Diffusion on an interval under linear moment conditions

Diffusion processes on an interval under linear moment conditions

Abstract.

We discuss a class of diffusion-type partial differential equations on a bounded interval and discuss the possibility of replacing the boundary conditions by certain linear conditions on the moments of order 0 (the total mass) and of another arbitrarily chosen order . Each choice of induces the addition of a certain potential in the equation, the case of zero potential arising exactly in the special case of corresponding to a condition on the barycenter. In the linear case we exploit smoothing properties and perturbation theory of analytic semigroups to obtain well-posedness for the classical heat equation (with said conditions on the moments). Long time behavior is studied for both the linear heat equation with potential and certain nonlinear equations of porous medium or fast diffusion type. In particular, we prove polynomial decay in the porous medium range and exponential decay in the fast diffusion range, respectively.

Key words and phrases:
Nonlocal conditions for PDEs, Porous Medium Equation, Heat equation, Subdifferentials
2010 Mathematics Subject Classification:
47D06, 47H20, 34B10, 76S05
Version of August 9, 2019
This article was written partially during a visit of the second author at the LAMAV in Valenciennes under the financial support of the Land Baden–Württemberg in the framework of the Juniorprofessorenprogramm – research project on “Symmetry methods in quantum graphs”.

1. Introduction

In [Váz83] J.L. Vázquez made the simple observation that possibly diffusion-type equations of the form

(1.1)

which are well-known to be associated with a well-posed Cauchy problem in if (and even for ), enjoy conservation of mass and barycenter. Here and in the following, is some constant strictly larger than : This result applies therefore to both the porous medium equation (PME) and the fast diffusion equation (FDE) along with the linear heat equation, corresponding to the cases of , and , respectively.

Vázquez’ assertion is easily explained: For a density distribution function denote by the -th moment (say, about 1), i. e.,

Then in particular and represent the total mass and the barycenter of the distribution described by , respectively. Now, differentiating with respect to time the moments of order 0 and 1 of a solution of (1.1) with initial data and integrating by parts with respect to space a simple localization argument shows that

Choosing boundary conditions judiciously, one can see that conservation of mass and/or barycenter may also hold in the case of a PME or a FDE on a bounded interval. Observe that conservation of mass and barycenter can also be defined for solutions so irregular that boundary conditions do not make sense – this is particularly relevant in the case of the PME and the FDE, which are typically solved in spaces of distributions ([Váz07, Chapt. 10]). May then the condition that mass and barycenter be conserved replace boundary conditions altogether?

Imposing conditions on the moments may appear bizarre. However, since certain moment conditions boil down to boundary conditions if solutions are regular enough ([MN11, Cor. 4.10]), at a second glance it looks reasonable to investigate well-posedness under such conditions in the case of a bounded domain. Beginning with [Can63], many authors have studied linear partial differential equations equipped with conditions on the moments complementing those on the boundary values. In [MN11] the present authors have gone on to observe that, in fact, in the case of the linear heat equation one can drop the boundary conditions and obtain well-posedness under a wide class of linear conditions on the moments of order 0 and 1 – and in particular, whenever both of them are assumed to vanish constantly, i.e.,

(1.2)

This special case had already been discussed by A. Bouziani and his coauthors starting with [BB96], see [MN11] for more detailed references. A somehow comparable approach has been followed in [Váz07, § 9.6], where an analysis based on mere finiteness of mass is performed.

There exist counterexamples showing that, in general, moments of higher order are not conserved under the evolution of (1.1) on , cf. [Váz07, § 9.6.4]. Hence it is not natural to expect well-posedness upon imposing a condition analogous to (1.2) for any two moments. Our aim in this article is to show that, however, suitable conditions on and a further moment suffice to obtain well-posedness of certain modified evolution equations – which can be looked at as PMEs or FDEs with a potential depending on .

It turns out that the analysis of diffusion equations on an interval under conditions on and for general can be performed closely following some techniques developed in [MN11]. The extension of such techniques to the more general setting of the present article is discussed thoroughly in Section 2. As it is, the well-posedness results presented in [MN11] are just a special case of those that we obtain in Section 3.

