On a fractional thin lm equation

This work is licensed under the Creative Commons Attribution alone 4.0 License. Adv. Nonlinear Anal. 2020; 9: 1516–1558 Antonio Segatti and Juan Luis Vázquez* On a fractional thin lm equation https://doi.org/10.1515/anona-2020-0065 Received December 27, 2019; accepted February 3, 2020. Abstract: This paper deals with a nonlinear degenerate parabolic equation of order α between 2 and 4 which is a kind of fractional version of the Thin FilmEquation. Actually, this one corresponds to the limit value α = 4 while the Porous Medium Equation is the limit α = 2. We prove existence of a nonnegative weak solution for


Introduction
In this paper we are mainly interested in the analysis of the following system of partial di erential equations where Ls := (−∆) s , s ∈ ( , ), is the fractional Laplacian (see, e.g., [51,65]), the dimension d ≥ , and the mobility function m is linear, namely m(u) = u. From a mathematical point of view, System (1.1) appears, at least formally, as an interpolation between the second-order nonlinear di usion model called Porous Medium Equation (case s = , described in the survey paper [5] and in the monograph [69], where complete references to origins, theory and applications are given) and and the fourth-order Thin Film Equation (case s = ) for which the theory of existence of weak solutions in one and in higher dimensions is quite advanced. Without claiming any completeness, we refer to [11,13,15,24,32] and to the review papers [10,56].
As for physical applications, the system has been analysed in dimension one for s = / and power law mobilities by Imbert-Mellet in [45] on a bounded interval with Neumann boundary conditions as a model for the dynamics of cracks. The study is continued in [46]. The one dimensional analysis for a general s ∈ ( , ) has been completed in [66] and [67]. Selected references to the applied literature are given in those papers. We recall that Barenblatt was quite involved in the mathematical modeling of hydraulic fractures, [7].
Another mathematical motivation comes from comparison with the system studied in the papers [16] in 1D and [21] in all dimensions, respectively. This system reads

