cλ -E 2 E log(λ/Λ) ≤ C.
Hence we conclude that A λ 2 f θ β ≤ Cλ -max(E,0) .

To treat the case q = 1 we argue as in the proof of Lemma 7.

Proofs of Theorem 6 and Theorem 1.

Proof of Theorem 6. By (2.5), Lemma 7 and Lemma 8, for λ 1 and t ≥ 0,

K(t, f ) := inf f = f 0 + f 1 f 0 θα + t f 1 θ β ≤ A λ 1 f θα + t A λ 2 f θ β ≤ C λ E 1 + tλ -E 2 .
Choosing, λ = t

1 E 1 +E 2 yields K(t, f ) ≤ Ct E 1 E 1 +E 2 = t η ∀t 1.
Since f θα ≤ C (as can be seen e.g. by choosing λ = Λ in Lemma 7) we have

K(t, f ) ≤ C f θα ≤ C ∀t ≥ 0. Hence, f belongs to the real interpolation space [L p , Ẇ 1,q ] θ 1 ,∞ , [L p , Ẇ 1,q ] θ 2 ,∞ η,∞ = [L p , Ẇ 1,q ] θ,∞ ,
where θ = (1 -η)θ α + ηθ β and the equality follows from the reiteration Theorem of real interpolation. We further note that this space contains W s,r for all s < s * = θ. This argument works for q > 1 and for q = 1 we may adapt it as in the proof of Lemma 7.

Proof of Theorem 1. We apply the kinetic formulation for (1.1) (cf. Appendix A), that is,

f = 1 0<v<u(t,x) -1 0>v>u(t,x) ,
satisfies, in the sense of distributions, (2.13)

∂ t f + a(v) • ∇ x f = ∂ v m + δ v=u S on [0, T ] × R n x × R v for some Radon measure m ≥ 0. We further note that f ∈ L 1 ([0, T ] × R n x × R v ) ∩ L ∞ ([0, T ] × R n x × R v ) and f ∈ L ∞ ([0, T ] × R n x ; BV (R v )). Hence, by interpolation, f ∈ L 2 loc ([0, T ] × R n x ; W σ,2 (R v )) for all σ ∈ [0, 1 2 ).
For a bounded interval I ⊆ R v let Z ∩I = {z 1 , . . . , z N }. By Proposition 12 below, |v -z i | α-1 m has locally finite mass for every α ∈ (0, 1) and i ∈ 1, . . . , N . It follows that dist(v, Z) -γ m has locally finite mass for any γ ∈ (0, 1). [START_REF] Bergh | Interpolation spaces. An introduction[END_REF] ). We now apply Theorem 6 with p = 2, q = 1, any σ ∈ [0, 1 2 ) and any γ ∈ [0, 1), to obtain, for all φ ∈ C ∞ c (R), ˆf φ dv ∈ W s,r loc for all s < s * , where r is as in the conclusion of Theorem 1, and the value of s * can be choosen, depending on σ and γ, in the way described by the conclusion of Theorem 6. It can be checked directly that taking σ arbitrarily close to 1 2 and γ arbitrarily close to 1 allows to take s * arbitrarily close to the value given by the conclusion of Theorem 1.

Let η ∈ C ∞ c (0, T ). Then f := f η satisfies (1.11) with g = mη, h = δ v=u Sη + ηf and f ∈ L 2 loc (R t × R n x ; W σ,2 (R v )) for all σ ∈ [0, 1

Appendix A. Kinetic solutions for scalar conservation laws with a force

In this section we present some brief comments on the extension of the concept of kinetic solutions and their well-posedness for scalar conservation laws with an L 1 -force (1.1). This proceeds along the lines of [START_REF] Chen | Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations[END_REF][START_REF] Perthame | Kinetic formulation of conservation laws[END_REF]. We will refer to kinetic solutions also as entropy solutions.