However, serious problems seem to arise in the truly nonlinear case, i.e., whenever we discuss (1.1) for – this is the topic of Section 4. Both the PME and the FDE with Dirichlet boundary conditions are well-known to be the flow of the gradient associated with a suitable energy functional (also known as the functional’s subdifferential in the language of nonlinear semigroup theory, see e.g. [Sho97, Example IV.6.B] and [Váz07, Chapt. 10], and more generally [Bré73] for the abstract theory) with respect to an -inner product. (Observe that a different, more involved but also mightier approach based on flows on Riemannian manifolds has been introduced by F. Otto in a celebrated article [Ott01]). In our setting the gradient flow structure is still present, but unlike in the linear case we are not able to show that for initial data smooth enough this evolution equation is just the PME or the FDE with a certain potential. Nevertheless, we obtain well-posedness of a certain -dependent nonlinear evolution equation that strongly resembles the PME or the FDE, and we are able to show that its long-time behavior depends on . In particular, we show that for all the decay of the -norm of the solutions is polynomial if and exponential if . Our analysis is made different from the classical case by the fact that for this new evolution equation integration by parts is not easily applied – this in turn prevents us from applying the classical method based on the weak formulation of the PME or the FDE, which are the backbone of many proofs in [Váz07].

2. The functional analytical setting

If we consider as the torus , then the test function set is in fact the set of smooth functions in such that the derivatives at all orders coincide at 0 and 1. In the same manner we will use the Sobolev space , by which we denote the subspace of those such that (i.e., of those -functions supported on the torus). We use as pivot space and denote by (resp., ) the dual of (resp., ). For , , we define the Sobolev spaces and likewise. In this paper we refrain from considering the case of (1.1) for , which is known to require a quite different approach, cf. [Váz06, § 2.2.1].

Remark 2.1.

It was already observed in [MN11, Lemma 2.1] that each element of can be identified with an element of , but the identified vector is not unique. We denote this non-injective identification operator by , and by its restriction to which, by [MN11, Lemma 2.4], is an isomorphism.

For all we denote by the linear functional defined by

which is bounded on (and even on for ).

For and we denote by the function defined by

and set for all

(2.1)

which is meaningful since .

We obtain the following analogue of [MN11, Lemma. 2.2], where the attention was devoted to only.

Lemma 2.2.

Let and . Then the following assertions hold.

  1. For all , can be written as

    (2.2)

    where

    In particular, whenever .

  2. Moreover, is a bounded linear operator from to and from to as well as from to .

  3. Finally, in for all .

Proof.

Let be fixed. Denote for a moment the right-hand side of (2.2) (which is by construction an element of ), i.e.,

By integrations by parts for all one has

Accordingly, for by Fubini’s Theorem we find

This proves that , hence (1), and furthermore .

In a second step we first easily check that for , is in with (it is even in ) and that

According to (2.1) we then get

Therefore

which proves that is a bounded operator from to due to the density of into . As the boundedness of from to has been observed above, point (2) is proved.

Finally, the assertion (3) follows the definition of and the fact that for all . ∎

In particular,

Observe that, unlike for , for one has in general

(2.3)

since .

Remark 2.3.

Recall that by [MN11, Lemma 2.3]

defines an equivalent inner product on , and in particular

(2.4)

In fact, we can say more. Taking into account Remark 2.4.(1) and reasoning just like in the proof of [MN11, Lemma 2.3] one can easily see that

Accordingly, we can even say that

defines an equivalent inner product on for all .

Remark 2.4.

Let .

  1. Observe that for all

  2. Let . Then by Lemma 2.2.(1) and integrating by parts

  3. Observe also that

    since for all .

  4. It is also important for the following that

    (2.5)

    as well as

    (2.6)

    both identities are direct consequences of (2.2).

  5. It is a straightforward observation that [MN11, Rem. 2.6] can be generalized as follows: For all , all and all , we have

    (2.7)

    and in particular by (3)

  6. Finally, observe that by Lemma 2.2 and (2.5) is bounded from to and from to .

The crucial point for our investigation is that an integration-by-parts-type formula holds. Recall that we are denoting by the isomorphism between and and let . Then for all and all , by Remark 2.3 one has

where

(2.8)

Hence, (2.6) and a standard integration by parts yield

(2.9)

since by Lemma 2.2.(3) .

