Entropy solutions of doubly nonlinear fractional Laplace equations

In this contribution, we study a class of doubly nonlinear elliptic equations with bounded, merely integrable right-hand side on the whole space $\mathbb{R}^N$. The equation is driven by the fractional Laplacian $(-\Delta)^{\frac{s}{2}}$ for $s\in (0,1]$ and a strongly continuous nonlinear perturbation of first order. It is well known that weak solutions are in genreral not unique in this setting. We are able to prove an $L^1$-contraction and comparison principle and to show existence and uniqueness of entropy solutions.


Introduction
Let N ∈ N. We consider the doubly nonlinear fractional Laplace equation b(u) + div(F (u)) + (−∆) where f ∈ L 1 (R N ) ∩ L ∞ (R N ) and b : R → R is Lipschitz continuous, nondecreasing with b(0) = 0, satisfying the growth condition b(u)u ≥ λ|u| 2 for some λ > 0. For example, these assumptions are fulfilled by b = Id + arctan, and it is also possible for b to be constant on finite intervals. The function F : R → R N is locally Lipschitz continuous with F (0) = 0 and (−∆) s 2 is the fractional Laplacian with s ∈ (0, 1] (see Section 2 for the precise definition).
The fractional Laplacian is a nonlocal generalization of the classical Laplacian which appears in many fields of analysis and probability theory. In the last two decades, there has been an intensive study of elliptic and evolutionary partial differential equations driven by the fractional Laplacian or related nonlocal operators, see, e.g., [15] for a list of interesting references. In applications from physics and finance, anomalous diffusion is often modeled by a fractional Laplace evolution equation (see [6], Appendix B for more details and references). In this contribution, we study doubly nonlinear elliptic equations of type (P f ) with bounded right-hand side in L 1 (R N ) on the whole space R N . The equation is driven by the fractional Laplacian (−∆) solutions of (P f ) even in the case b(u) = u and therefore one has to choose a more appropriate solution concept. The well-posedness of (P f ) for div F ≡ 0 has been studied in [3] in the framework of renormalized solutions. However, since (P f ) can be interpreted as a special case of a fractional Laplace evolution equation with a first-order convection term, it seems to be more natural to apply the notion of entropy solution in our general setting. In [5], entropy solutions have been introduced for elliptic equations with L 1 -data. In this contribution, we define the notion of entropy solutions for (P f ). Moreover, we show existence and the L 1 -contraction and comparison principles for entropy solutions. In particular, we obtain uniqueness of b(u) in this framework. We recall that existence and L 1 -contraction allows us to define the m-accretive, densely defined, multivalued operator A b in L 1 (R N ) by According to nonlinear semigroup theory (see, e.g., [4]), there exists a unique mild solution b ∈ C([0, T ]; L 1 (R N )) to the abstract Cauchy problem for A b for any given data (v 0 , f ) ∈ L 1 (R N ) × L 1 (0, T ; L 1 (R N )). In the next step, one wants to show that the mild solution is the unique entropy solution to the associated evolution equation b(u) t + div(F (u)) + (−∆) s 2 u = f in (0, T ) × R N u(0, ·) = u 0 (1.1) for appropriately chosen data (u 0 , f ) ∈ L 1 (R N )×L 1 (0, T ; L 1 (R N )). In this manner, our study of (P f ) serves as a basis for the investigation of (1.1), which will be subject of forthcoming work.
1.1. Outline. We organize this contribution as follows: We start in section 2 by introducing some basic notations which we will use throughout the paper and give some preliminary results that are used within the later sections. After that, in section 3, we establish the definition of an entropy solution to the equation (P f ) and formulate the two main theorems of this work. In section 4 we prove the L 1 -contraction and comparison principle with the help of Kruzhkov's method of doubling variables. Since this result is also crucial for the existence proof later on, it is proven before the existence of entropy solutions. Furthermore, the contraction principle gives us uniqueness of the saturation function b(u) and in some cases even the uniqueness of the entropy solution u itself. Finally, in section 5 we prove the existence of entropy solutions. For this, we apply the method of vanishing viscosity to be able to show that there exist weak solutions for a sequence of approximating problems of higher regularity. It is then left to show that these weak solutions converge to the entropy solution of the initial problem. In the appendix, for the sake of completeness, some of the technical results used in this work are proven.

