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Liviu I. Ignat and Diana Stan
Abstract

We consider a convection-diffusion model with linear fractional diffusion in the sub-critical range. We prove that the large time asymptotic behavior of the solution is given by the unique entropy solution of the convective part of the equation. The proof is based on suitable a-priori estimates, among which proving an Oleinik type inequality plays a key role.

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assertion\@definecounterconjecture\@definecounterdefinition\@definecounterhypothesis\@definecounterremark\@definecounternote\@definecounterobservation\@definecounterproblem\@definecounterquestion\@definecounteralgorithm\@definecounterexample Fractional diffusion-convection equations]Asymptotic behaviour of solutions to fractional diffusion-convection equations

1 Introduction and main results

We consider the convection diffusion equation

where , is the Fractional Laplacian operator of order and is a locally Lipschitz function whose prototype is with . This model has received considerable attention since the 1990s due to the interesting phenomena that appear: there is a competition between the effects of the diffusion and convection terms. Depending on the parameters and , the asymptotic behaviour is given by either the solution of the diffusion equation:

or the convective one

or by a self-similar solution of (CD) in a critical case. The classical case has been analysed for all in the quoted papers of Escobedo, Vázquez and Zuazua [21, 22, 23].

In the last twenty years there has been a great interest in models with nonlocal diffusion, specially fractional diffusion since the fractional Laplacian is the infinitesimal generator of a stable Levy process. There are many applications in physical sciences where models with anomalous diffusion are needed, see the survey [45] for a description of possible applications, and the lecture notes [42] for a presentation of recent models involving nonlocal diffusion.

We are interested in the large time asymptotic behavior of solutions to the initial value problem

(1.0)

The critical case makes the difference in the asymptotic behavior since equation (1) is invariant by scaling , and it admits self-similar solutions. In this case the asymptotic behavior of the solutions is given by the self-similar solution with the same mass as the initial datum (see [6]). In the supercritical range , the asymptotic behaviour is given by the fundamental solution of the diffusion model (D) multiplied by the mass of the initial datum (see [7] for ). We will provide more details in next section.

In this paper we consider the case and the nonlinearity in the subcritical range , which has been an open issue so far. The main result of this paper is the following theorem.

Theorem 1.1

For any , and nonnegative there exists a unique mild solution of system (1.0). Moreover, for any , solution satisfies

(1.0)

where is the mass of the initial data and is the unique entropy solution of the equation

(1.0)
  • Remark 1.   We believe that the -assumption on the initial data can be dropped. Through the paper we will consider nonnegative solutions. The general case of changing sign solutions can be analysed following the same arguments as in [14, Section 6]. We emphasise that since the nonlinearity should be locally Lipschitz we should impose . Since we are interested in the subcritical case where the convection is dominant we have to impose and hence should belong to the interval .

An interesting phenomenon happens: the diffusion is dominant over the convection for , having a regularizing effect on the solution. However, when in the asymptotic limit as time goes to infinity the solution approaches the unique entropy solution to the pure convective equation which is discontinuous and develops shocks. This phenomenon has been established for the local case by Escobedo, Vázquez and Zuazua in [21]. In this paper we prove that this behavior holds as long as . This is done using both parabolic and hyperbolic arguments and dealing with the difficulties created by the nonlocal operator and the nonlinearity of the convective term.

The organization of the paper is as follows. In Section 2 we give a panorama on previous results on the model both in local and nonlocal cases. Also we provide a reminder on the diffusion equation which will be useful throughout the paper. In Section 3 we are concerned with the existence and main properties of solutions. Entropy and mild solutions are introduced. The key estimate is given in Proposition 3.4 where we show that for any and any initial data uniformly bounded above and below by two positive constants, the solution of our problem satisfies an Oleinik type inequality, . We emphasize that this estimate does not require . In Section 4 we prove the asymptotic behavior of solutions stated in Theorem 1.1.