Definition 9. A kinetic/entropy solution to (1.1) is a function u ∈ C([0, T ]; L 1 (R n )) such that the corresponding kinetic function

f (t, x, v) = χ(v, u(t, x)) = 1 0<v<u(t,x) -1 0>v>u(t,x)
satisfies, in the sense of distributions,

∂ t f + a(v) • ∇ x f = ∂ v m + δ u=v S on (0, T ) × R n (A.1)
f |t=0 = χ(v, u 0 ) on R n ,
where a := A , m is a non-negative Radon measure and

ˆm(t, x, v) dtdx ≤ µ(v) ∈ L ∞ 0 (R),
where L ∞ 0 (R) denotes the space of all essentially bounded functions decaying to zero for |v| → ∞.

Remark 10. For a comparison of the concept of renormalized entropy solutions introduced in [START_REF] Bénilan | Renormalized entropy solutions of scalar conservation laws[END_REF] and kinetic solutions we refer to [START_REF] Ishikawa | The relationship between kinetic solutions and renormalized entropy solutions of scalar conservation laws[END_REF]. A proof that entropy solutions give rise to kinetic solutions is contained in the proof of Theorem 11 below.

Theorem 11. Let u 0 ∈ L 1 (R n ), S ∈ L 1 ([0, T ] × R n ).
Then there is a unique kinetic solution u to (1.1). For two kinetic solutions u 1 , u 2 with initial conditions u 1 0 , u 2 0 and forces S 1 , S 2 respectively we have

(A.2) sup t∈[0,T ] (u 1 (t) -u 2 (t)) + L 1 (R n ) ≤ (u 1 0 -u 2 0 ) + L 1 (R n ) + S 1 -S 2 L 1 ([0,T ]×R n ) .
Proof. Contraction: We first note that the function g(t, x, v) = 1 v<u(t,x) satisfies the same kinetic equation as f , since

f (t, x, v) -g(t, x, v) = -1 v>0 -1 v=0 1 u(t,x)≥0 -1 v<0 1 u(t,x)=v , (∂ t + a(v) • ∇ x )1 v<0 = 0 in D t,x,v ,
and 1 v=0 = 1 u(t,x)=v = 0 for a.e. (t, x, v).

The proof of the contraction inequality (A.2) relies on the identity ˆg1 (1 -g 2 )dv = (u 1 -u 2 ) + .

We introduce nonnegative mollifiers Φ ε (t, x) and let the subscript ε denote the convolution in (t, x) with Φ ε . In particular, we have

(∂ t + a(v) • ∇ x )g ε = ∂ v m ε + δ v=u(t,x) S(t, x) ε ,
where δ v=u(t,x) S(t, x) ε is the distribution given by δ v=u(t,x) S(t, x) ε , θ(t, x, v)

= ˆS(t, x) ˆθ(s, y, u(t, x))Φ ε (s -t, y -x)dsdydxdt.

We also introduce a nonnegative cut-off function χ(v). By dominated differentiation, for any ε 1 , ε 2 > 0 we have

T := ∂ t ˆg1 ε 1 (1 -g 2 ε 2 )χ(v)dv + ∇ x • ˆg1 ε 1 (1 -g 2 ε 2 )χ(v)a(v)dv = ˆχ(v)(1 -g 2 ε 2 )(∂ t + a(v) • ∇ x )g 1 ε 1 dv + ˆχ(v)g 1 ε 1 (∂ t + a(v) • ∇ x )(1 -g 2 ε 2 )dv = lim δ→0 T 1 δ + T 2 δ ,
where

T 1 δ (t, x) = ˆχ(v)(1 -g 2 ε 2 (t, x, w))(∂ t + a(v) • ∇ x )g 1 ε 1 (t, x, v)ρ δ (v -w)dvdw, T 2 δ (t, x) = ˆχ(v)g 1 ε 1 (t, x, w)(∂ t + a(v) • ∇ x )(1 -g 2 ε 2 (t, x, v))ρ δ (v -w)dvdw,
and ρ δ (v) is an even nonnegative mollifier. Using the equation satisfied by g 1 we have, for any nonnegative test function θ(t, x),

