A Proof of Theorem 2.6 (a)

# [

## Abstract

We study well-posedness and equivalence of different notions of solutions with finite energy for nonlocal porous medium type equations of the form

 ∂tu−Aφ(u)=0.

These equations are possibly degenerate nonlinear diffusion equations with a general nondecreasing continuous nonlinearity and the largest class of linear symmetric nonlocal diffusion operators considered so far. The operators are defined from a bilinear energy form and may be degenerate and have some -dependence. The fractional Laplacian, symmetric finite differences, and any generator of symmetric pure jump Lévy processes are included. The main results are (i) an Oleĭnik type uniqueness result for energy solutions; (ii) an existence (and uniqueness) result for distributional solutions with finite energy; and (iii) equivalence between the two notions of solution, and as a consequence, new well-posedness results for both notions of solutions. We also obtain quantitative energy and related -estimates for distributional solutions. Our uniqueness results are given for a class of functions defined from test functions by completion in a certain topology. We study rigorously several cases where this space coincides with standard function spaces. In particular, for operators comparable to fractional Laplacians, we show that this space is a parabolic homogeneous fractional Sobolev space.

u
###### :
3

On solutions with finite energy]On the well-posedness of solutions with finite energy for nonlocal equations of porous medium type \contact[felix.delteso@ntnu.no, jorgen.endal@ntnu.no, espen.jakobsen@ntnu.no]Department of Mathematical Sciences
Norwegian University of Science and Technology (NTNU)
N-7491 Trondheim, Norway

Dedicated to Helge Holden, who never stops inspiring us, on the occasion of his 60th birthday.

5A02, 35B30, 35D30, 35K55, 35K65, 35R09, 35R11

niqueness, existence, energy solutions, distributional solutions, nonlinear degenerate diffusion, porous medium equation, Stefan problem, fractional Laplacian, nonlocal operators, bilinear forms, Dirichlet forms, homogeneous fractional Sobolev spaces.

## 1 Introduction

In this paper we study uniqueness and existence of solutions with finite energy of the following two related Cauchy problems of nonlocal porous medium type,

 ∂tu−Aλ[φ(u)] =0 inQT:=RN×(0,T), (1.1) u(x,0) =u0(x) onRN, (1.2) and ∂tu−Lμ[φ(u)] =0 inQT, (1.3) u(x,0) =u0(x) onRN, (1.4)

where is the solution, , and are nonlocal (convection-) diffusion operators, the nonlinearity is any continuous nondecreasing function, and . The problems are nonlinear degenerate parabolic and include the fractional porous medium equations [26] where and for and . Included are also Stefan problems, filtration equations, and generalized porous medium equations, see the introductions of [26, 24, 22] for more information.

Both problems are connected to a bilinear energy form defined as

 Eλ[f,g]:=12∬RN×RN∖D(f(y)−f(x))(g(y)−g(x))Λ(dx,dy), (1.5)

where is the diagonal and is a nonnegative Radon measure on . The operator is the generator of defined by

 Eλ[f,g]=−∫RNfAλ[g]dx (1.6)

(see Corollary 1.3.1 in [29]), while for the special case where . In general is symmetric, -dependent, and has no closed expression, while is an -independent operator with integral representation

 Lμ[ϕ](x)=∫RN∖{0}(ϕ(x+z)−ϕ(x)−z⋅Dϕ(x)1|z|≤1)μ(dz), (1.7)

where is the gradient, an indicator function, and a symmetric (even) nonpositive Lévy measure satisfying . The operator is nonnegative and symmetric and the fractional Laplacian is an example.

A first warning is that is not a pure diffusion operator in general: Under density and symmetry assumptions on , will have an integral representation like (1.7) with -depending plus an additional drift term! A second warning is that the -dependence in is restricted, e.g. is not covered! We refer to Section 2.1 for precise assumptions and to Section 2.4 for a discussion and examples of .

The inspiration for this work were the two recent papers [24] and [22] which contain well-posedness results for energy (or weak) solutions of (1.1)–(1.2) and distributional (or very weak) solutions of (1.3)–(1.4) respectively. These very general results requires different techniques and formulations. The uniqueness argument of [22] is based on a complicated resolvent approximation procedure of Brézis and Crandall [18], while in [24] it is based on an easier and more direct argument by Oleĭnik et al. [32].