2 Preliminaries

2.1 Panorama: from local to nonlocal diffusion

We describe some of the results known so far for this convection-diffusion model. We try to cover all the ranges of parameters and finally to better place our contribution in this field.

The general model is

(2.0)

where is a Lévy type operator, , whose symbol is written in the form

Usually , is a positive semi-definite quadratic form on and is a positive Radon measure satisfying

Two particular cases are the Laplacian, and corresponding to , , and , , respectively.

Local Diffusion. The local diffusion case, i.e. , has been intensively studied for linear diffusion , see [23] for the supercritical and critical cases ( in ) and [21] for the subcritical case in dimension . The subcritical case in any dimension has been analysed in [22] for nonnegative solutions and for changing sign solutions in [13].

Nonlocal Diffusion. There is always a competition between the diffusion, which is differentiable of order , and the convection terms having one derivative. This implies the consideration of certain classes of solutions: entropy solutions, weak solutions, mild solutions. The study takes into consideration the fractional order , the nonlinearity , the dimension and the regularity of the initial data .

Existence of solutions. For all ranges or parameters , , the model admits a unique entropy solution. More precisely, for and locally Lipshitz, the existence and uniqueness of entropy solutions were proved by Droniou [17]. Then Alibaud [1] proved the same for . Cifani and Jakobsen [16] proved the existence of entropy solutions for the degenerate nonlinear nonlocal integral equation with and developed a numerical scheme that gives an idea of the asymptotic behavior of the solution.

The existence of entropy solutions for (2.0) with merely bounded (possibly non-integrable) data has been proved by Endal and Jakobsen [20]. If moreover , and then there exists a unique mild solution with good regularity properties, see Droniou, Gallouet, Vovelle [18].

When the diffusion is smaller, regularity is lost, since the convection has the effect of shock formation. There is non-uniqueness of weak solutions, as proved by Alibaud and Andreianov [2]. However, uniqueness holds in the class of entropy solutions.

Asymptotic Behaviour. Concerning the asymptotic behavior of solutions there are previous works in some ranges of exponents.

(i) Integrable data. When the data is there are previous works in the critical and supercritical cases. The critical case corresponds to when the equation (1) admits a unique self-similar solution with data For the critical case has been analyzed Biler, Karch and Woyczyński [8] who proved that the asymptotic profile as is given by the self-similar solution described above. When the critical exponent is less than one and the nonlinearity would not be Lipschitz which is out the scope of this analysis.

In the supercritical case , , the diffusion is dominant and then the asymptotic behavior of solutions to (1.0) with is given by , the solution of the linear diffusion problem with data (see Biler, Karch and Woyczyński [7, Th. 4.1, Lemma 4.1]). Some results in the one dimensional case were obtained by Biler, Funaki and Woyczyński [6]. The analysis of the linear semigroup generated by (1) shows that the first term in the asymptotic behaviour may be chosen as where is the fundamental solution of problem (1). See for instance [10, Theorem 6.3]. In Section 2.2 we present more details about the linear model (1) and its properties.

When all the nonlinearities considered here are super-critical since . The asymptotic behavior is given again by the linear semigroup. We state in the following theorem the result in the one-dimensional case.

Theorem 2.1

For any , , and there exists a unique entropy solution of system (1.0). Moreover, for any , solution satisfies

where is the unique weak solution of the equation

  • Proof.   The proof should follow as in [3, Th. 1.1, Th. 3.5] by using the technique of approximation with a vanishing viscosity term:

    The asymptotic behavior is proved first for this approximating problem and then by letting for the initial problem. We could also work directly with entropy solutions as in this present paper, but one should consider a parabolic scaling instead of the one used in Section 4. A detailed proof of these fact does not bring great novelty and we consider it is beyond the purpose of this paper.

In this work we make a step further by describing the asymptotic behavior of mild solutions in the subcritical case and dimension one, that is , for bounded integrable data.

