International Journal of Partial Differential Equations

Volume 2014 (2014), Article ID 450417, 8 pages

http://dx.doi.org/10.1155/2014/450417

Research Article

General Asymptotic Supnorm Estimates for Solutions of One-Dimensional Advection-Diffusion Equations in Heterogeneous Media

José A. Barrionuevo, Lucas S. Oliveira, and Paulo R. Zingano

Departamento de Matemática Pura e Aplicada, Universidade Federal do Rio Grande do Sul, 91509-900 Porto Alegre, RS, Brazil

Received 18 October 2013; Accepted 1 March 2014; Published 8 May 2014

Academic Editor: Chi K. Lin

Copyright © 2014 José A. Barrionuevo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We derive general bounds for the large time size of supnorm values of solutions to one-dimensional advection-diffusion equations with initial data for some and arbitrary bounded advection speeds , introducing new techniques based on suitable energy arguments. Some open problems and related results are also given.

1. Introduction

In this work, we obtain very general large time estimates for supnorm values of solutions to parabolic initial value problems of the form

for arbitrary continuously differentiable advection fields . Here, by solution to (1a) and (1b) in some time interval ,, we mean a function which is bounded in each strip , , solves (1a) in the classical sense for , and satisfies in as . It follows from the a priori estimates given in Section 2 that all solutions of problem (1a), (1b) are actually globally defined , with for each finite. Given , what then can be said about the size of supnorm values for ?

When for all , it is well known that, for each , is monotonically decreasing in , with

for some constant that depends only on ; see, for example, [1–5]. For general , however, estimating is much harder. To see why, let us illustrate with the important case , where one has

as recalled in Theorem 1. Writing (1a) as

we observe on the right hand side of (4) that is pushed to grow at points where . If this condition persists long enough, large values of might be generated, particularly at sites where . Now, because of constraint (3), any persistent growth in solution size will eventually create long thin structures as shown in Figure 1, which, in turn, tend to be effectively dissipated by viscosity. The final overall behavior that ultimately results from such competition is not immediately clear, either on physical or on mathematical grounds.

Figure 1: Solution profiles showing typical growth in regions with , where . After reaching maximum height, solution starts decaying very slowly due to its spreading and mass conservation (decay rate is not presently known).

As shown in (4), it is not the magnitude of itself but instead its oscillation that is relevant in determining . Accordingly, we introduce the quantity defined by

which plays a fundamental role in the analysis. Our main result is now easily stated.

Main Theorem. For each , one has ¹

where .

In particular, in the important case considered above, we obtain, using (3),

so that stays uniformly bounded for all time in this case.² Estimates similar to (6) can also be shown to hold for the -dimensional problem

but to simplify our discussion we consider here the case only. Our derivation of (6), which improves some unpublished results by the third author, uses the 1D inequality

where , and can be readily extended to other problems of interest like 1D systems of viscous conservation laws [6, Ch. 9] or the more general equation

with bounded values ; provided that we assume : using a similar argument, we get the estimate³ [7, Ch. 2]

for each , where

More involving applications, such as problems with superlinear advection or degenerate diffusion, which require considerable extra work, will be studied in the future.

2. A Priori Estimates

This section contains some preliminary results on the solutions of problem (1a) and (1b) needed later for our derivation of estimate (6), which is completed in Section 3. (Recall that a solution on some given time interval , , is a function which is smooth ( in , in ) in and solves (1a) there, verifying the initial condition in the sense of, i.e., as for each compact . Local existence theory can be found in, e.g., [8, Ch. 6].) We start with a simple Gronwall-type estimate for , . The corresponding result for the supnorm is more difficult to obtain and will be given at the end of Section 2; see Theorem 4.

Theorem 1. If solves problem (1a), (1b), then for each , and

for all .

Proof. The proof is standard, so we will only sketch the basic steps. Taking such that  for all , , for, let (given ), so that as , uniformly in. Let. Given , , let be the cut-off function for , for . Multiplying (1a) by if , or if , and integrating the result on , we obtain, letting and then , since ,

where , , and

By Gronwall’s lemma, (14a) and (14b) give, from which we obtain (13) by simply letting . This shows, in particular, that if . Now, to get , it is sufficient to show that, given and arbitrary, we can find large enough so that we have for any . Taking with and for all , for all , let be the cut-off function given by if , if , and if , if , if , where , are given. Multiplying (1a) by if , or if , and integrating the result on , , we obtain, as in (14a) and (14b), by letting , , that for all , provided that we take sufficiently large. This gives the continuity result, and the proof is complete.

An important by-product of the proof above is that we have (letting in (14a) and (14b), and using (13)), for each and ,

Therefore, if we repeat the steps above leading to (14a) and (14b), we obtain (letting , , , in this order, taking (13) and (15) into account) the identity

for every and , where

The core of the difficulty in the analysis of (1a) and (1b) is apparent here: under the sole assumption that is bounded, it is not much clear how one should go about the last term in (16) in order to get more than (13) above. Actually, it will be convenient to consider (16) in the (equivalent) differential form, that is,

for all , where has zero measure. We then readily obtain, using (9) and the one-dimensional Nash inequality [9]

where the value given above for is optimal [10], the following result.

Theorem 2. Let . If is such that , then

Proof. Consider (20a) first. From (5), (17), and (18), we have

This gives

or, in terms of defined by if , if ,

Using (19), we then get , which is equivalent to (20a). Similarly, (20b) can be obtained, using (9).