The first part of this paper is devoted to Oleĭnik type uniqueness arguments for (1.1)–(1.2). We try to push this argument as far as possible, and in the process we extend some of the results and arguments of [24]. E.g., we remove absolute continuity, symmetry, and comparability assumptions. We also discuss the applicability and limitations of the method. Our uniqueness results are given for a class of functions defined from test functions by completion in a certain topology. We study rigorously several cases where this space coincides with standard function spaces. In particular, for operators (globally) comparable to fractional Laplacians, we show that this space is a parabolic homogeneous fractional Sobolev space. In an appendix we also provide rigorous definitions and results of these spaces, some of which we were not able to find in the literature.

In the second part of the paper we study the equivalence between energy and distributional formulations in the setting of (1.3)–(1.4). A main result is a new existence result for distributional solutions with finite energy. This existence result and the uniqueness result of [22] is then transported from distributional solutions to energy solutions by equivalence, while the Oleĭnik uniqueness results of the first part is transported in the other direction. These result are all either new, or for the Oleĭnik results, represent a much simpler approach to obtaining uniqueness compared to [22]. At the end, we give several new quantitative energy and related -estimates for distributional solutions.

The type of bilinear form defined in (1.5) plays a central role in probability theory. It is associated with a Dirichlet form and a corresponding symmetric Markov process, see e.g. [29] for a general theory. The type of “nonlocal” bilinear form we consider here is similar to those studied in e.g. [35, 5]. In the linear case (), equations (1.1) and (1.3) are (at least formally) Kolmogorov equations for the transition probability densities of the corresponding Markov processes (see e.g. Section 3.5.3 in [4]).

Let us now give a brief summary of previous works on (1.1)–(1.2) and (1.3)–(1.4). We focus first on the -dependent equation (1.1). In the linear case there is a large amount of literature. Some of the main trends in the more PDE oriented community are described in the two surveys [31, 37] (along with extensions to other types of nonlinear equations). When is nonlinear, we are not aware of any other result than the ones presented in [24]. There the authors consider operators where the densities of the measures are comparable to the density of the fractional Laplacian. Existence and uniqueness is discussed in the first part, but the main focus of the paper is to prove continuity/regularity and long time asymptotics for energy solutions.

There is a vast literature on special cases of (1.3)–(1.4). In the linear fractional case for , we have well-posedness even for measure data and solutions growing at infinity [6, 13]. If we replace by an operator whose measure has integrable density, well-posedness results can be found in [17]. In the case of the fractional porous medium equation (see above), existence, uniqueness and a priori estimates are proven for (strong) -energy solutions in [25, 26]. We also mention that there are results for that equation in weighted -spaces [14], with logarithmic diffusion () [27], singular or ultra fast diffusions [11], weighted equations with measure data [30], and problems on bounded domains [12, 15, 16]. There are other ways to investigate these equations: In [10, 19, 38, 9, 40], the authors consider a so-called porous medium equation with fractional pressure, and in [3] they consider bounded diffusion operators that can be represented by nonsingular integral operators on the form (1.7). Finally, we mention that in the presence of (nonlinear) convection in (1.3)–(1.4), additional entropy conditions are needed to have uniqueness [1, 20, 21]; a counterexample for uniqueness of distributional solutions is given in [2].

### Outline

In Section 2 we state the assumptions and present and discuss our main results. The main uniqueness result is proven in Section 3. Properties such as equivalence of distributional and energy solutions, existence of distributional solutions with finite energy, and energy and -estimates are finally proven in Section 4. In Appendices A, B, and C we give rigorous results on the Sobolev spaces we use in this paper along with the proofs of characterizations of the uniqueness function class in terms of common function spaces.

### Notation

We use the same notation as in [22] except for the ones we explicitly mention here: The (Borel) measure is said to be even if for all Borel sets . We say that the (Borel) measure is symmetric if . A kernel on satisfies: (i) is a positive measure on for each fixed ; and (ii) is a Borel measurable function for every . An operator is symmetric on if . From the bilinear form defined in (1.5) we define a seminorm (the energy) and a space,

 |f|2Eλ:=Eλ[f,f]andEλ(RN):={f\textupismeasurable:|f|Eλ<∞},

and the related parabolic (energy) seminorm and space,

 |f|2T,Eλ:=∫T0|f(⋅,t)|2Eλdt, L2(0,T;Eλ(RN)):={f:QT→R\textupismeasurable:|f|T,Eλ<∞}.

The Cauchy-Schwartz inequality holds in this setting (cf. Lemma 3.1):

 ∣∣∣∫T0Eλ[f(⋅,t),g(⋅,t)]dt∣∣∣≤|f|T,Eλ|g|T,Eλ.

## 2 Main results

In this section we give the assumptions, main results, and a discussion of these. There are two sections with results. Section 2.2 contains a sequence of uniqueness results for energy solutions of (1.1)–(1.2), while Section 2.3 contains results about (1.3)–(1.4). There we prove the equivalence of energy and distributional solutions with finite energy, the existence of the latter type of solutions, and transport uniqueness and existence results between the two formulations. The results we obtain are either new or represent a much more efficient way to obtain such results compared to previous arguments.

### 2.1 Assumptions

We start by the bilinear form defined in (1.5). To have a more practical formulation of the assumptions, we first rewrite (1.5): We assume that has as kernel with respect to , change variables , and set to obtain

 Eλ[f,g]=12∫RN∫|z|>0(f(x+z)−f(x))(g(x+z)−g(x))λ(x,dz)dx. (2.1)

Our assumptions on can then be formulated as follows:

• has as kernel on ,

 Λ(dx,dy)=~λ(x,dy)dx.
• The translated kernel satisfies

• (i)  ; and

• (ii)  .

• is symmetric,

 ∫A∫BΛ(dx,dy)=∫B∫AΛ(dx,dy)for all BorelA×B⊂RN×RN∖D.

In some results, we need to strengthen assumption (A).

• Assumption (A) holds and in addition

• (i)  ; and

• (ii)   is locally shift-bounded: For some constant ,

 λ(x+h,B)≤Cλ(x,B)for allx,h∈RN, |h|≤1, Borel B⊂B(0,1)∖{0}.
• Assumption (A) holds and in addition

 Extra open brace or missing close brace

for some , , and every .

The remaining assumptions we will use in this paper are given below.

• is an even Radon measure on satisfying

 ∫|z|≤1|z|2μ(dz)+∫|z|>11μ(dz)<∞.
• is continuous and nondecreasing.

• .

###### Remark 2.1.
1. By (A) and (A), is well-defined on , nonnegative and symmetric,

 Eλ[f,f]≥0andEλ[f,g]=Eλ[g,f]forf,g∈C∞\textupc(RN).

Moreover, by Example 1.2.4 in [29], is a closable Markovian form on and its closure a regular Dirichlet form.

2. It is easy to check that (A”) (A’) (A), see also the remarks on locally shift-bounded kernels in Section 2.4. Assumption (A”) implies that is comparable to , and local shift-boundedness in (A’) is used to show that functions with finite energy can be approximated by test functions (cf. Theorem 2.6).

3. By (), the operator defined by (1.7) is well-defined on , nonpositive and symmetric. The generator of any symmetric pure jump Lévy process is included, like e.g. the fractional Laplacian and symmetric finite difference operators.

4. If , then

 {(\textupAμ)}⟹{(Aλ0)},{(Aλ1')}, \ and \ {(Aλ2)}.

The first two trivially hold, while (A) holds by e.g. Lemma 6.4 in [22].

5. Without loss of generality we can assume (by adding a constant).

### 2.2 Uniqueness results for energy solutions

In this section we give several uniqueness results for energy (or weak) solutions of (1.1)–(1.2). These results follow from an extension of the Oleĭnik argument.

###### Definition 2.2 (Energy solutions).

A function is an energy solution of (1.1)–(1.2) if

1. ; and

2. for all

 Missing \left or extra \right
###### Remark 2.3.
1. The integrals in (ii) are well-defined by (A), (A’), (), and the regularity of and . From (ii) it follows that the initial condition is assumed in the distributional sense ( is a weak initial trace):

 \textupesslimt→0+∫RNu(x,t)ψ(x,t)dx=∫RNu0(x)ψ(x,0)dx∀ψ∈C∞\textupc(RN×[0,T)).
2. By the support of the test functions, we could take  in (i).

To state the uniqueness results, we will introduce spaces in which the Oleĭnik argument works. A particular requirement is that test functions are dense in these spaces w.r.t. to the weakest convergence that can be used in the proof. This is encoded in the following space:

 X:= {f∈L∞(QT)∩L2(0,T;Eλ(RN)): there exists {ψn}n∈N⊂C∞\textupc(RN×[0,T)) such that |ψn−f|T,Eλ→0asn→∞, and ∬QTψnϕdxdt→∬QTfϕdxdtfor allϕ∈L1(QT)asn→∞}.

Below we show that limits can be avoided to get more useful characterizations of such spaces if we (i) go to subspaces, e.g.

 X∩L2(QT)=L2(QT)∩L∞(QT)∩L2(0,T;Eλ(RN)); (2.2)

or (ii) restrict the operator by assuming (A”) which implies

 X=L∞(QT)∩L2(0,T;Eμα(RN)). (2.3)

We refer to Theorem 2.6 below for precise statements.

Our most general uniqueness result applies to energy solutions in the following class of functions:

 UC:={u∈L1(QT)∩L∞(QT):φ(u)∈X}.
###### Theorem 2.4 (Uniqueness 1).

Assume (), (A), (A), and (). Then there is at most one energy solution of (1.1)–(1.2) in .

A proof can be found in Section 3.

###### Remark 2.5.

A similar but less general uniqueness result is given by Theorem 1.1 in [24]. They assume that is absolutely continuous with a density comparable to the Lévy measure of the -stable process, and hence is comparable to . In this case (A’) is satisfied in view of the discussion in Section 2.4.

Note that in general the uniqueness class is smaller than the natural existence class

 EC:={u∈L1(QT)∩L∞(QT):φ(u)∈L2(0,T;Eλ(RN))}.

This is an intrinsic problem with the Oleĭnik argument when it is extended to such general settings as we consider here, and it is also observed in [24]. However, the two classes may coincide under additional assumptions, e.g. if also belongs to or if is comparable to . This is a consequence of the following result.

###### Theorem 2.6.

Assume (A) and (A).

1. If (A’) holds, then (2.2) holds.

2. If (A”) holds, then (2.3) holds.

The proofs are given in Appendices A and C respectively. See also Section 2.4 for a possible alternative based on recurrence. By Theorem 2.4 and Theorem 2.6, we now have:

###### Corollary 2.7 (Uniqueness 2).

Assume (A), (A), (), and () hold.

1. If (A’) holds, then there is at most one energy solution of (1.1)–(1.2) such that and .

2. If (A”) holds, then there is at most one energy solution of (1.1)–(1.2) such that .

###### Remark 2.8.

When the operator is comparable to the fractional Laplacian for (i.e. (A”) holds), the uniqueness and existence classes coincide, and if they satisfy

 UC=EC={u∈L1(QT)∩L∞(QT):φ(u)∈L2(0,T;˙Hα2(RN))}. (2.4)

The latter space is often used in the porous medium setting [43, 26], see also [24]. See Appendix B for rigorous definitions and properties of the homogeneous fractional Sobolev spaces and , some of these we were not able to find in the literature.

Note that if () holds and , then . Now let and assume is locally -Hölder continuous at :

 sup|s|0. (2.5)

Then, since and ,

 u∈L2β(QT)⟹φ(u)∈L2(QT).

By interpolation, functions belongs to for . This leads us to our next result:

###### Corollary 2.9 (Uniqueness 3).

Assume (), (A), (A’), (A), and () hold. If in addition (2.5) holds for some , then there is at most one energy solution of (1.1)–(1.2) such that .

Now we specialize to the case and . Equation (1.1) then becomes equation (1.3). From all the above uniqueness results and Remark 2.1 (d) we obtain the following uniqueness results for (1.3)–(1.4).

###### Corollary 2.10 (Uniqueness 4).

Assume (), (), and () hold.

(a) There is at most one energy solution of (1.3)–(1.4) in .

(b) There is at most one energy solution of (1.3)–(1.4) such that and .

(c) If in addition (2.5) holds for some , then there is at most one energy solution of (1.3)–(1.4) such that .

### 2.3 Equivalence with distributional solutions and consequences

In this section we study the connection between distributional (or very weak) solutions and energy (or weak) solutions. We focus on the simpler case where , and hence the measure is independent of . In other words, we consider the Cauchy problem (1.3)–(1.4). In general, will have an additional drift/convection term compared to , see Section 2.4. This gives rise to a nonlinear convection term in the equation and the possibility that solutions develop shocks (cf. e.g. [20] and references therein). Whether this happens or not here is not known and another reason to avoid this case now.

We state an equivalence result for the two solution concepts, existence and uniqueness results for distributional solutions with finite energy, and then transport these results from distributional solutions to energy solutions. The uniqueness results of the previous section are transported in the opposite direction, and the different uniqueness results are then compared. We also give quantitative energy and related -estimates for distributional solutions.

###### Definition 2.11 (Distributional solutions).

A function is a distributional solution of (1.3)–(1.4) if

 ∬QT(u∂tψ+φ(u)Lμ[ψ])dxdt+∫RNu0(x)ψ(x,0)dx=0 ∀ψ∈C∞\textupc(RN×[0,T)).

The integral is well-defined under the assumptions (), (), and () if also (which is the case when ). This weaker notion of solutions does not require finite energy, but when the energy is finite, the two notions of solutions will be equivalent.

###### Theorem 2.12 (Equivalent notions of solutions).

Assume (), (), , and . Then the following statements are equivalent:

1. is an energy solution of (1.3)–(1.4).

2. is a distributional solution of (1.3)–(1.4) such that

 φ(u)∈L2(0,T;Eμ(RN)).

We prove this result in Section 4.1. In the setting of this paper, it turns out that there always exists distributional solutions with finite energy.

###### Theorem 2.13 (Existence 1).

Assume (), (), and (). Then there exists a distributional solution of (1.3)–(1.4) satisfying

1. ; and

This is one of the main results of this paper and will be proven at the end of Section 4.2. For such solutions we have a new uniqueness result by equivalence, Theorem 2.12, and the uniqueness result for energy solutions in Corollary 2.10.

###### Corollary 2.14 (Uniqueness 5).

Assume (), (), and () hold.

(a) There is at most one distributional solution of (1.3)–(1.4) in .

(b) There is at most one distributional solution of (1.3)–(1.4) such that and .

(c) If in addition (2.5) holds for some , then there is at most one distributional solution of (1.3)–(1.4) such that .

Note that we have uniqueness in a smaller class than we have existence for by Theorem 2.13. This uniqueness result should also be compared to our recent general uniqueness result from [22].

###### Theorem 2.15 (Uniqueness 6, Theorem 2.8 in [22]).

Assume (), (), and (). Then there is at most one distributional solution of (1.3)–(1.4) satisfying

 u∈L1(QT)∩L∞(QT)∩C([0,T];L1\textuploc(RN)).

In particular, any solution from Theorem 2.13 is unique. This result is more general than Corollary 2.14, but the proof is also more complicated. When Corollary 2.14 applies, a greatly simplified uniqueness argument is available (as we have seen).

In view of the equivalence in Theorem 2.12, we can also transport results in the other direction: from distributional solutions to energy solutions. First we obtain a new existence result as an immediate consequence of Theorem 2.13.

###### Corollary 2.16 (Existence 2).

Assume (), (), and (). Then there exists an energy solution of (1.3)–(1.4) satisfying

 u∈L1(QT)∩L∞(QT)∩C([0,T];L1\textuploc(RN)).

In the case of -independent operators, this existence result is much more general than the result given in Theorem 1.1 in [24]. Uniqueness results for energy solutions of (1.3)–(1.4) are given in Corollary 2.10. These results hold for a smaller class of functions than the above existence results. However, a (more) general uniqueness result can be obtained from the result for distributional solutions in Theorem 2.15 and equivalence.

###### Corollary 2.17 (Uniqueness 6).

Assume (), (), and (). Then there is at most one energy solution of (1.3)–(1.4) satisfying

 u∈L1(QT)∩L∞(QT)∩C([0,T];L1\textuploc(RN)).

The proof is immediate. The solutions of Corollary 2.16 are therefore unique, and this result is stronger than the Oleĭnik type result Corollary 2.10. In view of the well-posedness of both energy and distributional solutions and the equivalence between the two notions of solutions, we now have a full equivalence result under assumptions (), (), and ().

###### Corollary 2.18 (Equivalent notions of solutions 2).

Assume (), (), (), and . Then is an energy solution of (1.3)–(1.4) if and only if it is a distributional solution.

We end this section by new quantitative energy and related -estimates for the unique distributional solution provided by Theorems 2.13 and 2.15. This type of estimates are widely used for different local and nonlocal equations of porous medium type, see the discussion in Section 2.4. All proofs are given in Section 4.2. Now, define by . Then we have:

###### Theorem 2.19 (Energy inequality).

Assume (), (), and (). Then the distributional solution of (1.3)–(1.4) satisfies

 ∫RNΦ(u(x,τ))dx+|φ(u)|2τ,Eμ≤∫RNΦ(u0)dx% forτ∈(0,T].

Since , we immediately have a quantitative bound on the energy.

###### Corollary 2.20.

Assume (), (), and (). Then the distributional solution of (1.3)–(1.4) satisfies

 |φ(u)|2T,Eμ≤∥φ(u0)∥L∞(RN)∥u0∥L1(RN)<∞.

There is also a second type of energy inequality that implies -bounds.

###### Theorem 2.21 (Lp-bounds).

Assume (), (), and (). Then the distributional solution of (1.3)–(1.4) satisfies, for ,

 ∫RN|u(x,τ)|pdx≤∫RN|u0(x)|pdxfor allp∈[1,∞),

and in the case ,

 ∥u(⋅,τ)∥L∞(RN)≤∥u0∥L∞(RN).

### 2.4 Remarks

#### Locally shift-bounded kernels

Let be a nonnegative locally finite Borel measure on and a measurable function satisfying

 0

Then the kernel

 λ(x,dz)=j(x,z)μ(dz)

is not only locally, but also globally, shift-bounded in the sense that for all and Borel ,

 λ(x+h,B)λ(x,B)≤Mm.

Examples of are Lévy measures of Lévy processes, e.g. for the -stable process () with the fractional Laplacian as generator. The latter case corresponds exactly to assumption (A”).

#### Recurrence and alternative characterization of X

In Theorem 2.6 (a) approximation by test functions is obtained by an additional assumption on the function class. Alternatively, as in part (b), we can keep the original function class, but restrict the bilinear form (and hence the generator ). In the elliptic setting such results are given in Theorem 3.2 in [36] under the assumptions that (A), (A’) and (A) hold and the closure of is recurrent. A condition ensuring recurrence for symmetric Lévy processes is given in Section 37 in [34]. E.g. the fractional Laplacian for is recurrent if – which is a rather restrictive assumption! Similar results are true in our parabolic setting. Assuming recurrence, or rather, assuming existence of the sequence of cut-off functions mentioned in Lemma 3.1 in [36], we get

 X=L∞(QT)∩L2(0,T;Eλ(RN)).

The proof is an easy modification of the proof of Theorem 3.2 in [36] if we assume (A), (A’), and (A) hold (as in Theorem 2.6 (a)) and, in addition, . Note that the latter condition implies for any such that and . However, this extra condition excludes all Lévy processes and all -independent generators.

### Integral representations of the operators Aλ

In general the operator is abstractly defined from by formula (1.6). However explicit integral representation formulas exist under additional assumptions on the kernel (cf. (2.1)). We follow [35] and assume (A) and (A) hold and

 λ(x,dz)=~λ(x,x+dz)=j(x,x+z)dz,

where is a symmetric measurable function on such that

 ∫|z|≤1|z|∣∣j(x,x+z)−j(x,x−z)∣∣dz<∞.

