NON-CRITICAL FRACTIONAL CONSERVATION LAWS IN DOMAINS WITH BOUNDARY

. We study bounded solutions for a multidimensional conservation law coupled with a power s ∈ (0 , 1) of the Dirichlet laplacian acting in a domain. If s ≤ 1 / 2 then the study centers on the concept of entropy solutions for which existence and uniqueness are proved to hold. If s > 1 / 2 then the focus is rather on the C ∞ -regularity of weak solutions. This kind of results is known in R N but perhaps not so much in domains. The extension given here relies on an abstract spectral approach, which would also allow many other types of nonlocal operators.


1.
Introduction. This short note reports on some joint work with Nathael Alibaud on the perturbation of scalar conservation laws by nonlocal operators in domains. Complete proofs and technical details will appear in two forthcoming papers [3] [4]. The purpose of this note is rather to give an overview of the general program of study, which covers problems of both hyperbolic and parabolic nature.
In an open bounded subset Ω ⊂ R N (N ≥ 1), we consider the Cauchy problem for a scalar conservation law perturbed by a fractional power (0 < s < 1) of the Dirichlet laplacian in Ω, i.e.    ∂ t u(t, x) + div x f (u(t, x)) + (−∆ x ) s u(t, x) = 0 (t, x) ∈ I × Ω, u(0, x) = u 0 (x) x ∈ Ω, u(t, x) = 0 (t, x) ∈ I × ∂Ω. (1) Here the time interval I = [0, T ] is finite (0 < T < ∞) and the flux function f : R → R N is assumed smooth (say f ∈ C ∞ like in Burgers's equation). The problem is set in the framework of bounded real-valued solutions u = u(t, x) of time and space. The Dirichlet condition u = 0 applied on the boundary ∂Ω is for the time being only formal.
To the best of our knowledge, (1) has mainly been investigated so far for homogeneous spaces Ω = R N only. Our present goal is to extend some existing results on R N to domains by means of abstract functional analysis on the power, rather than through an explicit representation of (−∆) s as a space integral as is used in most works on the subject.
In terms of differential order, 0 < s < 1/2 resp. 1/2 < s < 1 corresponds to a hyperbolic resp. parabolic problem, while the value s = 1/2 of the power is critical in the sense that it is related to a change of PDE class. our method consists in replacing the Dirichlet laplacian by a general second-order elliptic operator with nonconstant coefficients. This can be done at the price of very minor changes at the start and does not affect the final results. A less academic question is to enlarge the class of powers of second-order elliptic operators by identifying clearly the few qualitative operator-properties of the resolvent (λ − ∆) −1 really needed for our techniques. Even the (integro)differential character of ∆ does not seem essential, if (λ − ∆) −1 has some 'good' properties. Highlighting these properties is still an open question out of the scope of this note.
It is kwown that s-subordination applied to the original semigroup defines an analytic C 0 -semigroup of operators on E with the same uniform bound Remark 1. The analyticity of S s holds even if S is not analytic itself.
At the level of resolvents, there is also a nice formula based on some convex combination These two identities (2)-(3) in L(E) provide an easy way to pass from to the pure case s = 1 to the fractional case s ∈ (0, 1) as far as semigroups and resolvents are concerned. On the contrary, the infinitesimal generator A s of the semigroup (S s (t)) t≥0 is less easy to handle, in the sense that the corresponding integral representations u − K λ u dλ λ 1−s (4) are more singular, and only make sense for special elements u of E. Typically, the belonging u ∈ D(A) to the domain of A itself is a sufficient condition for both integrals to converge in E, whatever s ∈ (0, 1) be.
We now turn to deeper considerations on the characterization of the domain of A s , and recall its link with interpolation theory. Let the domain D(A) be endowed with the graph norm u D(A) := u E + Au E . In real interpolation theory as developped and reviewed in Chapter III of [6] for instance, the K-method consists in defining a large variety of "intermediate spaces" between E and D(A) by means of the so-called K t functional Among those numerous spaces are two extreme examples A far-reaching result on the characterization of domains for general powers asserts that D(A s ) for the graph norm u D(As) := u E + A s u E is nested between those two extreme spaces, in the sense of the double inclusion This result is essentially contained in the last page of Ch.II in [6], which describes what the exact domain of the power is in terms of abstract convergence of integrals. Furthermore, a general comparison theorem (see e.g. Thm 3.2.12 and Cor 3.2.13 in [6]) between the spaces obtained by real interpolation gives an easy continuous imbedding To sum up: a relation that will allow us to have a rather good idea of what D(A s ) is, even if an exact characterization is missing. The tendency to prefer spaces of the type [E; D(A)] θ,1 (0 < θ < 1) will be explained later on by a simple coincidence of this scale of spaces with a well-known Sobolev class.