Notations and preliminary results
Let us introduce some notations and functions that will be frequently used. For any real number r ∈ R and k > 0, we define the functions For all u ∈ S(R N ), the Schwartz space of rapidly descreasing functions, and all s ∈ (0, 2), we define the fractional Laplacian (−∆) with Γ being the gamma function. The constant C(N, s) is motivated by an equivalent definition of the fractional Laplacian via Fourier transform, i.e. by (−∆) where it naturally occurs. We further define the Gagliardo-seminorm by As it is well-known, the fractional Sobolev space is a Hilbert space, if endowed with the natural scalar product which induces the norm We point out that the fractional Sobolev space H s 2 (R N ) is strictly related to the fractional Laplacian (see [13] Proposition 3.6). An important tool will be a decomposition of the fractional Laplacian which was introduced by Droniou and Imbert in [10] Theorem 1, where the authors split the fractional Laplacian into a regular and a singular part.
for any ϕ ∈ D(R N ). The right-hand side can also be associated to a bilinear form which is well-defined on H A useful convergence result, which we will use later on, can also be found in the work of Droniou and Imbert (see [10], Proposition 1).

Concept of solution and main results
Now, we introduce the notion of entropy solutions which is adapted from the work of N. Alibaud (see [1]) and point out where it fits within the well-known concepts of classical and weak solutions.
The main goal of this paper is to prove the following theorems, concerning the existence and uniqueness of entropy solutions.
Remark 3.5. If f =f almost everywhere, it follows directly from Theorem 3.4 that b(u) = b(ũ) almost everywhere holds. If the nonlinearity b is strictly monotone, we further obtain u =ũ almost everywhere. Therefore, we know that the entropy solution u of (P f ) is unique. In general, i.e. for just nondecreasing b, we can not expect uniqueness of the entropy solution, but only for the saturation function b(u).

L 1 -contraction principle
In this section, we will prove the L 1 -contraction principle by applying Kruzhkov's method of doubling variables (see [14]). The L 1 -contraction principle and L 1comparison principle (see (4.3)) will play an important role in the proof of the existence theorem.

Proof of Theorem 3.4:
For all k ∈ R we choose the entropy-flux-pair (η k , φ k ), which is defined by for all a ∈ R. Since these entropies are not smooth enough, we have to show first that we can use them in the entropy inequality (3.1). To this end, we approximate η k in the following way: Then, for all n ∈ N, η n k ∈ C 2 (R) is convex and such that (η n k ) ′′ has compact support. Now we can use them in the entropy inequality (3.1) and prove, with the help of Lebesgue's dominated convergence theorem, that (η k , φ k ) are valid entropy-fluxpairs for all k ∈ R. We can now apply the method of doubling variables. For the entropy solution u of (P f ) let y ∈ R N be fixed but arbitrary and choose If we apply these in (3.1) and integrate with respect to y over R N , we get where the symmetric function ζ is given by Analogously, for the entropy solutionũ of (Pf ) and x ∈ R N fixed but arbitrary, we choose ϕ(y) = ϕ x (y) = ψ(x, y) as above, η = η k with k = u(x), apply these in (3.1) and integrate with respect to x over R N . If we add these inequalities, we get In the next step we pass to the limit with r → 0.
Proof: Without loss of generality let r < 1. We show that the integrand belongs to L 1 (R N ×R N ×B 1 (0)). The claim then follows by Lebesgue's dominated convergence theorem. Similar to the proof of Theorem 3.1 i.) we get ) .