2)
This model has been widely studied and has interesting applications [16,39,40,44]. The di erence with system (1.1) clearly lies in the constitutive law that relates the density u with the pressure p, that implies that the order of di erentiation is (formally) − s. Consequently, (1.2) can be seen as an interpolation between the porous medium equation (2nd order) and the (0-th order) superconductor model analysed by Ambrosio and Serfaty in [4]. The model is called in [71] "Porous Medium Di usion with Nonlocal Pressure". On the other hand, the present Model (1.1) has formally order of di erentiation + s ∈ ( , ). Our aim in this paper is to develop a basic theory for System (1.1). As a rst issue, we prove existence of suitably de ned weak solution in the general multidimensional setting for linear mobilities (see the Section Open Problems 7 for a discussion on this topic). A remarkable feature of the weak solutions we construct is positivity. This property is proved in our general setting along the lines of papers [13] and [45] and it is a nontrivial e ect of the degeneracy of the mobility. We also show that weak solutions originating from initial conditions with nite rst moment, keep their rst moment nite during the whole evolution. Based on this estimate, we will also prove (for a particular class of weak solutions, see below and Subsection 5.1) that also the second moment, if nite at t = , remains nite.
The investigation of the intermediate range is thus quite important from the mathematical point of view since both borderline cases belong to very di erent types of equations. We point out that uniqueness is not proved, it seems to be a di cult problem.
A second issue of our analysis concerns the existence of self-similar solutions. Our strategy has some similarity with the analysis in [22] and, in general, with the analysis of the long time behavior of the porous medium equation (see [23]). In particular, we show that self-similar solutions to (1.1) with the regularity provided by the existence Theorem 3.1 are related to stationary solutions of a nonlocal Fokker-Plank type equation. More precisely, starting from a weak solution u of (1.1), if function v is implicitly de ned as  The reason for looking at this particular class of stationary solutions is motivated by the fact that these are the stationary solutions that emerge in the long time behavior of (1.5) as solutions with zero dissipation (see Section 5). Recalling that we are looking for positive solutions, (1.6) reduces to the free boundary problem ∇ Ls v + β |y| = on P := {v > }.
(1. 7) In principle, the geometry of the positivity set P can be quite complicated (see [38] and the PhD Thesis [50] for the Thin Film case). In particular, P can be disconnected. However, restricting to solutions with connected support, we have a quite complete picture of the self-similar solutions to (1.1). More precisely, we can show that solutions to (1.7) are indeed solutions of an obstacle problem (the obstacle being the zero level set) for the energy E (v) := ˆR d |L s/ v| dy +ˆR d β |y| − vdy, (1.8) thus showing that self-similar solutions are somehow minimal for the energy E . Remarkably, the self-similar solutions are radially decreasing, compactly supported and with explicit form given by formula (1.3), with α and β as in (1.4). Moreover, the stationary pro le v has the form v(y) = (C − C |y| ) +s + , (1.9) where C = C (d, s) > (see (4.43) for the exact value) while C > is a free constant that allows to adjust either the mass of the solution or the radius of its support. Showing that from the minimizers of (1.8) one can obtain solutions with the explicit form (1.9) and with the free constant C requires some work. In particular, our analysis relies on the following steps. At rst, by scaling and comparison and relying on the results of Dyda ([30]) we show that the solution of the obstacle problem for (1.8) has the explicit form (the D in the subscript refers to Dyda) v D (y) := λ s κ s,d ( − λ |y| ) +s + , (1.10) where λ := β/( γ s,d ) and is supported in B R D where R D = /λ. In a second step, by a further scaling, we nally obtain (1.9) These questions are discussed in full detail in Subsection 4.2.1.
Note that the limit cases s = and s = , are known and agree with this formula. For s = we get the well-known Barenblatt pro le v = (C − C |y| )+ (1.11) for the porous medium case, that was found around 1950 in papers by Zeldovich-Kompanyeets [72] and Barenblatt [6] (they deal with general power-like mobilities m(u) = u k ). For s = we get the zero-angle pro le v = (C − C |y| ) + (1.12) for the corresponding Thin Film equation (see [60], [14] and [32]). The similarity exponents α and β also agree, being based only on dimensional considerations. It is interesting to note that these results are somewhat similar to the ones obtained in [22] for the porous medium with fractional pressure, which is a quite di erent setting. Remarkably, the self-similar solutions of that problem follow formulas (1.3) and (1.9) with s ∈ ( , ) replaced by −s, cf. [16], [17] and [70]. In this way we get a panorama of related self-similar patterns for equations of (formal) order ranging from 0 to 4. In all cases the self-similar solutions are of the type called source-type solutions, which means that the initial data of u(x, t) is necessarily a point mass distribution, i.e., a Dirac delta. This property follows easily from the conservation of mass due to the divergence form of the equation and the compact expanding support that shrinks to a point as t → . Actually, all of these solutions have free boundaries of the form |x| = R t β . The study of the behaviour and regularity of free boundaries for solutions with general initial data is a di cult topic (see Section 7 in this paper for some discussion).
Our analysis is purely variational and uses symmetrization comparison arguments to prove the compactness and radial symmetry of the support. Moreover, the analysis works in any dimension of space and for any s ∈ ( , ). We must point out that our analysis is restricted to a linear mobility function. The general case of power function mobility is considered, with a di erent analysis, only in dimension one and for s = / in the paper [46]. In particular, the self-similar solution (1.9) corresponds to the solution of the "Zero Toughness Case" for dimension one in [46] with a linear mobility function and s = / .
A third issue of the paper is the long-time behavior of the weak solutions to (1.1). As in [22] and in [23], this is done by working on the Fokker-Planck equation (1.5).
As we have already mentioned, if we rescale according to (1.3) a weak solution of (1.1) we get a weak solution of the Fokker-Planck equation (1.5) that preserves mass and positivity. The aim is then to prove that the rescaled orbits converge to our selected class of self-similar solutions. This is achieved at the prize of accepting some regularity assumption that restricts the class for which we can justify the classical study of long time behaviour. We explain the problem at the beginning of Section 5. Let us now say that a main ingredient in the proof of the needed energy-dissipation estimate is the following equality (see Lemma 5.4) This identity furnishes the exact balance between the second moment and the fractional energy of Section 5. At present, we are able to prove (5.1) for functions that satisfy a suitable decay at in nity. As the proof will show, this is needed to ensure the niteness of the righthand side of (5.1). Therefore, we investigate the long time behaviour for weak solutions for which the right hand side is nite (we refer to 5.4 for the precise assumption). It is important to observe that weak solutions with compact support actually satisfy (5.1). It is an open problem to prove that (5.1) holds for all weak solutions. It is interesting to note that an analogous identity holds also the weak solutions of the fractional porous medium equation (1.2) constructed in [21] (see Lemma 5.5). In this case the term in the left hand side is the energy for which (1.2) is a Wasserstein gradient ow (cf. [52]).
The rst step in the long time behaviour analysis is to prove (see Theorem 5.7) that for large times the weak solutions to the Fokker-Planck equation approach the stationary solutions, up to the extraction of a subsequence. It is interesting to observe that the above energy E (1.8) decreases on weak solutions to the Fokker-Planck equation (namely, properly rescaled weak solutions to (1.1)), thus suggesting that the longtime behaviour of the weak solutions of (1.1) can be described by the constructed self-similar solutions. This is indeed the case, as we prove in this paper, under a connectedness condition on the positivity set of the cluster points for large times of the weak solutions of the nonlocal Fokker-Planck equation.
Due to our success in constructing self-similar solutions for Equation (1.1), and also the interest in treating nonlinear parabolic equations of even higher order, we devote another section to discuss the existence of selfsimilar solutions for equations of the type (1.14) with < s < (hence the total order of the equation goes from 4 to 6). We nd explicit compactly supported and nonnegative self-similar solutions with a Barenblatt pro le of the type similar to (1.9), that is solutions u(x, t) of the self-similar form (1.3) with adjusted similarity exponents 15) and pro le of the Barenblatt type: v(y) = (C − C |y| ) +s + , (1.16) This holds for all < s < , the constant C = C (s, d) is xed and C > is a free constant. See whole details in Section 6 where parameter C is explicitly computed. Its value is consistent with corresponding self-similar solution for the Thin Film equation in one dimension mentioned in [14]. It is worthwhile commenting on the repeated appearance of the Barenblatt pro les, that looks surprising. We recall that these pro les appear in the Porous Medium equation (1.17) in all the range of exponents < σ < ∞ since σ = /(m − ), and they are quite relevant at all levels of the theory, as amply documented in [69]. As a consequence of our results in our paper, we nd that they appear as relevant self-similar solution for the nonlocal equations (1.1), (1.2) and (1.14), and they are expected to play a big role in the theory. As a further observation, notice that the solution pro le (1.9) coincides with the PME solution pro le (1.17) for the precise choices m − = /( + s), while (1.16) leads to a similar identi cation with the PME when m − = /( + s) (see [26] and [55] where this similarity between the Porous Medium equation and the Thin Film equation is noticed and used).
O . We gather preliminary material in Section 2. In Section 3 we discuss the existence of a suitably de ned weak solution The very important topic of existence of self-similar solutions is settled in Section 4, and the long-time convergence to a stationary solution is studied in Section 5. We develop the higher order application in Section 6. A nal section contains a number of open directions.