T 1 δ , θ = -ˆm1 ε 1 (dt, dx, dv)θ(t, x)χ (v) ˆ(1 -g 2 ε 2 (t, x, w))ρ δ (v -w)dvdw -ˆm1 ε 1 (dt, dx, dv)θ(t, x)χ(v) ˆ(1 -g 2 ε 2 (t, x, w))(ρ δ ) (v -w)dvdw + ˆθ(t, x) ˆS1 (s, y)Φ ε 1 (t -s, x -y)χ(u 1 (s, y)) • ˆ(1 -g 2 ε 2 (t, x, w))ρ δ (u 1 (s, y) -w)dwdsdydtdx.
The second term on right-hand side is nonpositive since w → (1g 2 ε 2 (t, x, w)) = (1 w≥u 2 (t,x) ) ε 2 is nondecreasing. Moreover, since ρ δ is even, for any (t, x, v) we have as δ → 0,

ˆ(1 -g 2 ε 2 (t, x, w))ρ δ (v -w) dw = ˆΦε 2 (t -s, x -y) ˆ1w≥u 2 (s,y) ρ δ (v -w) dwdsdy → ˆΦε 2 (t -s, x -y) sgn + 1 2 (v -u 2 (s, y)) dsdy = [sgn + 1 2 (v -u 2 )] ε 2 (t, x),
where sgn

+ 1 2 (z) = 1 (0,∞) (z) + 1 2 1 {0} (z). Hence, we find lim sup δ→0 T 1 δ , θ ≤ ˆθ(t, x)|χ (v)|m 1 ε 1 (dt, dx, dv) + ˆθ(t, x) ˆS1 (s, y)Φ ε 1 (t -s, x -y)χ(u 1 (s, y)) • [sgn + 1 2 (u 1 (s, y) -u 2 )] ε 2 (t, x) dsdydtdx. A similar computation shows lim sup δ→0 T 2 δ , θ ≤ ˆθ(t, x)|χ (v)|m 2 ε 2 (dt, dx, dv) -ˆθ(t, x) ˆS2 (s, y)Φ ε 2 (t -s, x -y)χ(u 2 (s, y)) • [sgn + 1 2 (u 1 -u 2 (s, y))] ε 1 (t, x) dsdydtdx.
By Fatou's lemma these inequalities imply

T, θ ≤ ˆ|χ (v)|m 1 ε 1 (•, •, dv), θ + ˆ|χ (v)|m 2 ε 2 (•, •, dv), θ + S 1 χ(u 1 ) sgn + 1 2 (u 1 -u 2 ) ε 2 ε 1 , θ -S 2 χ(u 2 ) sgn + 1 2 (u 1 -u 2 ) ε 1 ε 2 , θ .
Next we "integrate" this inequality in x, that is, we apply it to a test function θ(t, x) = ζ(t)K(x) ≥ 0 and let K(x) approach K ≡ 1. Note that since ˆ|g

1 (s, y, v)| • |1 -g 2 (s , y , v)| dv = (u 1 (s, y) -u 2 (s , y )) + , for any θ(t, x) ∈ L ∞ and ψ(v) ∈ L ∞ loc we have ˆ g 1 ε 1 (1 -g 2 ε 2 )θ(t, x)ψ(v)χ(v) dtdxdv ≤ ψ L ∞ (supp χ) θ L ∞ u 1 L 1 t,x + u 2 L 1 t,x .
Using this together with S j ∈ L 1 and ´mj (dt, dx, dv)|χ (v)| < ∞, and letting K(x) approach K ≡ 1 nicely enough, we obtain

∂ t ˆg1 ε 1 (1 -g 2 ε 2 )χ(v) dxdv ≤ ˆ|χ (v)|m 1 ε 1 (•, dx, dv) + ˆ|χ (v)|m 2 ε 2 (•, dx, dv) + ˆ S 1 χ(u 1 ) sgn + 1 2 (u 1 -u 2 ) ε 2 ε 1 dx - ˆ S 2 χ(u 2 ) sgn + 1 2 (u 1 -u 2 ) ε 1 ε 2 dx.
The same integrability properties also allow to let ε 1 , ε 2 → 0 and to find