Remark 2.5.

Note that

because

– this is the duality between and , with the function given by

For , and we get , but in any case, this is does not really matter since is multiplied by in (2.9).

Indeed, we can still improve the formula in (2.9) for .

Theorem 2.6.

Let and . Then

Proof.

The case has already been discussed. For , our starting point is again (2.9). From the property and an integration by parts, we can write

Hence we deduce that for , the constant in (2.8) is given by

Now, applying (2.5) we find

Applying the previous identities to (2.9) we obtain

and the claimed formula follows. ∎

If , this is the formula already obtained in [MN11, Lemma 2.13]. Otherwise, we have obtained a more general identity that allows us to extend the study of diffusion processes under linear conditions on and (as e.g. in [BB96, MN11]) to linear conditions on and for general .

For each subspace of and each we can consider the reflexive Banach space

Lemma 2.7.

Let . Then for all with pairwise distinct entries, the operator

is surjective.

Proof.

We are going to prove more, namely that the vector valued mapping

is surjective from the space of all polynomials of degree at most to . For fixed, it is well known that there exists a invertible matrix such that

where is the Legendre-type polynomial of degree defined on through the standard Legendre polynomial of degree defined on by

Setting

we deduce that

(2.10)

As the Legendre polynomials are pairwise orthogonal, the mapping

is trivially surjective. By (2.10) and the invertibility of , we conclude that

is also surjective. ∎

We denote by the Hilbert space

(2.11)

It has been shown in [MN11] that an equivalent inner product is given in either case by

(2.12)
Lemma 2.8.

The space is dense in for each subspace of .

Proof.

By construction, and hence the inclusion is clear.

The proof of the converse inclusion is divided in several steps, but follows closely [MN11, Cor. 2.4 and Thm. 4.2]. We first prove the assertion for .

(1) To begin with, observe that it follows from (2.12) that

(2.13)

and

(2.14)

To prove the claimed inclusion, we show that each that is orthogonal to for the inner product is identically zero. In fact, let be orthogonal to . Then it satisfies

But according to its definition (2.1) we get equivalently

(2.15)

Now define

(2.16)

This space is dense in since for , belongs to . Hence for all there exists a sequence of such that

and therefore with

Now for , we take . By construction

Plugging this identity in (2.15) yields

(2.17)

But due to the fact that , we deduce that

As is dense in , we conclude that .

If , we see that

and the assertion follows.

(2) We consider the remaining cases. More precisely, it suffices to study the case of 1-dimensional with , because once proved this for and any 1-dimensional subspace we have

Hence, assume that there exists such that is the set of all satisfying

(2.18)

Let be such that

(2.19)

Since , one has

and reasoning as in (1) we deduce that (2.17) holds for the space defined in (2.16). Since is dense in , this is equivalent to

or again

(2.20)

Coming back to (2.19) and taking into account Remark 2.3 we get

(2.21)

Now, by Lemma 2.7 there exists such that

where is as in (2.18). Hence, we can plug such a in (2.21) and we find , hence, by (2.20), . This concludes the proof. ∎

Lemma 2.9.

Let . Then the vector space

(2.22)

and hence also

are dense in

Proof.

The proof is based on Lemma 2.8, i.e., on the validity of the same assertion in the special case of . Let : we are looking for a sequence here and below in the proof, is seen as a mapping from into ) that approximates in . By Lemma 2.8, we already know that this is possible for : that is, there exists a sequence that approximates in . Now, for observe that the binomial formula yields

with , and accordingly

and in particular

Thus, by continuity,

We distinguish the two cases and .

  1. If , it suffices to take

    (observe that ), so that for all and moreover

  2. If , write , with such that , i.e., . Now, set

    and observe that for one has , hence owing to the first case. As we conclude that as well.

This concludes the proof. ∎

Also the following holds.

Lemma 2.10.

Let and be a subspace of with and . Then the vector space