Preliminary Material
In this section we collect some of the material that is needed for our analysis.
First of all, we recall that the Fractional Laplacian (-∆) s (s ∈ ( , ) is the nonlocal operator de ned, at least for functions in the Schwartz class S(R d ), as then the Fractional Laplacian can be equivalently de ned as the operator with symbol |ξ | s , namely For a function in S(R d ) and for s ∈ ( , ) we de ne, component-wise, the operator L s/ • ∇ by Note that using the Fourier transform we have that ∇ and L s/ commute. More precisely there holds for any j = , . . . , d Moreover, again using the Fourier transform, we can express the seminorm in H +s (R d ) (s ∈ ( , )) using this operator. Namely, u

Existence of a weak solution
We discuss the existence of nonnegative weak solution to system (1.1): Weak solutions are de ned as follows.
De nition 3.1. Given u ∈ L loc (R d ) and nonnegative, we say that u is a weak solution of (1.1) if

The following relation holds for any test function
Here is the Existence Theorem.  where the vector eld ξ ∈ L ( , +∞; L (R d )) satis es

Entropy Estimate
Important functional remark. The vector eld ξ emerges as a weak L limit of a sequence, in the approximation scheme we introduce for the proof of the Existence Theorem, and it is related to the product u / ∇p. In particular, due to the nonlinear and degenerate structure of the system (1.1) we are not able to rigorously identify ξ = u / ∇p, as the formal estimates suggest. However, thanks to the characterization (3.6) we can conclude that in the regions of R d × ( , +∞) in which u > we have ξ = u − / (∇(up) − p∇u) and, if p were regular enough to give a pointwise meaning to ∇p, we would have (still in the regions where u is nonzero) the plain expression ξ = u / ∇p. On the contrary, in the regions where u ≡ , (3.6) gives no detailed information on ξ , we know that ξ ∈ L in space and time.
We then consider the following nondegenerate approximate problem To be precise the problem above should be intended in the distributional sense on R d × ( , T). Note however that since < ε ≤ m M ε ≤ M, the rst equation is not degenerate and with bounded mobility and this will imply that the approximate solutions uε (for notational simplicity, when no confusion arises we do not write the dependence on M) are regular enough for positive times to justify all the estimates we perform at the approximate level. See details below.
. . Existence for Problem (3.8) The existence of an approximate solution follows from a nested approximation scheme. Given a bounded domain Ω, for any s ∈ ( , ) we introduce the Hilbert space We let τ > and R > and we consider the following stationary problem: This problem is related (see below) to the implicit Euler scheme for the evolution Now we discuss, using the Leray-Schauder xed point Theorem, the existence of a solution of (SP). To this end, we let σ ∈ [ , ] and we implement the following scheme 1. Givenū ∈ Xs(B R ( )), we let p ∈ H (B R ( )) the weak unique solution of (3.10) 2. Given p from step , we let u ∈ Xs(B R ( )) the unique solution of (3.11) Therefore, the procedure above produces a map A : Xs(B R ( )) × [ , ] → Xs(B R ( )) such that where (u, p) ∈ Xs(B R ( )) × H (B R ( )) is the unique solution of (3.10)-(3.11). We can check that the map A has the following properties (1) A(u, ) = for any u ∈ Xs(B R ( )).
(3) There exists M > such that u Xs(B R ( )) ≤ M, ∀(u, σ) satisfying u = A(u, σ). (3.12) Then, the Leray-Schauder Fixed Point Theorem (see [41,Theorem 11.6]) gives the existence of a xed point for the map A u = A(u, ), for u ∈ Xs(B R ( )), namely a solution of (3.9). The rst two properties listed above are evident. In particular, the second comes from fractional elliptic regularity. Concerning this last point, note that from (3.11) and the fact that p ∈ H (B R ( )) ⊂ L (B R ( )) we conclude that, at least, u ∈ H s/ −δ (R d ) for any δ > (see [2,Corollary 1.1]). In particular, this last space is compactly embedded in Xs(B R ( )). More regularity can be obtained by bootstrapping but we will not use it.
We still have to verify the boundedness property (3.12). To this end, let u ∈ Xs(B R ( )) such that u = A(u, σ). Recall that < ε ≤ m M ε ≤ M and thus m M ε ∇p ∈ H (B R ( )). Then we have If σ = there is nothing to check, hence we can assume σ > . We take ϕ = p in the rst equation (we still denote with p the truncation to of p outside B R ( )), and ψ = (u − v)/σ in the second equation. All this is justi ed in the above mentioned regularity framework. We thus obtain that easily implies (3.12). S 2. Next, we tackle the evolution process. Given u ∈ H s (R d ), we consider a smooth function that is supported in B R ( ) and such that u (R) Then, we introduce the uniform partition P of ( , +∞), i.e., Then, we iteratively solve (3.9) with v = u , u , . . . , u k− , . . ., where u k is a solution of (3.9) with v = u k− . In a standard way we introduce the piecewise-linear (û k ) and the piecewise-constant (ū k ) interpolants of the discrete values u k . We setû (3.13) Now, in order to pass τ → we perform some a priori estimates onû k andū k . First of all, sinceû k ≡ in R d \B R ( ), we have that ∂ tûk ≡ in R d \B R ( ) and thus the second equation in (3.13) gives, for any t ∈ ( , +∞), Therefore, xing T = τN for some N ∈ N and integrating the above relation on ( , Moreover, the rst equation in (3.13) giveŝ and thus we have the estimate on the discrete solution This estimate is the core of the existence theory for (3.8) and produces one of two estimates available for (1.1). Note that, for any xed ε > , and thus a comparison in the rst equation gives that the time derivative ∂ tû k is bounded in L ( , T; W − ,q (B R ( ))), uniformly in M and in R, for some q > . In particular, since ε > is kept xed, the bounds above are su cient to pass to the limit with respect to τ via standard compactness arguments and nd in the limit a solution u R of the following problem: (3.16) Note that for u R we have, (3.17) , the estimate above is uniform w.r.t. to R. Then, again as before we can easily pass to the limit in (3.16) and obtain a solution u ε,M of (3.8) with, at least, the energy regularity (3.18) and the companion estimate (3.19) The above estimate and a comparison in (3.8) guarantees that ∂ t u ε,M is bounded, uniformly in M, in L ( , T; W − ,q (R d )) for some q > . Note that for any xed ε and M we also have that The membership of ∂ t u ε,M to this space of distributions is clearly not uniform with respect to ε and M as it heavily depends on the boundedness and nondegenerate character of m M ε .
At the end, we can let M → +∞ and obtain a (weak) solution uε of where we recall mε(y) := y + + ε.