An analogous calculation for the term
With similar calculations we can also pass to the limit with r → 0 in the regular term of the nonlocality. If we combine these results, we obtain, for r → 0: It yields D 2 Φ n → 0 pointwise and locally uniformly in R N for n → ∞. For Φ = Φ n in (4.1), we can make use of Proposition 2.2, the growth condition of b, and Lebesgue's theorem of dominated convergence to get This completes the proof of Theorem 3.4.

Extensions and Remarks.
Remark 4.2. Similar to the proof of Theorem 3.4, we can show the following L 1 -comparison principles: To prove this, we apply the method of doubling variables again, but with entropies respectively.
of (P f ) andũ of (Pf ) respectively. Then we can still prove the " local" inequality (4.1) for the contraction and comparison principle.
, we can also pass to the limit in (4.2) for the comparison principle, to show that (4.3) holds.
We are now able to prove the following L ∞ -estimate.
and u be an entropy solution to (P f ), almost everywhere which completes the proof.

Existence of entropy solutions
5.1. The vanishing viscosity method. We are now turning our attention to the existence proof for entropy solutions to problem (P f ). For this, consider the following modified problems for ε > 0: to be a classical solution to (P ε f ). We obtain a weak formulation of (P ε f ) by multiplication with a test function ϕ ∈ D(R N ) and subsequent integration over R N . [8] Corollaire 4.34 (ii)), all integrals are well-defined.
First, we want to prove the existence and uniqueness of weak solutions to the modified problem (P ε f ), with the help of Zarantonello's theorem (see [11], Theorem 3.5.2). By a fixed-point argument, we will then prove that there exist also weak solutions to the modified, doubly nonlinear problem which we will define later on. These solutions turn out to be of higher regularity and are therefore entropy solutions. Finally, we will show that the weak solutions to the modified, doubly nonlinear problems converge to the entropy solution of problem (P f ).
Proof: We define Then a(u, ·) is linear and bounded for every fixed u ∈ H 1 (R N ), i.e., it is an element in H −1 (R N ). Consider now and we claim that A is Lipschitz continuous and strongly monotone. It then follows by the Theorem of Zarantonello that A is bijective and therefore, for all . This completes the proof. Let u 1 , u 2 ∈ H 1 (R N ). Then it holds where L b is the Lipschitz constant of b and therefore, A is Lipschitz continuous. Moreover, A is also strongly monotone.