∂ t ˆg1 (1 -g 2 )χ(v) dxdv ≤ ˆ|χ (v)|m 1 (•, dx, dv) + ˆ|χ (v)|m 2 (•, dx, dv) + ˆS1 χ(u 1 ) sgn + 1 2 (u 1 -u 2 ) dx -ˆS2 χ(u 2 ) sgn + 1 2 (u 1 -u 2 ) dx.
We apply this inequality to a nonnegative test function ζ(t) and choose χ = χ n for a sequence χ n → 1 a.e. with χ n (v) ≡ 0 for |v| ≤ n and |χ n | ≤ 1. Then the first two terms in the right-hand side are estimated by

ζ L ∞ • sup |v|>n µ 1 (v) + µ 2 (v) ,
which tends to 0 as n → ∞ since µ j ∈ L ∞ 0 . The two last terms converge by dominated convergence, which yields

∂ t ˆg1 (1 -g 2 ) dxdv ≤ ˆ(S 1 -S 2 ) sgn + 1 2 (u 1 -u 2 ) dx.
Applying this to a nonnegative test function ζ approaching ζ = 1 [0,t] and using that u

j ∈ C([0, T ], L 1 (R n x )), we conclude that ˆ(u 1 (t) -u 2 (t)) + dx ≤ ˆ(u 1 0 -u 2 0 ) + dx + S 1 -S 2 L 1 ((0,t)×R n ) .
Interchanging the roles of u 1 , u 2 thus yields [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF] and [16, Corollary 2.5] there is a unique entropy solution u ε ∈ C([0, T ]; L 1 (R n )) to (1.1) with initial condition u ε 0 and force S ε . To see that u ε is also a kinetic solution we follow the arguments given in [START_REF] Perthame | Kinetic formulation of conservation laws[END_REF]: Let

(A.3) sup t∈[0,T ] u 1 (t) -u 2 (t) L 1 (R n ) ≤ u 1 0 -u 2 0 L 1 (R n ) + S 1 -S 2 L 1 ([0,T ]×R n ) . Existence: Let u ε 0 ∈ W 1,∞ (R n ), S ε ∈ W 1,∞ ([0, T ] × R n ) with u ε 0 → u 0 in L 1 (R n ) and S ε → S in L 1 ([0, T ] × R n ). By
f ε (t, x, v) = χ(v, u ε (t, x)) = 1 0<v<u ε (t,x) -1 0>v>u ε (t,x)
and define the distribution m by

m ε := ˆv 0 ∂ t f ε dw + ˆv 0 a(w) • ∇ x f ε dw - ˆv 0 δ u ε =w S ε dw.
Hence, f ε satisfies (A.1) and it remains to show that m ε is a nonnegative measure. Given η ∈ C ∞ c (R) convex we obtain, in the sense of distributions, (A.4)

-ˆmε η (v) dv = ˆ∂t f ε η (v) dv + ˆη (v)a(v) • ∇ x f ε dv - ˆη (v)δ u ε =v S ε dv = ∂ t η(u ε ) + divA η (u ε ) -η (u ε )S ε .
Since u ε is an entropy solution, the right hand side is non-positive. This implies that m ε is a non-negative distribution and thus a measure. An analogous equality is satisfied for (u ε (t) -k) -. Hence, with

µ ε (k) :=1 k≥0 ( u ε (0) -k) + L 1 x + sgn + (u ε -k)S ε L 1 t,x ) +1 k≤0 ( u ε (0) -k) -L 1 x + sgn -(u ε -k)S ε L 1 t,x )
we have µ ε ∈ L ∞ 0 (R) by dominated convergence, and ˆmε (t, x, k) dtdx ≤ µ ε (k).

Hence, u ε is a kinetic solution.

By (A.3) we have

u ε -u δ C([0,T ];L 1 (R n )) ≤ u ε 0 -u δ 0 L 1 (R n ) + 2 S ε -S δ L 1 ([0,T ]×R n ) .
Since u ε and S ε were chosen to converge in L 1 (R n ) and L 1 ([0, T ] × R n ), this implies that u ε is a Cauchy sequence in C([0, T ]; L 1 (R n )). Hence, there is a u ∈ C([0, T ]; L 1 (R n )) and a sequence ε k converging to zero such that u ε k → u in C([0, T ]; L 1 (R n )) and almost everywhere in [0, T ] × R n . Moreover, by (A.5) we obtain that m ε k has locally uniformly bounded mass. Thus, choosing a diagonal sequence we obtain a subsequence (again denoted by ε k ) such that m ε k * m in the space of measures on R + × R n × R. It is then easy to see that u is a kinetic solution to (1.1).