. Uniform estimates with respect to ε: Energy and Entropy Estimate
In this Subsection we derive the two basic estimates, uniform on the approximate parameter ε, on the solution uε of the approximate problem (3.21), namely the Energy Estimate and the Entropy Estimate which correspond to the estimate (3.5) and (3.4) in the limit ε → , respectively. Energy Estimate The Energy Estimate follows by semicontinuity from the analogous estimate (3.18). Note valid for a.a. t > . Moreover, we observe that uε is indeed a bit more regular in space. In fact, being pε = . We will show that this estimate produces, in the ε → limit, the Energy Estimate (3.5).
Entropy Estimate System (3.21) admits a further estimate that is in principle not uniform with respect to the parameter ε. But this estimate will produce the Entropy Estimate (3.4) in the limit ε → . We will need this observation: for any ε > , we consider the smooth and positive real function fε such that Without loss of generality we can choose fε in such a way that fε( ) = f ε ( ) = . Thus, An important property of fε is that, when y < , there holds To prove the above estimate we observe that for y < we have Thus, setting g(w) :=´ w r + +ε dr, we immediately have that Moreover, when w < , there holds g(w) ≥ ε w.
In order to fully justify the argument, we work at the approximate level of problem (3.8). Therefore we let u ε,M be a solution of (3.8) and we consider, for any M > , a positive real function f ε,M such that Note that f ε,M is de ned as in (3.25) and that f ε,M , at least, together with ∂ t u ε,M . Thus, the computations are justi ed. Therefore, using that p ε,M = Ls u ε,M and that f ε,M = m ε,M , integrating with respect to time we get . Passage to the limit: Proof of Theorem 3.1 The energy and the entropy estimate give some important uniform estimates (with respect to ε) on the approximate solutions uε. We work on bounded time intervals ( , T) with T > for compactness reasons. First of all, the energy estimate (3.22) gives that the sequence L s/ uε is bounded, uniformly with respect to ε, in . The entropy estimate (3.28) gives that L s/ ∇uε is bounded in L ( , T; L (R d )). Hence, we gain some spatial regularity for uε and for pε = Ls uε, namely (3.29) with C possibly depending on T. The energy estimate (3.22) gives that .
Consequently, a comparison in the equation (3.21) gives the estimate on the time derivative ∂ t uε, namely Then, the Aubin-Lions compactness lemma gives that for any δ > , uε is strongly compact in L ( , T; H +s−δ (K)), for any compact K ⊂ R d . Thus, there exists u ∈ L ( , T; H +s loc (R d )) and a subsequence of ε for which we conclude that, denoting with p the weak-star limit of pε in L ∞ ( , T; H −s (R d )), we have at least in the sense of distributions. Actually much more is true. In fact, the estimate (3.29) gives that pε is bounded in L ( , T; H −s (R d )). Thus, we have that and that the relation p = Ls u holds, at least, almost everywhere in R d × ( , T). We have all the ingredients to pass to the limit in the following weak formulation of (3.21) We note that the rst term in the left-hand side converges to the expected limit thanks, e.g., to the dominated convergence. Now we pass to the limit in the nonlinear term. Since (see [ The second term I ε tends to zero when ε ↘ . In fact, for a constant C that depends on φ, we have, thanks to the Schwarz inequality, thanks to (3.22). The term I ε tends to the expected limit since we have that, for any compact K ⊂ R d , and pε → p weakly in L ( , T; L (K)).
As a result, we have that u veri es, for Moreover, we have that by passing to the limit in (3.28) we obtain (3.4) thanks to semincontinuity. Finally, since (3.22) implies that ξε := m / ε (uε)∇pε is bounded uniformly in L ( , T; L (R d )) we have that there exists a vector eld ξ ∈ L ( , T; L (R d )) to which ξε weakly converges and such that (3.5) holds.
Thus, it remains to identify ξ as in (3.6). To this purpose, we introduce the vector eld ζε := mε(uε)∇pε = mε(uε) / ξε and we note that, on the one hand, ζε weakly converges to some ζ ∈ L ( , T; L (R d )) and ζ = u / ξ . On the other hand, we have that, since ζε = ∇(pε mε(uε)) − pεH(uε)∇uε, In particular, for those points in which u > we can express ξ in terms of ζ as ξ = u − / ζ In order to prove that u is indeed a solution of (3.1) it remains to show that u ≥ almost every where in we get, by semicontinuity of norms, that ∀T > , Moreover, this estimate is also uniform with respect to time and thus

. . Nonnegativity
To prove positivity we exploit the entropy estimate (3.28). More precisely, the positivity of u follows from the fact that sup combined with (3.26). We aim at proving that u ≥ for almost any (x, t) ∈ R d × ( , T). To this end, we x t ∈ ( , T), a compact subset K of R d and we assume, by contradiction, that the set has positive measure. Since this implies that that for some xed λ > the set Severini-Egorov Theorem furnishes that for any η > there exists a measurable set Gη ⊂ K such that |K \Gη| ≤ η and such that We x η and we ndε > such that if ε <ε there holds Thus, thanks to Fatou Lemma we get This is in contradiction with (3.42), which would imply that for all t ≥ lim sup Hence the positivity is proved. As a consequence we have that u is a solution of (3.1).

. . Conservation of mass
We take a smooth cut-o function g : [ , +∞) → [ , ] such that (3. 43) and such that There holds that for any Since u, ∇u and p are, at least L loc (R d × ( , T)) functions we have Thus, since, for any compact K ⊂ R d there holds that u ∈ L ( , T; L (K)), ∇u ∈ L ( , T; L (K)) and p ∈ L ( , T; L (K)), we have that (3.46) Moreover, as u ≥ in R d × ( , T), the monotone convergence Theorem gives that, for almost any t ∈ ( , T), by passing to the limit R → +∞ in (3.45) we obtain, for almost any t < T that gives the desired conservation of mass.