(ii) Step-like data. There is an interesting phenomenon when supplemented by a step-like initial datum approaching the constants , , as , respectively. For in [31] the authors study the one dimensional case and they prove that the limit profile is given by a rarefaction wave, that is the unique entropy solution of the Riemann problem

When the convection is negligible and the asymptotic behavior is given by the solution of the diffusion problem (1) with the same initial initial data as above. This is proved in [3] in dimension one. The two-dimensional case of the above results has been analysed by Karch, Pudelko and Xu [32]. The characterization depends on the fractional order and on the direction of the convective nonlinearity in (2.0).

Remarks. (i) There is a connection with Hamilton-Jacobi equations. By considering the integrated solution , it follows that solves the equation , which is a type of Hamilton-Jacobi equation with fractional diffusion. The problem admits classical solutions when ([19, 28]). For this is related to drift-diffusion equations ([38]).

(ii) There is a considerable interest in nonlocal equations with zero-order operators , where is a non-singular, integrable kernel with mass one. This is a quite different topic, since the nonlocal operator does not provide any regularity for the solution, as it happens in the fractional derivative case, and then other techniques must be used. When , the first author considers the model in [14]. The asymptotic behavior is given by the solution of (1.0). The case has been analyzed in [34] and in [26]. There are situations when the convection is also nonlocal, . We refer to [27] for the supercritical case and [25] for the critical case . However, for the subcritical case, i.e. there are no results on the long time behavior of the solutions.

(iii) The case of nonlinear local diffusion also brings considerable difficulties, for instance for porous-medium type diffusion and convection the model becomes . The third parameter of the nonlinearity changes the behaviour of the solution. For slow diffusion and slow convection we refer to Laurençot and Simondon [35]. See [33] for fast convection and slow diffusion . The asymptotics of both fractional and nonlinear diffusion, plus convection has not been considered as far as we know.

2.2 Reminder on linear fractional diffusion

We recall some useful results concerning the associated diffusion problem (1), that is the Fractional Heat Equation for . We consider the initial value problem

(2.0)

This problem has been widely studied and many results are known (see [4, 5, 9] for the probabilistic point of view, [41] for a nice motivation of the model and the recent survey [10] for a complete characterization). Some useful properties are proved in [18, Section 2]. For initial data the solution of Problem (2.0) has the integral representation

where the kernel has Fourier transform If , the function is the Gaussian heat kernel. We recall some detailed information on the behaviour of the kernel for . In the particular case , the kernel is explicit, given by the formula

Kernel is the fundamental solution of Problem (2.0), that is solves the problem with initial data Dirac delta It is known [9] that the kernel has the self-similar form

for some profile function, . For any the profile is , positive and decreasing on , and behaves at infinity like . Moreover, the solution of Problem (2.0) behaves as time as , where is the total mass:

See for instance [10, Theorem 6.3]. Throughout the paper we will need the following time decay estimates on the fractional derivatives of the kernel.

Lemma 2.2

For any , and the kernel satisfies the following estimates for any positive :

(2.0)
(2.0)
(2.0)

We used the notation . The proof of these estimates is given in the Appendix.

3 Existence of solutions and main properties

3.1 Concept of solution: entropy and mild solutions

We now recall some classical results for systems (1.0) and (1.0). In the case of the conservation law (1.0) the entropy formulation is as follows.

  • Definition 1.    ([36]) By an entropy solution of system (1.0) we mean a function

    such that:

    C1) For every constant and , , the following inequality holds

    C2) For any bounded continuous function

The existence of a unique entropy solution of system (1.0), as well as its properties were deeply analysed in [36]. For system (1.0) has an unique entropy solution , see [36, Section 2], which is given by the -wave profile

with .

Let us first recall the representation of the fractional Laplacian in [19]. For any : there exists a positive constant such that for all , all and all the following holds

(3.0)

Using this representation, we introduce, according to [1], the following definition of the entropy solution for system (1.0).