Given a bounded domain Ω ⊂ R N with regular boundary, we shall apply this general construction to the Dirichlet laplacian ∆ properly defined on L p (Ω) to create a generator (1 ≤ p ≤ ∞): denotes the subspace of continuous functions in Ω with null limit at every point of ∂Ω. For references on this classical subject see e.g. Ch VII in [18]. All these realizations give rise to an analytic C 0 -semigroup S(t) = e t∆ of contractions (M = 1) on E. Moreover, their action is coherent within these functional spaces, in the sense that the construction on L 1 (Ω) extends the others. Application of the general s-power to this setting yields an analytic C 0semigroup S s (t) = e t∆s of contractions on L 1 (Ω), which are also contractions on L p (Ω) for any 1 ≤ p ≤ ∞ and particularly contractions on L ∞ (Ω). The opposite of the corresponding generator −∆ s =: (−∆) s is known as the Dirichlet fractional laplacian on Ω. To make it clearer, we choose to define one single operator ∆ s to be the realization of the s-power in the functional space E = L 1 (Ω): it is indeed the 'largest' version of the s-power one can hope for.

Remark 2.
Another equivalent way to get ∆ s on L 1 (Ω) is to build it as the adjoint of the power defined in the C 0 0 (Ω)-setting. This seems less natural even if it could lead to a more general action on measures.
The main interest for switching from the original space L ∞ (Ω) to the wider space L 1 (Ω) is that ∆ s may now operate on functions u ∈ L ∞ (Ω) which are not necessarily zero on the boundary. More specifically, we have for any 0 < s < 1/2, for any 1/2 < s < 1, and in particular D(∆ s ) contains any constant function whenever s < 1/2 (this phenomenon may appear as rather surprising ...). The proof is essentially a combination of (5) with the trivial imbeddings W 2,1 0 (Ω) ⊂ D(∆) ⊂ W 1,1 0 (Ω). Indeed, this gives 1 , where the spaces on the left and on the right are known to be standard Sobolev spaces, namely (Ω) for any 1 ≤ p < ∞ and any 0 < θ < 1, provided that mθ is not an integer. The statement for 1 < p < ∞ may be found in books by H. Triebel or P. Grisvard, the case p = 1 being excluded for the sake of simplicity. In fact, p = 1 has nothing special in this result, even if it is really hard to find a reference in the literature.
Working in L 1 (Ω) is especially useful for this kind of simplification.

Remark 5.
More generally, the domain of the s-power built on L p (Ω) is nested between W 2s,p 0 (Ω) and W s,p 0 (Ω) for any 1 ≤ p < ∞, by the same argument (again whenever s = 1/2).
Coming back to the L ∞ -setting, the preceding discussion sheds some light on the action of ∆ s on (regular) functions with arbitrary behavior on the boundary: Note again that this result has been allowed by a fundamental change of functional spaces. In other words, using only L ∞ -norms would make it impossible to drop the Dirichlet boundary condition on u and still let ∆ s u well-defined. Roughly speaking, for the term ∆ s u to make sense, it was "natural" to think that u should meet some requirement on the boundary ∂Ω in addition to some local regularity in Ω, but this belief is only true for s > 1/2 and not so much for s < 1/2. In any case, even if the Dirichlet boundary condition disappears from the domain D(∆ s ) when s < 1/2, it is still encoded in the operator itself (which really depends on the Dirichlet realization of the laplacian).
We now return to the solutions u ∈ L ∞ (I × Ω) to the formal Cauchy problem where ∆ s has been built through (4) i.e.
Remark 6. When Ω = R N , the group invariance due to translations implies that is a convolution operator (by some Bessel-type function scaled by λ), so that (6) becomes in this case for some (irrelevant) constant c depending on N and s. This representation of ∆ s as a 'principal value integral' in space is just one realization of ∆ s among many others.