. . First Moments estimate
Lemma 3.2. Let u be a weak solution as constructed before. Then, We take as a test function in (3.48) the function ϕ(x) = |x|ϕ R (x) where ϕ R is a smooth cut-o function (see (3.43) in Subsection 3.3.2) such that To be precise, to obtain a smooth ϕ one should also round o the function |x| around the origin. The proof is analogous and for the sake of simplicity we use φ = |x|ϕ R . We have Due to the regularity of the weak solution, the rst integral is clearly bounded by a constant that depends on the nal time T. Being ∇ϕ R supported on R ≤ |x| ≤ R, thanks to |∇ϕ R | ≤ C/R, we have that also the second integral is bounded by a constant possibly depending on the nal time T.
We bound the second integral in the right hand side of (3.48 Now, for |x| > we have that ϕ R |x| ≤ and thus, the second integral is bounded using that up ∈ L (R d × ( , T)).
Regarding the rst integral we rst note that u ∈ L ( , T; L q * (R d )) and that p ∈ L ( , T; L r * (R d )) due to Sobolev inequality. More precisely, Thus, de ning q ≥ in such a way that we have that q < d and thus |x| − ∈ L q (B ( )). Therefore, the Young inequality shows that Collecting all the above estimates we have (3.47).

Self-similar Solutions
In this Section we construct self-similar weak solutions of System (1.1) (in the sense of De nition 3.1). More precisely, we look for solutions of the form where the pro le function v : R d × R → R is to be appropriately determined and the parameters α and β are given by due to the constraints that we will nd below. In what follows, we will set It is interesting to observe that the pro le function v will have compact support. Hence, the self-similar solutions will have compact support as well (in the space variable). As it is now customary (see [23] and [69] and references therein), the self-similar solutions of (1.1) are related to stationary solutions of a nonlinear (and nonlocal in this case) Fokker-Planck type equation solved by the pro le v. Thus, as a rst step, we look for an equation to be satis ed by v. Clearly, since v is related to a weak solution u by the relation (4.1), it has the very same (low) regularity. Thus, the following computations are only formal at this moment. Therefore, assuming all the regularity needed to justify the computations, we have Moreover, Thus, the problem rewrites as (4.5) Now, the choice made above for α and β implies the algebraic relation that allows us to eliminate the time factors in the above equation. We thus obtain an expression involving only the rescaled variables τ and y. Namely, (4.7) Moreover, since also impose a second relation α = βd, equation (4.7) can be written in divergence form, so that conservation of mass will be guaranteed (at this stage only formally). More precisely, the system contains the following nonlinear and nonlocal Fokker-Planck type equation: (4.8) . The structure of the stationary solutions Self-similar solutions are thus related to stationary solutions of (4.8). Therefore, we rst analyse the structure of the stationary solutions.
(i) First of all, we make a reduction in the set of possible solutions and concentrate on those stationary solutions of (4.8) such that v∇y w As in the parallel study made in [22] for negative values of s, this reduction must be justi ed by the later analysis of the long-time behavior and the asymptotic convergence to a self-similar pro le.
(ii) Assuming for the moment that we have continuous solutions,¹ if we denote by P the positivity set of v, i.e. the set P := y ∈ R d : v(y) > , (4.10) Thus, on the every connected component C i of P = i∈N C i , there exists a constant c i such that The above problem can be rewritten as (χ i is the characteristic function of C i ). Necessarily, the constants c i cannot be all negative, otherwise P = ∅ thanks to the maximum principle. Note that since the operator is nonlocal we do not claim positivity of all constants. In any case, this fact will not be important for our results.
(iii) Now we restrict to look at continuous solutions for which P is connected. Let us denote with v a solution of (4.12), namely a particular solution of (4.11). In this case, problem (4.12) becomes (C ∈ ( , +∞)). By construction, v is strictly positive on P. Beside v , let us denote with v a solution of the problem (4.14) It is necessarily positive, thanks to the maximum principle [28,Theorem 1.8]. Therefore, by linearity the (continuous) solutions of (4.11) for which the positivity set P is connected have the form In fact, let v denote a particular solution of the linear equation Then, all the solutions to the equation (4.11) are given by (4.15) provided v solves the homogeneous problem which corresponds to (4.14). We will relate the v component of the solution (4.15) to an obstacle problem for which we prove existence and uniqueness of a smooth (C ,α , α ∈ ( , s), according to the obstacle problem regularity theory), radially decreasing solution. In such a way we construct a kind of minimal energy solution. As a consequence, we will conclude that the positivity set P of v is a ball.
Following the analysis, the v component of v in the decomposition (4.15) is a kind of correction of v . The function v solves (4.14) in a ball, and is explicitly obtained as a rescaling of the solution (the subscript G refers to Getoor) Proof. First of all, we prove that the energy is bounded below. Let us x R := /β and note that (4.20) Let vn ∈ K be a minimizing sequence, that is E (vn) In particular, vn we use Fatou's Lemma. We let R = / β and use the decomposition Thanks to the strong convergence in L p on compact sets we have that Thus, we conclude that ( β |y| − )v ∈ L (R d \ B R ) and therefore v ∈ K . Now, to conclude that v is indeed a minimum for the energy, we observe that the semicontinuity of the H s seminorm with respect to the weak convergence and (4.23) imply that namely the minimality of v. The uniqueness follows from the strict convexity of the energy.
In the next Theorem we prove some important properties of the solution of the obstacle problem (4.18). To this purpose, we prepare the following Ls v = P in R d with P a polynomial, forces u to be a ne and P to be equal to zero, which is clearly not the case in our situation.
To conclude that v O is radially symmetric and has compact support, we argue as follows. We denote with v * : R d → R the symmetric decreasing rearrangement of v O . Lemma 4.2 gives that with γ s,d := + s d > . Notice that Lsṽ (y) is positive for small |y| but negative for |y| ∼ . Next, we need to change the constant γ s,d into β/ in the last formula, and this is done by rescaling as follows: we introduce a parameter λ > and set v D (y) := λ sṽ (λy) = λ s κ s,d ( − λ |y| ) +s + , (4.32) Fixing the value λ := β/( γ s,d ) , and setting R D = /λ we observe that: (ii) we have the regularityṽ ∈ C ,s (R d ), and (iii) for every y ∈ B R D ( ) we have Thus, v D solves the problem  (i) to deal with the supports we argue by contradiction. Assume that their supports are di erent. This means that we may suppose that R > R D (the opposite situation can be treated with the very same argument). We letṽ (y) := in such a way thatṽ is the unique solution of but this is impossible due to the C ,s regularity of w.
(ii) Now, we call v := v O − v D and we observe that it solves

. . Adjusting mass and constant.
The fact that v D = v O permits to construct a solution v C of (4.13) for any parameter C > simply by rescaling the solution v O of the obstacle problem (4.18). The free constant C > allows to x at will either the radius of the support or the mass of the self-similar solution. More precisely, we have the following result.