Proposition 12. Let u 0 ∈ L 1 (R n ), S ∈ L 1 ([0, T ] × R n ) and u be the corresponding entropy solution to (1.1). For each α ∈ (0, 1) and each v 0 ∈ R the measure |v -v 0 | α-1 m has locally finite mass.

Proof. Let α ∈ (0, 1). Let φ be a non-negative smooth compactly supported function in (0, T )×R n ×R. Then (A.1) implies, with f := φf , m := φm,

(A.6) ∂ t f +a(v)•∇ x f = ∂ v m-(∂ v φ)m+φδ u=v S +∂ t φf +(a(v)•∇ x φ)f.
By translation we may assume v 0 = 0. We next choose a sequence of smooth, compactly supported functions η ε such that sgn(η ε (v)) = sgn(v), η ε (v) ≤ 1 α |v| α and (η ε ) ↑ |v| α-1 pointwise. Multiplying (A.6) by η ε and integrating yields ˆ(η ε f )(t) dxdv + ˆt 0 ˆ(η ε ) m dxdv dr = ˆx,v (η ε f )(0)

+ ˆt 0 ˆ -η ε (∂ v φ)m + η ε φδ u=v S + η ε ∂ t φf + η ε (a(v) • ∇ x φ)f dxdv dr.
Since m ≥ 0, by Fatou's Lemma we may pass to the limit ε → 0 to obtain ˆt 0 ˆ|v| α-1 m dxdv dr ≤C < ∞ for some constant C depending on u 0 L 1 loc , m M loc , S L 1 loc .

  8) with α = 1/2, β = 1 and κ = τ = 1 (cf. Example 5 below). (5) Our model one-dimensional velocity field a(v) = v satisfies (1.5)-(1.8) with α = 1/ , β = 1 and κ = τ = -1.

Example 5 .

 5 Consider (1.1) with flux A

  A standard approximation argument (cf. e.g.[START_REF] Perthame | Kinetic formulation of conservation laws[END_REF], Proposition 3.2.3]) allows to choose η (v) = sgn + (v -k) in (A.4), which yields (A.5) ˆ(u ε (t) -k) + dx + ˆmε (t, x, k) dtdx = ˆ(u ε (0) -k) + dx + ˆsgn + (u ε -k)S ε dtdx.

Appendix B. A basic estimate

From [START_REF] Gess | Sobolev regularity for the porous medium equation with a force[END_REF] we recall the following basic L p -estimate for certain Fourier multipliers. This result generalizes [START_REF] Tadmor | Velocity averaging, kinetic formulations, and regularizing effects in quasi-linear PDEs[END_REF]Lemma 2.2] by taking into account possible v-regularity of f . This allows to avoid bootstrapping arguments in the application to scalar conservation laws. This is crucial in the case of scalar conservation laws with L 1 -forcing, since in this case bootstrapping arguments do not apply.

Lemma 13. Let m(τ , ξ , v) := iτ + ia(v) • ξ , ϕ, φ be bounded, smooth functions, ψ be a smooth cut-off function and M ψ be the Fourier multi-

where

Lemma 13 relies on the fact that ψ iτ +ia(v)•ξ δ is a bounded L p (and M) multiplier uniformly in v ∈ I and δ > 0 (the truncation property in [START_REF] Tadmor | Velocity averaging, kinetic formulations, and regularizing effects in quasi-linear PDEs[END_REF]). This can be deduced, arguing as in [START_REF] Diperna | L p regularity of velocity averages[END_REF], from the invariance of the L p multiplier norm under partial dilations and the Marcinkiewicz multiplier theorem. For details we refer to [START_REF] Gess | Sobolev regularity for the porous medium equation with a force[END_REF]Lemma A.3].