Definition 3.1. We say that u ∈ L ∞ (I × Ω) is a semigroup-solution to problem (P) when the equality being inderstood in the L ∞ (I × Ω)-sense (i.e. almost everywhere in I × Ω).
Here σ s (t) stands for the formal operator S s (t) • div extended as a bounded operator acting on the whole of L p (Ω) for any 1 ≤ p ≤ ∞ thanks to the following result: The proof of Lemma 3.2 consists in applying the s-subordination to the local case (s = 1), for which one can prove directly by different methods from local PDEs that the operator σ(t) := S(t) • div extends from C ∞ c (Ω) to L p (Ω) into a bounded operator satisfying for some constant C independent of t (and p). In fact, the same property holds for any first-order derivative applied on the left or on the right of S, namely Here the effect of s-subordination is just to exhibit the power t 1/2s out of √ t. The existence and uniqueness of semigroup-solutions results from the fact that the mapping has a unique fixed point. The proof of this consists in showing that F has some strictly contractive iterate, by using some combinatorial properties of iterated convolutions (on R * + ) with the locally integrable power t −1/2s given by Lemma 3.2. In semigroup theory, performing fixed points in Duhamel's formula is a very common idea: the existence-uniqueness of semigroup-solutions claimed here is not new and may be read as Thm 3.3.3 of [14] for instance.
In the same spirit, F can be studied w.r.t. the finer norms u → u L ∞ (I×Ω) + t 1/2s ∇ x u L ∞ (I×Ω) and u → u L ∞ (I×Ω) + t γ ∂ t u L ∞ (I×Ω) for any fixed 0 < γ < 1. As a result, the fixed point u of F is shown to have the additional regularity t 1/2s ∇ x u ∈ L ∞ (I × Ω) for any u 0 ∈ L ∞ (Ω) and t γ ∂ t u ∈ L ∞ (I × Ω) for any u 0 ∈ L ∞ (Ω) s.t. ∆ 1−γ s u 0 ∈ L ∞ (Ω). Given t > 0, Lemma 2.1 then applies to u(t, ·) ∈ W 1,∞ (Ω) for any arbitrarily small value of the power s(1 − γ) (fix γ close to 1), to the effect that ∂ t u ∈ L ∞ ([ε, T ] × Ω) (for any ε > 0). The consequence of this first step is that u has first order-derivatives in t > 0 and x ∈ Ω. This shows also that ∆ s u = ∂ t u + div(f (u)) makes sense for any t > 0, and accordingly that u evolves in the domain of the s-power viewed on any L p (Ω), since In particular, u(t) ∈ ∩ 1<p<∞ W s,p 0 (Ω) by Remark 5, and the Dirichlet boundary condition persists for any positive time: For any u 0 ∈ L ∞ (Ω), the semigroup-solution u is continuous in t > 0 and x ∈ Ω, and satisfies the Dirichlet boundary condition pointwise on ∂Ω, i.e. u ∈ C 0 (I * × Ω) : ∀t > 0 u(t) ∈ C 0 0 (Ω). A second step in the analysis consists in iterations of the additional regularity argument explained before, in order to get higher and higher regularity for u w.r.t. time and space. With this goal, it is tempting to differentiate the equation of u w.r.t. space, and try to check that the first-order derivatives ∇ x u satisfy a similar equation as u. This 'standard' way does not seem to work at all, since commuting symbols in is a really serious obstacle: quantities like f (u).∇ x u (and later on f i (u)∂ i ∂ j u + f i (u)∂ i u∂ j u ... for higher-order derivatives) do not satisfy Dirichlet boundary conditions any more. For a precise statement of the 'standard' procedure, the reader may refer to Exercises 1 and 2 p.77 of [14], which fail to apply here, not because of a loss in local regularity but because of a loss in boundary conditions (in the notation of [14], the scale of spaces (X t ) t∈R+ is destroyed by the nonlinearity u → f (u).∇u, even if its 'regularity' looks good enough). From the viewpoint of abstract functional analysis, this phenomenon is essentially due to the fact that the operator (div) governing the scalar conservation law is not obtainable by functional calculus on the generator ∆. Besides, truncations do not appear easy to handle due to the nonlocality. Instead, our idea is rather to focus on time derivatives and show that all make sense. This cascade of regularity relations is enough to make u infinitely smooth: For any u 0 ∈ L ∞ (Ω), the semigroupsolution u is infinitely regular in t > 0 and x ∈ Ω, i.e. u ∈ C ∞ (I * × Ω).