Proposition 4.5. For any C > there exists a unique solution v
v C > , and Ls v C = C − β |y| a.e. in B R C ( ), (4.37) which is supported in the ball of radius This constant coincides with known values for the limit cases s = and s = agree. More precisely, for s = we get the self-similar solution of Barenblatt type for the PME with value C = / in 1D and C = / (d + ) in higher dimensions. For s = (Thin Film) it is known that C = / in 1D, while we get C = / (d + )(d + ) for d ≥ .

. . Self-similar weak solutions with a connected positivity set.
Now, we address the question of the existence of self-similar weak solutions to (1.1) with a connected positivity set. As we will see, this is a regularity question about the solutions (4.15). More precisely, we remark that we look for weak solutions in the sense of De nition 3.1. This means that u belongs to H +s (R d ) for a.e. t ∈ ( , +∞). The same regularity holds also for the self-similar pro le v given by (4.1). Therefore, the arbitrary constant K in the decomposition formula (4.15) must vanish. In fact, as we have already observed, the v component of the general solution (4.15) is indeed a rescaled version of the Getoor solution v G (y) = κ − s,d ( − |y| ) s + , y ∈ R d and we have v G ∉ H +s (R d ). (4.44) It is interesting to observe that with minor modi cations one can also prove that in general that v ∉ when P is smooth, bounded and satisfying the internal ball condition.
To prove (4.44) we can reason as follows. On the one hand, we observe that if s ∈ ( , / ] then ∇v G is neither in L (R d ). In fact, we have On the other hand when s ∈ ( / , ), we observe that v G (y) := g(|y|) with g(t) = κ − s,d ( − t ) s + . The function g does not belong to H +s (R). In fact, if g ∈ H +s (R) then we would have that g ∈ H s (R) ⊂ C (R), thanks to Sobolev embeddings. This is impossible since g → −∞ for t → − . Now, since (see e.g. [42]) (4.45) where ω d is the measure of the unitary sphere in R d , d ≥ . Thus, since we already know that g ∈ L (R d ), the last inequality would imply that g ∈ H +s (R d ), absurd. Therefore, we must take K = in (4.15) and thus the self-similar solutions complying with the regularity prescribed by Theorem 3.1 are rescaled version of the model solution v D in (4.32). In particular, (see Lemma 4.5) the constant C is xed according to the mass law (4.39). Moreover, the fact that v D is indeed the solution of an obstacle problem re ects in a kind of minimality, with respect to the energy (4.17), of the self-similar solution.
The following Theorem clari es the situation. , is a weak solution (in the sense of De nition 3.1) of (1.1) with mass M > , and it satis es (4.48) and we obtain, by a direct computation that it is a distributional (self-similar) solution of (1.1) such that (4.49)

Long time analysis
In this Section we address the long time behavior of the weak solutions constructed in Theorem 3.1. Our rst result on the long-time behavior is Theorem 5.7 in which we prove that the set of cluster points for τ → +∞ (that is, the ω-limit set de ned in (5.35) below) of the weak solution to the Fokker-Planck equation (5.25) (see below for the de nition) is not empty and that its elements are indeed weak stationary solutions of (5.25). This proof needs an extra assumption on the regularity of the class of weak solutions, see (5.7), that seems technical to us. Let us brie y explain the problem: Unfortunately, the basic energy estimate (see (3.5)) available for the weak solutions of (1.1) does not rescale directly to an analogous energy estimate (see (5.33)) for the weak solutions of the Fokker-Planck equation. This estimate should contain, as a formal computation reveals, both the second moment and the fractional energy ´R d |L s/ u| and appears to be given by a proper balance between these two terms. We obtain (5.33) by rescaling an improved energy estimate for the weak solutions of (1.1) that contains both a fractional energy This identity resembles a Pohozaev identity and furnishes the exact balance between the second moment and the fractional energy. As the proof of Lemma 5.4 reveals it holds for functions with some decay at in nity in order to guarantee that the right hand side makes sense. Therefore, this is the class for which we address the long time behaviour. It is important to observe that weak solutions with compact support actually satisfy (5.1). It is an open problem to prove that (5.1) holds for all weak solutions.
Our second result is Theorem 5.9 in which we are interested in relating the long-time dynamics of (1.1) with the self-similar solutions constructed in Section 4. At this stage our analysis needs some connectedness assumption on the elements of the ω-limit set of a weak solution v. This assumption permits to conclude that the only stationary solution that attracts the dynamics for large times is the compactly supported self-similar solution v C constructed in Theorem 4.6 with the constant C adjusted to match the mass constraint. As a result, we will obtain the long-time asymptotics As a starting point we prepare the following technical Lemma Thus, the second integral is bounded bŷ Collecting all the estimates we have the thesis.

. Moments and Re ned Energy Estimate
In this Subsection we show that if the initial condition u has nite second moments, then the weak solutions starting from u maintain their second moments nite. This fact combined with the energy estimate (3.5) gives a re ned energy estimate that turns to be fundamental for the long time behaviour analysis. The estimate on the second moments and the re ned energy estimate hold for those weak solutions constructed in Theorem 3.1 that verify that for a.a. t ∈ ( , +∞) Under this condition we prove (see Lemma 5.4 The condition (5.7) is not optimal in terms of regularity and serves to guarantee that the right hand side of (5.8) makes sense. In particular, what seems to be needed for the proof is a good decay at in nity for the solutions.
It is interesting to observe that (5.8) holds, without invoking (5.7), for weak solutions with compact support. It is an interesting open problem to verify its validity for all the weak solutions given by Theorem 3.1.