  • Definition 2.    ([1]) Let . We define an entropy solution of Problem (1.0) as a function such that for all , all non-negative , all smooth convex functions and all such that , ,

  • Remark 2.   In the above definition it is sufficient to consider the particular entropy-flux pairs, , , for any real number .

For any and locally Lipschitz there exists a unique entropy solution of Problem (1.0). Entropy solutions belong to . If , then so does , for all , and moreover . All these properties have been proved in [18, 1]. In the above papers the authors introduce a splitting in time approximation in order to prove the existence of an entropy solution. In fact for any they define the approximation in the following way: let ; for all , on the time interval , is the solution of with initial condition , and on the time interval , is the entropy solution of with initial condition . For any initial data in the approximation converges in , , to the entropy solution of Problem (1.0).

In [18], for , and [1] for , the authors prove that the entropy solutions in the sense of Definition 3.1 are solutions in the sense of distributions. Moreover when , Droniou [18] proved that this distributional solution is the unique mild solution in the sense of Definition 3.1 below.

  • Definition 3.   Let and or . We say that a mild solution of Problem (1.0) is a function which satisfies for a.e. ,

    (3.0)

The existence and regularity of the mild solution are given in the following Proposition.

Proposition 3.1

For any there exists a unique global mild solution of Problem (1.0). Moreover satisfies:

(i) .

If then

(ii) . Moreover, .

(iii) for any and solution satisfies and .

  • Remark 3.   Since we have for any that for any . Moreover for any , the map is continuous. The last property also guarantees that various integrations by parts used in the paper are allowed.

  • Proof.   The global existence, uniqueness and the first two properties are proved in [18]. We now prove property (iii). Its proof relays on a classical bootstrap argument: one starts with some regularity of in the right-hand side and obtain that this right hand side term is slightly better than the hypothesis. For a nice review of the method we refer to [40, Ch. 1.3, p. 20]. Let us fix . We first remark that since we have that belongs to the same space. Moreover, it is sufficient to prove that for any , with a norm that is bounded in any interval with .

    The main steps of the proof are as follows: we first prove that for the right hand side in (3.0) belongs to for any , . The next step is to use this new regularity to prove the same for . The last step, the most technical one, is to extend the regularity up to .

    Step I. We first prove that we gain some regularity for , for any and . Let . We have

    (3.0)

    Using the decay of the derivative of in (2.0), (2.0) and that we find that for any the following holds for any :

    Let us now explain why identity (3.0) holds. We know that and by Lemma 2.2 kernel satisfies for any . Hence . Let us now prove that for a.e. the following holds

    (3.0)

    For any , the Tonelli-Fubini theorem can be applied to obtain that

    (3.0)

    Indeed, (3.0) is true since we avoid the singularity of at . Moreover, as , , using (2.0) we obtain that for any the following holds

    Similarly, using (2.0) it follows that

    Therefore we obtain that

    and

    (3.0)

    in any , . In view of (3.0) and (3.0) we obtain that belongs to for any and moreover (3.0) holds in , , so for a.e. .

    This type of arguments apply also in the rest of the paper, whenever one needs to commute with the integral .

    Step II. In order to extend the range of we first recall the chain rule for fractional derivatives (see [24, Prop. 5 (a)], [15, Prop. 3.1]). For any and the following inequality holds

    (3.0)

    where , and .

    Let us now choose two positive numbers and such that , and denote . Applying estimate (3.0) to with , , we obtain

    Assuming that for all we obtain that for any we have

    This means that we always we can gain up to derivatives with respect to the initial assumption.

    Repeating the above argument and using Step I we obtain that for any and any we have for all and

    Moreover, using the properties of the Hilbert transform we also obtain for any and any

    Step III. Let us now consider the case . We write the equation for :

    Let us consider with and . Thus

    Leibniz’s rule ([24, Th. 3], [15, Prop. 3.3]) gives us that