We close this part with a simple remark on semigroup and weak solutions. Combining Theorem 3.4 and Proposition 1, we get: (i) the existence of C ∞smooth solutions satisfying classically the evolution equation as well as the Dirichlet boundary condition (only the initial value u 0 ∈ L ∞ has to be interpreted weakly) and (ii) the uniqueness of weak solutions satisfying the problem in the sense of distributions (with boundary terms). This most favorable picture explains why the concept of entropy solutions, which yet makes sense for any s ∈ (0, 1), is more relevant to the hyperbolic case.
4. The hyperbolic case (2s < 1). Suppose 2s < 1 to fix ideas (in fact this section applies to any 0 < s < 1 as explained in the introduction).
for which holds the following convexity principle: This convexity principle is obviously the analog of Kato's inequality for the laplacian, namely With this end in mind, we introduce dy dλ λ 1−s , as well as a := b r + c r , and we refer to the formal identity −∆ s = a + B r + C r as the r-decomposition of ∆ s . To sum up: a is a multiplicative operator and B r is a mild integral operator, while C r acts as a kind of derivative of order 2s.
Where each term is meaningful in L 1 (Ω) individually. In general, a ∈ L 1 loc (Ω) is a nonnegative function whose global integrability is not garanteed, however the product aφ always falls in L 1 (Ω).
Remark 8. In R N , it is standard to analyze the integral representation (7) of ∆ s according to the proximity to the singularity, i.e.
Here, in some sense, a similar decomposition is made in Ω, but at a spectral level. Note also that the multiplicative part due to a vanishes in R N because of the corresponding value of the marginal R N k λ (x, y)dy = 1 in this case.

Entropy solutions.
We are now in a position to introduce the concept of entropy solution for problem (P) .
Definition 4.2. We say that u ∈ L ∞ (I × Ω) is an entropic solution to (P) if and only if (i) ∀ϕ ∈ D + (I × Ω) ∀r ∈ R * where Φ is the η-flux (Φ = η f ) and where (a, B r , C r ) is the r-decomposition of Section 4.1.
Recall that D + (I × Ω) is the positive cone of the space of Definition 3.5. Note that each term in (i) makes sense because of the boundedness of all quantities in u and because of the integrability of all quantities in ϕ (including the product aϕ of Proposition 2). Note also that the restrictions on the geometry of ∂Ω and on the BV -regularity of the solution only appear through the BLN -conditions of (ii). For BLN -conditions in purely local conservation laws, see the original paper [7].
The notion originates from the corresponding study [1] in R N . It has been designed mainly to lead to the following well-posedness and stability result: Note that the 'finite-infinite speed' estimate is a refined L 1 -contraction principle involving some localisations in balls. Therein, the symbol T s (t) refers to the semigroup generated by the s-power defined on the whole of R N and not only in Ω as before (i.e. T s (t) ∈ L[L ∞ (R N )] and S s (t) ∈ L[L ∞ (Ω)] should not be confused). For further discussion, see [1] and also [13].
The existence may be proved by a method of splitting like in [1] and [12]. The uniqueness results from the L 1 -contraction principle which can be established by Kruzhkov's method of doubling variables. For this, the general idea is to follow the argument of [1] at a spectral level (applying Lemma 4.1 to C r and optimizing in r → ∞), in order to infer from (8) the so-called Kato inequality: ∀ϕ ∈ D + (I × Ω) (9) where F (u, v) := sign(u − v)(f (u) − f (v)). Afterwards, it turns out that local and nonlocal terms do not really interfere: the BLN-condition (ii) is only used to deal with the boundary terms that come out from testing (9) against a special approximation of the constant in space (I = I Ω ) of the type φ µ := K µ I ∈ D(∆) ↑ I as µ ↑ ∞.
We close this part by a simple remark connecting Definition 4.2 to weak solutions: