Existence of weak solutions for a general porous medium equation with nonlocal pressure  

Existence of weak solutions for a general porous medium equation with nonlocal pressure  

Abstract

We study the general nonlinear diffusion equation that describes a flow through a porous medium which is driven by a nonlocal pressure. We consider constant parameters and , we assume that the solutions are non-negative and the problem is posed in the whole space. In this paper we prove existence of weak solutions for all integrable initial data and for all exponents by developing a new approximating method that allows to treat the range that could not be covered by previous works. We also consider as initial data any non-negative measure with finite mass. In passing from bounded initial data to measure data we make strong use of an - smoothing effect and other functional inequalities. Finite speed of propagation is established for all , which implies the existence of free boundaries. The authors had already proved that finite propagation does not hold for .

Keywords: Nonlinear fractional diffusion, fractional Laplacian, existence of weak solutions, energy estimates, speed of propagation, smoothing effect, numerical simulations.

2000 Mathematics Subject Classification. 26A33, 35K65, 76S05.

Addresses:
Diana Stan, dstan@bcamath.org, Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009 Basque-Country, Spain.
Félix del Teso, felix.delteso@uam.es, Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009 Basque-Country, Spain.
Juan Luis Vázquez, juanluis.vazquez@uam.es, Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049 Madrid, Spain.

1 Introduction

In this paper we study the following evolution equation of diffusive type with nonlocal effects

(1.1)

for , exponents , , and space dimension . We will only consider nonnegative data and solutions on physical grounds. The problem will be posed in the whole space, with and . Here denotes the inverse of the fractional Laplacian operator as defined in [33].

Our aim is to construct weak solutions for all initial data and for all the stated range of parameters. Model (1.1) formally resembles the Porous Medium Equation when , but here we allow for a new dependence via the inverse fractional Laplacian operator, with , which accounts for nonlocal effects in the diffusive process. We will call this intermediate variable the pressure, though it is not in agreement with the usual PME convention unless .

The problem for was studied by Caffarelli and Vázquez starting with [9, 10], followed by [7, 8, 11]. Our model is a particular case of the general equations proposed in [17, 18] in statistical physics, which take the form . There is also a physical motivation in the theory of dislocations proposed by Head, that has been investigated by Biler, Karch and Monneau [3] for in one space dimension. However, the extension of the dislocation model to several dimensions leads to a more complicated system that falls outside of the present investigation. Finally, we point out that the gradient flow structure for (1.1) with has been recently developed in [22] using Wasserstein metrics in the style of [1]. Uniqueness is still an open problem for all these models in several space dimensions, but it holds for according to [3]. There are recent uniqueness results if the initial data are very smooth, see [37]. They obtain unique local-in-time strong solutions in Besov spaces; thus, for initial data in if and with

Existence of constructed weak solutions for was proved by the same authors in [30, 32] under some extra decay conditions on the initial data. In that paper we employed a rather standard regularization of the singular operator by considering where is a suitable smooth kernel. Energy estimates allowed us to obtain compactness, but only in the range of . New methods seemed to be needed for the more degenerate case . A further discussion on this issue can be found in Section 6. The main step we take here in order to prove existence of weak solutions solutions of (1.1) is a novel approximation method. This consists in interpreting model (1.1) in the form

Then we approximate the operator by

Considering this approach to model (1.1) allows us to prove certain -estimates. These are an essential tool in order to derive convergence of the solutions of approximating problems.

We start by assuming initial data , and we prove existence of a class of weak solutions that we construct using an approximating method. The paper combines a great variety of compactness techniques and the detailed proofs show how the available energy estimates can be used step by step as we pass to the limit in the approximating models. The main difficulties of the construction are: the nonlocal and nonlinear character of the equation, absence of comparison principle, absence of explicit self-similar solutions (except very particular cases, c.f [31]).

A second contribution of the paper is the generality of the initial data. We may take , the space of nonnegative Radon measures on with finite mass. This covers in particular the case of merely integrable data . We cover that issue in Section 5 where we obtain existence of weak solutions for the whole range , generalizing the results of [9] and [32], where the cases and were covered respectively. This rounds up the existence theory.