5.1.1.
Control of the second moment. We state such control for a special class of weak solutions satisfying the just stated assumptions.
To ease the presentation and to convey the main ideas, we work at rst with smooth solutions with a good decay at in nity. Thus, we let u be a smooth solution of We have d dt Thus, thanks to Lemma 5.4 below we get d dt This identity is interesting since similar relations are available ( [68]) for other evolutions of gradient ow type such as the Porous medium equation, both the classical one and both the fractional one (see, for this last case equation 5.24 below). The computation above is of course only formal since we can not use |x| as a test function in the de nition of weak solution. However, thanks to the estimate on the rst moments, we have the following Lemma 5.2 (Second Moments). Let u be a weak solution such that (5.7) holds, then that is the thesis.
Now we prove the validity of the key equality (5.8).
The rst step in the long time analysis is the following Theorem in which we prove that the set of the cluster points for large times of the weak solutions to (5.25) is not empty and its elements are indeed stationary solutions. More precisely, we set It remains to show that the limit v∞ is indeed a stationary solution. To this end, we standardly de ne vn(·) := v(· + τn). For any n ∈ N, vn is a weak solution in the sense of De nition 5.1 with initial condition vn( ) = v(τn).
Therefore vn satis es both the estimates (5.32) and (5.33  namely the mass conservation. The convergences above are enough, as in the proof of Theorem 3.1 to pass to the limit in the weak formulation (5.27) and obtain thatv∞ is indeed a weak solution of (5.25). Moreover, (5.33) gives that, for any T > , Therefore, thanks to the above proved weak and strong convergences we conclude that +∞ ˆR d |H∞| dydr = , (5.54) where the vector eld H∞ ∈ L ( , +∞; L (R d )) is the weak limit of Hn and satis es v / ∞ H∞ = ∇ (w∞ + β |y| )v∞ − w ∞ + β |y| ∇v∞, Therefore (5.54) gives that H∞ = almost everywhere in R d × ( , +∞). Thus, (w∞ :=w∞ for any φ ∈ C c (R d × ( , T)) and thus we have thatv∞ is constant in time, hencev∞ = v∞ for all τ ≥ . In particular, we conclude that v∞ satis es (5.38).
We have the following Proposition 5.8. Let v be a weak solution of (5.25) constructed according to Proposition 5.6 and let v∞ ∈ ω(v Thus, from it follows that, in any connected component of P∞, we get thatw∞ is constant. In fact, for any δ > let us consider the set Due to the Sobolev regularity of v∞ this set is quasi open (see [49] for the de nition). Now, for any xed R > and x ∈ R d , thanks to (5.56), we have that Consequently, (5.56) implies that ∇w∞ = almost everywhere in B R (x ) ∩ P δ , for any x ∈ R d , for any R > and for any δ > which implies thatw∞ is constant on any connected component of P∞.
We can now state the main result of this Section.
where v C is the solution of the obstacle problem provided by Theorem 4.6 with the constant C determined by the mass law (4.39). Therefore (recall (4.1), (4.46) and (5.26)), in terms u we have the following large times convergence Proof. The Proposition above shows that v∞ solves Thus, the assumption of connectedness of P∞, gives that v∞ can represented as in (4.15) with K = due to the regularity. In this way, thanks to Lemma 4.4, we conclude that v∞ is the obstacle solution v C with the constant C given according to the mass law (4.39). Therefore, up to a subsequence (see (5.51) Then, the uniqueness of the solution of the obstacle problem, gives that the convergence above holds not only for a subsequence of times and therefore (5.60) is satis ed. The convergence for u follows from the de nition of v in (4.1).
We conclude this Section with some comments. Both Theorems 5.7 and 5.9 work for those weak solutions that satisfy the extra assumption (5.7). As we observed, the proof that all weak solutions satisfy (5.7) constitutes a challenging open problem. We observe that we can actually dispense with this assumption at the price of introducing an extra approximation at the level of the Fokker Planck equation. This approximation is analogous to the approximation we used for proving existence in Theorem 3.1 and produces weak solutions of the Fokker Planck equation that satisfy the estimates (5.32) and (5.33). This would correspond in studying as a rst the long time behavior of these weak solutions of the Fokker Planck equation and then in obtaining as a second step the convergence to the self-similar solution of the weak solutions of the thin lm equation by rescaling. Unfortunately, this procedure has a potential oddity since, due to nonuniqueness, the weak solution of the thin lm equation that we obtain from rescaling back the weak solution of the Fokker Planck is not necessarily one of the weak solutions we construct in Theorem 3.1.
This explains why we chose to include (5.7) as a suitable extra regularity assumption.

An extension to higher order problems with similar structure
An important feature of equation (1.1) is its conservation law structure, that we may display as The particular equation depends on the closing relationship between u and p. For instance, to obtain equation where K can be a local or nonlocal operator, even of higher order than . The case K = (−∆) m with m > has been rst studied to our knowledge in [13] and then in [25,[33][34][35] and others. Work is mostly done in one space dimension. Self-similar higher order solutions. As an advance to the theory of higher order equations, we contribute here the calculation the regular self-similar solution for the equations of the form where s ∈ ( , ), A + s = (−∆) +s = Ls • (−∆) and Ls, is the fractional Laplacian as in previous section. The dimension d ≥ . The order of the equation is then + s ∈ ( , ). The theory of existence for general equations of the type (6.2) has not been done but it should follow the steps of Section 3.
(i) If again we look for solutions of the self-similar form where the function v : R d × R → R is to be appropriately determined and the parameters α and β are now given by α = d d + ( + s) , β = d + ( + s) . (6.4) due to the constraints that we will nd below. We set Assuming all the regularity needed to justify the computations, and after calculations that have no novelty, we arrive following nonlinear and nonlocal Fokker-Planck type equation: We make a reduction in the set of possible solutions and concentrate on those stationary solutions of (6.5) such that v∇y w As in the parallel study made in [22] for negative values of s, this reduction must be justi ed by the later analysis of the long-time behavior and the asymptotic convergence to a self-similar pro le. Obtaining a solution is then reduced to the famous complementarity rule: either v = or ∇y w + β |y| = . Furthermore, and the second condition will be simpli ed to nding a ball where w = C − β |y| for some C ∈ R.
. Explicit form. The solution of the stationary self-similar problem can be explicitly computed as follows.  The splitting into these two functions will be very convenient. Note that −∆V = outside of the support, and −∆V is a smooth function globally, since there is no delta function (measure) at the support boundary because the normal derivative of V at r = R is zero.
(ii) Next we prepare some very precise calculations. It is convenient to de ne with γ s,d := + s d > . Notice that Ls v (y) is positive for small |y| but negative for |y| ∼ . Next, we need to change the constant γ s,d into β/ in the last formula, and this is done by rescaling as follows: we introduce a parameter λ > and set v λ (y) := λ s v (λy) = λ s κ s,d ( − λ |y| ) +s + , (6.12) For every y ∈ B /λ ( ) we have Ls v λ (y) = λ s (Ls v )(λy)λ s = (Ls v )(λ y).
(iii) In this step we proceed towards the solution V by adjusting F (y) in (6.9) to formula (6.14). Forgetting for the moment about A and K which are the free constants, we determine the main constant a > by the relationship V . For s = and d = we are dealing with the Thin Film equation in one dimension, and then a = / that is consistent with the explicit solution found by Bernis-Peletier-Williams in [14].
Moreover, the free constants K > and A > are related by

Open problems
In this nal Section we collect some open problems that we nd worth considering.
• G F . An interesting open problem, motivated by the decaying of the energy E de ned by (1.8), is whether the evolution (1.1) is a Wasserstein gradient ow for E . This is the case indeed for the related model (1.2), which was shown in [52] to be a Wasserstein gradient ow for the · H −s (R d ) -norm, and for the Thin Film equation (s = ) (see [55]). More in general, an interesting problem is to understand whether (1.1) with a concave mobility m(u) = u γ is indeed a gradient ow for E with respect to a weighted Wasserstein distance of the type of [29].
• C . In Section 4 we have constructed self-similar solutions with compact support. These are weak solutions (for t ≥ ) according to De nition 3.1 that originate from a Dirac Mass located in t = − . For the moment these are the only solutions we are able to construct that are compactly supported. It is clearly interesting to understand whether compactly supported initial conditions generate compactly supported solutions. This is indeed a quite complicated question since the equation is formally of order + s and thus we do not have comparison arguments at our disposal. If the solutions are compactly supported a free boundary appears and must be studied. This is a di cult open problem that was been thoroughly investigated for the PME, see for instance [69] and the recent work [48], where extensive references are given. The topic has also attracted lot of attention for the Thin Film equation, see without any claim of completeness [12], [43] and [37]. A general reference for the mathematics of free boundaries is [18].
• S -. As discussed in the paper, the self-similar solutions of equation (1.1) are given by the Barenblatt pro les v(y) = (C − C |y| ) +s + , (7.1) which coincides with the Barenblatt pro le for the standard Porous Medium equation with m = s+ s+ . We nd this coincidence quite interesting and worth to be further analysed. Note on this regard that when s = (hence m = / ), namely thin lms with linear mobility, this observation has been already successfully used in [24] for the long time behaviour of the thin lm equation. • U . So far we have proved existence of a weak solution. A natural question is to understand whether some uniqueness holds, at least in -D. This is an interesting problem already for s = , namely the Thin Film equation (see [54] and references therein). In particular, it would be interesting to see if there is uniqueness when there is a Dirac Mass as initial data. This uniqueness result, if true, would be important in the convergence to self-similar solutions as in the so called "three steps method" for the classical porous medium equation, see [69,Chapter 18 ].) • M -B . Theorem 5.9 requires the hypothesis of connectedness of the omega-limit set of a weak solution v of (5.25). An interesting problem is clearly to understand if this assumption is really necessary. In particular, it would be interesting to exclude the presence stationary states with disconnected support or to provide examples of multi-bump asymptotic limits. This problem is clearly related to the construction of self-similar solutions for which the positivity set is disconnected.
• S . As we have already pointed out, Equation (1.1) interpolates between the Porous Medium equation (s = ) and the Thin Film equation (s = ). A natural question is to investigate these singular limits for the constructed solutions, and rigorously relate these three equations.
• P L . The analysis of (1.1) has been restricted to a linear mobility function. The case of a power law mobility function of the type m(u) = u n is, to the best of our knowledge, open in dimension d ≥ (see [66] for the one dimensional case in a bounded interval with Neumann boundary conditions) and deserves to be studied. In particular, it would be interesting to understand the relation (if any) between the order of fractional di erentiation s, the exponent n and the dimension d for the existence of nontrivial compactly supported self-similar solutions. When s = and d = a quite complete picture is given in [14], while for s = (PME) the situation is understood in all dimensions [69]. For the porous medium equation with nonlocal pressure, case − < s < , this is studied in [62][63][64], and for s = − in [58].
• R C H E . The analysis of (1.1) suggests that it would be interesting to consider the following evolution ∂ t u = div (m(u)∇p) in R d × ( , +∞) w = Ls u + f (u), in R d × ( , +∞), (7.2) where f : R → R. The equation above can be considered as fractional version of the Cahn-Hilliard equation with nonconstant mobility, and to the best of our knowledge, it has been studied only in [1] for bounded domains with Neumann boundary conditions and with m independent of u. The Cahn-Hilliard equation plays a central role in material science and its analysis (see, among the others, [8], [9], [31]) suggests that there should be a precise relation between the mobility function and the nonlinearity f .
• I . A transformation that has been very useful in the study of similar equations of order from 0 to 2 in one space dimension is the integration transformation v(x, t) =ˆx −∞ u(y, t) dy.
This allows to pass from equation (1.1), i.e., uu = (u(p(u)x)x, to v t = vx p(vx)x, which for p(u) = Ls u gives v t = vx (Ls v)xx .
Our results can be transferred to the latter equation but otherwise no more seems to be known. Let us point out that the study of that equation for − < s < has been very fruitful thanks to the maximum principle that allows for the theory of viscosity solutions and comparison results, cf. [16,62].
• N . The theoretical results would greatly bene t from the development of e cient numerical methods for (1.1), in particular in dimension one, in view of the potential application to cracks dynamics (see [45] and [46] and the references therein.)