Symmetric here means that . Note that now in (1.5). By Theorem 2.2 in [35], it then follows that

 Aλ[ϕ](x):=∫|z|>0(ϕ(x+z)−ϕ(x)−z⋅Dϕ(x)1|z|≤1)j(x,x+z)dz+12∫|z|>0z1|z|≤1(j(x,x+z)−j(x,x−z))dz⋅Dϕ(x)

for . Compare with (1.7) and note that the second integral is like a drift term that vanishes if . Under slightly stronger assumptions, this coincides on with the generator of the closure of in – see Proposition 2.5 in [35].

Let us simplify and assume that

 j(x,y)=j1(x,y)μ(x−y)

for symmetric, , and even. This is symmetric and . Taking and , the Lévy density of the fractional Laplacian, we get an -depending fractional Laplace like operator:

 Aλ1[ϕ](x)= −a(x)(−Δ)α2ϕ(x) +∫|z|>0(ϕ(x+z)−ϕ(x)−z⋅Dϕ(x)1|z|≤1)a(x+z)cN,α|z|N+αdz.

From this example we also learn that our class of operators does not include the simplest and most natural -depending fractional Laplace operator,

 −a(x)(−Δ)α2ϕ(x),

since it only satisfies the symmetry assumption on (or (A)) if is constant!

#### On Lp-estimates

If and , then by [26] the estimate corresponding to Theorem 2.21 takes the form

 ∫RN|u(x,τ)|pdx+∫τ0∫RN∣∣∣(−Δ)σ4|u|p+m−12∣∣∣2dxdt≤∫RN|u0(x)|pdx. (2.6)

Note the additional energy term. A closer look at our proof, see Corollary 4.12 and the proof of Theorem 2.21, reveals that we could also have an -estimate with some energy. However, this energy is only a limit and hard to characterize under our weak assumptions.

Such type decay estimates are an essential tool for nonlinear diffusion equations of porous medium type. They imply that belongs to some Sobolev space. This estimate and the Nash-Gagliardo-Nirenberg inequality can be used in a Moser iteration argument to obtain an smoothing effect and then existence of energy solutions with initial data merely in [42, 43, 26, 27, 24]. The other main application of the -energy estimates is as key steps in Sobolev or Simon type compactness arguments. Such arguments are used in [42, 10, 9, 38, 39, 40] to prove existence of energy solutions through the resolution of a sequence of smooth approximate problems and passing to the limit in view of compactness.

## 3 Proof of uniqueness for energy solutions

In this section we prove Theorem 2.4. We start by some preliminary results.

###### Lemma 3.1 (Cauchy-Schwartz).

Assume (A). If , then

 ∣∣∣∫T0Eλ[f(⋅,t),g(⋅,t)]dt∣∣∣≤|f|T,Eλ|g|T,Eλ.

The proof is as for the classical Cauchy-Schwartz and we omit it.

###### Lemma 3.2.

Assume (A). If and , then .

###### Proof.

By Jensen’s inequality and Tonelli’s lemma,

 |g|2T,Eλ ≤∫T012∬(T−t)∫Tt|f(x+z,s)−f(x,s)|2dsλ(x,dz)dxdt =∫T0(T−t)(∫Tt|f(⋅,s)|2Eλds)dt,

and the result follows. ∎

Since an energy solution has some regularity, the weak formulation of the equation will hold also with less regular test functions. We will now formulate such a type of result in the relevant setting for the Oleĭnik argument.

###### Lemma 3.3.

Let be an energy solution of (1.1)–(1.2). If , , and , then for any ,

 ∫T0(−∫RNuϕdx−Eλ[φ(u),∫Ttϕ(⋅,s)ds])dt +∫RNu0(x)(∫T0ϕ(x,s)ds)dx =0.

In other words, we may take in Definition 2.2 for . Note that the integrals are well-defined: see Lemma 3.2. From the proof below it follows that the choice of space is (close to) optimal.

###### Proof.

From the definition of there is