Another positive property of this approach is that it can be successfully generalized to more general equations of the form

where is a regular function with at most linear growth at the origin.

A remarkable property of many diffusive PDE’s of degenerate type is the finite speed of propagation. When we combine degenerate nonlinearities (powers with ) and nonlocal effects it is not clear whether finite propagation will hold or not. The property was first observed by Caffarelli and Vázquez in [9] for the model with , see also [3] for . In [32] we discovered that the nonlinearity has a strong influence on the speed of propagation property of solutions independently of . Indeed, we proved two different behaviors depending on the exponent : finite speed of propagation for and infinite speed of propagation for . A numerical simulation using [12] pointed us to this change in the positivity property of the solution. We establish here the property of finite propagation for all . See Figure 10.

Let us comment on some related literature. Another possible generalization of the model studied by Caffarelli and Vázquez in [9] has been considered in [2, 3, 20]. They assume that . In this case, there exists a weak solution with finite speed of propagation for the whole range . Moreover, they find explicit Barenblatt self-similar profiles.

We finally recall that there is another model of nonlocal porous medium equation:

(1.2)

with and for which the theory has been developed in [23, 24, 5, 36], see also the survey paper [35]. Infinite propagation holds for this model even if . A very interesting result is the connection between model (1.1) and model (1.2). In [31] we found an exact transformation formula between self-similar solutions of the two models, but it only applies to the range of our present model.

2 Precise statement of the main results

We recall that all data and solutions are nonnegative and we will stress this fact when convenient. In this section will only present the results for integrable and bounded initial data since establishing the existence and main properties in this case contains the main difficulties. For clarity of exposition, we delay to Section 5 the case of measure data since it is an independent contribution of the paper.

Definition 2.1.

Let and nonnegative. We say that is a weak solution of Problem (1.1) if:
(i) , (ii) , (iii) and

for all test functions .

We state our main results on the existence and qualitative properties of solutions.

Theorem 2.2.

Let , , and let and nonnegative. Then there exists a weak solution of Problem (1.1) such that , , and for all . Moreover, has the following properties:

  1. (Conservation of mass) For all we have

  2. ( estimate) For all we have .

  3. ( energy estimate) For all and we have

    (2.1)
  4. (Second energy estimate) For all we have

    (2.2)
Remark 1.

(a) The a priori estimates 1, 2, 3 and 4 for Problem (1.1) can be derived in a formal way as in Section 3 of [32]. A rigorous proof for 1, 2 and 4 when can be found in that paper. The approximation used there does not allow to cover the whole range because of the lack of an type energy estimate like (2.1). However, 1 and 2 follow as in [32] and therefore they will not be discussed here.

(b) We would like to note that estimates (2.1) and (2.2) do not present any special form or extra difficulty when , or , as it happened with the First Energy Estimate (6.1) used in [32] and [9]. See Section 6 for a more detailed discussion about this fact.

Theorem 2.3 (Smoothing effect).

Let be a weak solution of Problem (1.1) with nonnegative initial data as constructed in Theorem 2.2. Then,

(2.3)

where , .

Proof.

We combine (2.1) with the Nash-Gagliardo-Niremberg Inequality (7.2) applied to the function to get a starting point for a Moser iteration. Then we continue as in Theorem 8.2 of [24] where the authors consider the model for . From here, the proof is straightforward. ∎

Remark 2.

In the limit , Theorems 2.2, 2.3 (and also Theorem 5.2) recover some of the results of the linear Fractional Heat Equation (cf. [4]).

Theorem 2.4.

Let , , . Let be a weak solution of Problem (1.1) as constructed in Theorem 5.2 with compactly supported initial data . Then is compactly supported for all , i.e. the solution has finite speed of propagation.

Proof.

Once we construct a weak solution of Problem (1.1), we apply the results from [32]. The proof is based on a careful construction of barrier functions, called true supersolutions in [9]. ∎

3 Functional setting

3.1 The fractional Laplacian and the inverse operator

We remind some definitions and basic notions for the functional setting of the problem. We will work with the following functional spaces (see [16]). Let denote the Fourier transform. For given we consider the space

with the norm

For functions , the fractional Laplacian operator is defined by

for , where Then,

For functions that are defined on a subset with on the boundary , we will use the restricted version of the fractional Laplacian computed by extending the function to the whole with in The same idea is used to define the norm for functions defined in .