As a consequence, for the unique weak solution to (P
in the distributional sense, hence ∆u ∈ L 2 (R N ) and from classical regularity results for the Laplacian (see e.g. [12], Chapter 6.3.1) it follows that u ∈ H 2 loc (R N ) . Now, we also want to consider the nonlinearity div F (u) in our equation. Since we can not show the existence of weak solutions with such a term directly, we need to look at approximating problems. To this end, for every R > 0, let ̺ R ∈ C ∞ (R N ) be equipped with the following properties: for a constant C > 0, independent of R. Let us define where F ε = F • T 1/ε with T 1/ε the truncation function at level 1/ε and w ∈ H 1 (R N ) fixed, but arbitrary. This map is linear and continuous, i.e. an element of H −1 (R N ). If we consider the family of integral equations for all v ∈ H 1 (R N ), we already know from Theorem 5.1 that there exists a unique u = u R,w ∈ H 1 (R N ) which satisfies (5.1) for all v ∈ H 1 (R N ) and thus, we can define the following map: w → the unique solution u ∈ H 1 (R N ) to (5.1) .
If we take u itself as a test function in (5.1) and exploit the properties of ̺ R , we get the following a-priori estimate: Now, consider the set M = {u ∈ H 1 (R N ) : u H 1 ≤ K ε,R } which is nonempty, bounded, closed and convex. If we restrict Ψ R to the set M and prove that there exists a fixed-point, we obtain the existence of a solution u to (5.1) with w = u.
To achieve this, we have to prove that Ψ R is weakly sequentially continuous, for the claim then follows by the fixed-point theorem of Schauder-Tikhonov (see [16], Corollary 9.7).
Lemma 5.5. The map Ψ R : M → M is weakly sequentially continuous.
Proof: For all n ∈ N, let w n , w ∈ H 1 (R N ) such that w n ⇀ w in H 1 (R N ) for n → ∞.
Since the sequence of solutions u n = Ψ R (w n ) is bounded in H 1 (R N ), it is sufficient to prove that every weakly convergent subsequence of (u n ) n converges weakly to Ψ R (w). Now, let (u n ) n be a not relabeled subsequence such that u n ⇀ u in H 1 (R N ) for n → ∞. By continuous and compact embeddings we can assume, without loss of generality, that (w n ) n and (u n ) n converge for n → ∞ in the following way: w n −→ w in L 2 loc (R N ) and almost everywhere in R N , Since u n is a solution to (5.1) with w = w n , for all n ∈ N and all ϕ ∈ H 1 (R N ), we have Now we can, thanks to (5.2)-(5.4), pass to the limit in all integrals and obtain u = Ψ R (w).
We have therefore proved that, for all R > 0, there exists u = u R ∈ H 1 (R N ) such that for all v ∈ H 1 (R N ). Our next goal is to pass to the limit with R → ∞. First, we need a technical result which allows us to get rid of the convection term. From the divergence theorem of Gau we can show the following Lemma.
Proof: See Appendix 6 Hence, if we apply u = u R ∈ H 1 (R N ) as a test function in (5.5) we get for all R > 0, where we used Lemma 5.6. Thus, there exists a subsequence, still denoted by (u R ) R , such that we can now pass to the limit with R → ∞ in (5.5): . This means, we have shown the existence of a weak solution to for all ε > 0 and all f ∈ L 1 (R N ) ∩ L ∞ (R N ).

Proposition 5.2.
For the weak solution u ∈ H 1 (R N ) to (P ε f ) the following holds: Proof: i.) Let k > 0. Applying the test function 1 k T k (u) ∈ H 1 (R N ) in the weak formulation of (P ε f ), we get If we use Lemma 5.6 and the positivity of the terms on the left-hand side, we obtain Since for the test function it holds 1 k T k (u) → sign u almost everywhere in R N for k → 0, with Fatou's Lemma we get that This implies b(u) ∈ L 1 (R N ) and thanks to the growth condition of b also u ∈ L 1 (R N ). ii.) Let now k, l > 0. This time, we use the test function 1 k T + k (u − l) ∈ H 1 (R N ) in the weak formulation of (P ε f ). We then get With the positivity of the integrands, Lemma 5.6 and the growth condition of b, we obtain Let l ≥ f + ∞ λ , we then get for k → 0: Since the left integrand is positive, it follows that (u − l) + = 0 almost everywhere in R N and therefore u ≤ l almost everywhere in R N . Analogously, we can show that there existsl > 0 such that −l ≤ u almost everywhere in R N . Thus, the claim follows.
Remark 5.7. For any ε > 0, the unique weak solution u ε of (P ε f ) is in is Lipschitz continuous, from the chain rule for Sobolev functions it follows that

With the same arguments as in the proof of Lemma 5.3 it follows that (−∆)
s 2 u ε ∈ L 2 (R N ). Therefore, and we may conclude that u ε ∈ H 2 loc (R N ). Lemma 5.8. For any ε > 0, the unique weak solution u ε of (P ε f ) satisfies the entropy inequality for all convex entropies η ∈ C 2 (R), . . , N , all ϕ ∈ D(R N ) with ϕ ≥ 0, and all r > 0: Proof. Since the weak solution u ε of (P ε f ) is in H 2 loc (R N ), the equation (P ε f ) is satisfied pointwise almost everywhere. Similar to the proof of Proposition 3.1 i.) (see Appendix 6), we are able to show the following inequality for all convex entropies η ∈ C 2 (R), φ ε = (φ 1 ε , . . . , φ N ε ) with (φ i ε ) ′ = η ′ (F i ε ) ′ for i = 1, . . . , N , all ϕ ∈ D(R N ) with ϕ ≥ 0, and all r > 0: It is left to show that we can transfer the fractional derivative in the last term on the left-hand side onto the test function ϕ. To this, we can choose a sequence (u n ε ) n ⊆ C 2 b (R N ) such that u n ε → u ε in L 2 loc (R N ) and a. e. in R N , ∇u n ε → ∇u ε in L 2 loc (R N ) and a. e. in R N , D 2 u n ε → D 2 u ε in L 2 loc (R N ) and a. e. in R N . From Lebesgue's dominated convergence theorem, the proof of Proposition 3.1 i.) (see Appendix 6) on the level of the approximating sequence for every n ∈ N, and again, Lebesgue's dominated convergence theorem, we have

5.2.
Convergence results for the approximating solutions. In the next step, we want to pass to the limit with ε → 0. We apply the weak solution u = u ε ∈ H 1 (R N ) as a test function and use Lemma 5.6 to get Therefore, we get From (5.8)-(5.11) we get, for a proper subsequence that is still denoted by (u ε ) ε , the following convergence results for ε → 0: Since we know, from Proposition 5.2 ii.), that (u ε ) ε is uniformly bounded in L ∞ (R N ), we can choose the subsequence such that for ε → 0. We now consider the entropy inequality for the approximating solutions u ε , i.e.
and it remains to prove that we can pass to the limit with ε → 0 in (5.17) using (5.12) -(5.16). We would then obtain u ∈ H s 2 (R N ) ∩ L ∞ (R N ) which satisfies the entropy inequality and is therefore the desired entropy solution of problem (P f ) (Notice that b(u) ∈ L 1 (R N ) follows from Proposition 5.2 i.), (5.15) and Fatou's Lemma). With the above convergence results, we can calculate for ε → 0. Furthermore, we have ε∇(η(u ε (x))) · ∇ϕ(x) = ε∇u ε (x)η ′ (u ε (x)) · ∇ϕ(x) and from this, we get for the second integral, together with (5.14), that ε R N ∇(η(u ε (x))∇ϕ(x) dx → 0 for ε → 0. For the nonlocal terms we argue as follows: First we get for almost all x ∈ R N , z ∈ B r (0) c and ε → 0. Further calculations show that for a constant C > 0, not depending on ε, since (u ε ) ε is uniformly bounded in L ∞ (R N ). With Lebesgue's dominated convergence theorem, we then get for ε → 0. For the singular part we also get that for almost all x ∈ R N , z ∈ B r (0) and ε → 0. We also notice that |ϕ(x + z) − ϕ(x) − ∇ϕ(x) · z| ≤ D 2 ϕ L ∞ |z| 2 for all x, z ∈ R N holds. Therefore, the integrand is dominated by the function for a constant C > 0, not depending on ε, where R > 0 is such that supp ϕ ⊆ B R (0). Again, by Lebesgue's dominated convergence theorem, we get that for ε → 0. Since, again, (u ε ) ε is uniformly bounded in L ∞ (R N ), there exists ε 0 > 0 such that u ε ∞ ≤ 1 ε ∀0 < ε < ε 0 . Therefore, we have T 1/ε (u ε ) = u ε ∀0 < ε < ε 0 and in this case we get for ε → 0. This follows by applying Lebesgue's dominated convergence theorem, since almost everywhere in R N for ε → 0 from (5.15), φ(u ε ) is uniformly bounded in L ∞ (R N ) and the integral is taken over the compact support of ϕ. This completes the proof of Theorem 3.3.
Proof of Lemma 5.6. By the divergence theorem of Gau it is well-known that, on every bounded Lipschitz domain Ω ⊆ R N , it holds Ω F ε (u(x))∇u(x) dx = 0 for all u ∈ H 1 0 (Ω). Let (u n ) n ⊆ C ∞ c (R N ) be such that supp u n ⊆ B n (0) =: Ω n for every n ∈ N and u n → u in H 1 (R N ) .