If , the inverse operator coincides with the Riesz potential of order . It can be represented by convolution with the Riesz kernel :

where Notice that . When and we have to consider the composed operator . This operator use to be called nonlocal gradient and is denoted by (c.f. [2, 32]). See Section 4.6 for a more detailed discussion of this range.

3.2 Approximation of the fractional Laplacian

Let and . We define the operator

(3.1)

for We will use the notation

This kind of zero-order operators has been considered in the literature, see e. g. [19, 27]. For any , is an integral operator with non-singular kernel and pointwise in as for suitable functions . This approximation can also be seen as a consequence of the fact that the fractional Laplacian can be computed by passing to the limit in the representation of the solution of an harmonic extension problem (using the explicit Poisson formula), as proved by Caffarelli and Silvestre in [6].

We can define the bilinear form

and the quadratic form

The bilinear form is well defined for functions in the space , which is the closure of with respect to the Gagliardo seminorm given by . We define

(3.2)

The space is endowed with the standard norm

Clearly,

(3.3)

We refer to [15] for a precise discussion of these spaces in a more general framework.

Lemma 3.1.

Let . Then, for every , we have that

Moreover,

Proof.

It is clear that since is integrable at infinity and nonsingular at the origin. Let , then

and

The restricted operator. For smooth functions such that on we extend on . Then is well defined by definition (3.1). Let . We take to be the closure of with respect to the quadratic form . Then is well defined on .

Square root. The operator has a square-root in the Fourier transform sense (see [14] Lemma 3.7), that we denote by . We have that

This implies that

where the second identity is obtained by symmetry. We get the following characterization of :

(3.4)
Theorem 3.2 (Generalized Stroock-Varopoulos Inequality for ).

Let . Let such that and . Then

(3.5)

where .

Proof.

We have that:

Now, we use that if is such that and , then

For convenience, we give the proof of this pointwise inequality based on the Fundamental Theorem of Calculus and the Cauchy-Schwarz Inequality:

We deduce, using (3.4), that

Remark 3.

(i) We refer to [15] for a related result with more general nonlinearities and nonlocal operators.
(ii) Note that we recover the classical Stroock-Varopoulos Inequality for by taking :

We refer to Stroock [34], Liskevich [21] where this kind of inequality is proved for general sub-markovian operators.

3.3 Approximation of the inverse fractional Laplacian ,

As a consequence of (3.1) we naturally derive an approximation for the inverse fractional Laplacian and the nonlocal gradient that will play an important role in the sequel to solve the difficulties created by estimates like (6.1) in the range .

Lemma 3.3.

a) Let , and . Then for every such that we have that

b) Let , . Then for every such that we have that

Proof.

a) Given any operator , let be the Fourier symbol associated to the operator whenever it is well defined. Now, we employ Plancherel’s Theorem to obtain:

We want to pass to the limit as in . For that purpose we need to find an dominating function for . We recall that for we have that

(3.6)

Moreover . Note that for every . Then

We conclude that since and , the Schwartz space of rapidly decaying functions. Moreover, we can see from (3.6) that pointwise as . Then we use the Dominated Convergence Theorem to conclude that as

b) The proof follows as above noting that and .

4 Existence of weak solutions via approximating problems

In order to prove existence of weak solutions of Problem (1.1) we proceed by considering an approximating problem. We regularize the degeneracy of the nonlinearity, the singularity of the fractional operator, we also add a vanishing viscosity term to get more regularity and we restrict the problem to a bounded domain. We write the equation in the form

The idea is to consider the approximation of the given by (3.1), that is

defined for functions in the space . We consider the approximating problem

with parameters . We use the notation . The initial data is a smooth approximation of .

Definition 4.1.

We say that is a weak solution of Problem (4) if: (i) , (ii) , (iii) and

(4.1)

for smooth test functions that vanish on the spatial boundary and for large .

An important tool in the proof of existence of weak solutions is the concept of mild solution of Problem (4) , i.e. fixed points of the following map given by the Duhamel’s formula