Thus, we can use (20a) and (20b) when is not decreasing. If it is decreasing, (18) becomes useless but at least we know in such case that is not increasing, which should be useful too. Different values of have different scenarios, which we will have to piece together in some way. The next result shows us just how. To this end, it is convenient to introduce the quantities and defined by

given arbitrary.

Theorem 3. Let . For each , we have

for all .

Proof. Set . There are three cases to consider.

Case I. for all . By (20a), Theorem 2, we must then have for all , so that is monotonically decreasing in . In particular, in this case, and (26) holds.

Case II. and for some . In this case, let be such that we have for all , while . We claim that for every : in fact, if this were not true, we could then find with such that for all , . By (20a), Theorem 2, this would require for all , so that could not increase anywhere on . This contradicts , and so we have for every , as claimed. On the other hand, by (20a), has to be monotonically decreasing on , just as in Case I. Therefore, we have in this case again, which shows (26).

Case III. Consider . This gives for every , by repeating the argument used on the interval in Case II. It follows that we must have in this case, and the proof of Theorem 3 is complete.

An important application of Theorem 3 is the following result.

Theorem 4. Let , . Then

for any , where and are given in (24) and (25) above.

Proof. Let , . Applying (26) successively with , we obtain

where

Now, for ,

by Young’s inequality (see, e.g., [11, page 622]); in particular, we get, from (28a) and (28b),

since for all . Letting , (27) is obtained.

It follows from Theorems 1 and 4 that is globally defined (). Now, from (27), we immediately obtain, letting ,

for any , where and are given by

Taking such that and , and applying (31) with for each , we then obtain, letting ,

where and are given by

3. Large Time Estimates

In this section, we use the results obtained above to derive two basic large time estimates (given in Theorems 5 and 6) for solutions of problem (1a), (1b), which represent important intermediate steps that will ultimately lead to the main result stated in Theorem 7.

Theorem 5. Let , and be as defined in (35). Then

where is the constant in the Nash inequality (19).

Proof. We set and assume that is finite. As in the proof of Theorem 2, we take given by if , if . It follows that

Therefore, from (18), we have, for some null set ,

for all , and so, by (19),

This gives, by Young’s inequality ([11, page 622]), for all ,

Setting

we claim that

In fact, let us argue by contradiction. If (42) is false, we can pick and a sequence , , such that (for all ) and for all . From (20a), Theorem 2, it will then follow that

In fact, suppose that (43) were false, so that we had for some . Taking with , we could then find such that for all , while , and so there would exist with positive at . By (20a), we would have , but this would contradict the fact that everywhere on . Thus, we conclude that (43) cannot be false, as claimed. We then obtain, from (19), (40), and (43),

for all . Recalling that , , for  all , this gives

for some constant independent of , which cannot be, since this implies

This contradiction shows (42), which is equivalent to (36), and the proof is complete.

Applying (36) successively with , we get

for arbitrary, where . Letting , this suggests

where

(cf. (6) above), as long as the limit processes , can be interchanged. That this is indeed the case is a consequence of (34) and the following result.

Theorem 6. Let . Then

where , are the constants given in (19) and (9).

Proof. Again, assuming finite (otherwise, (49) is obvious; cf. endnote⁴), we introduce, as in the previous proof, given by if , and if . Thus, (40) is valid, and setting , by

we have that (49) is obtained if we show that

We argue by contradiction and assume that (51) is false. Taking then so that and hold for all , we get, by (9) and (40),

for all . Since , , this gives

for some constant independent of . As before, this implies that for all , which is impossible because is finite. This contradiction establishes (51) above, completing the proof of Theorem 6.

We are finally in good position to derive (6), (48a), and (48b). Combining (34) and (49) above, we obtain

for each , so that we have, in particular,

for each . By (47), we then get

for all . Letting , Theorem 7 is obtained, and our argument is complete.

Theorem 7. Let . Assuming , then (6), (48a), and (48b) hold.

It is worth noticing that the corresponding estimate for the -dimensional problem (8), namely,

where is similarly defined, can be also derived in arbitrary dimension .

4. Concluding Remarks

We close our discussion of the problem (1a), (1b), given , , indicating a few questions which were not answered by our analysis:

(a)characterize all for which it is true that (as ) for every solution of problem (1a) and (1b);(b)same question as (a) above, but requiring only that (as ) for every solution of problem (1a) and (1b), in case ;⁵(c)given , characterize all such that (as ) for every solution of problem (1a) and (1b);(d)same question as (c) above, but requiring only that (as ) for every solution of problem (1a) and (1b);(e)for , characterize all such that (as ) for every solution , where is the solution mass;(f)for , and not satisfying property (e), what are the values of in case of initial states that change sign?

These questions can be similarly posed for solutions of autonomous problems

where does not depend on the time variable. For (58), question (e) has been answered in [12] (see also [13]). Another interesting question is the following:

(g)when (58) admits no stationary solutions other than the trivial solution , is it true that for every solution ?

Moreover, for solutions of (1a) and (1b) or (58) with as , there is the question of determining the proper decay rate.⁶ As suggested by Figure 1, solution decay may sometimes happen at remarkably slow rates.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to thank CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico, Brazil) for their financial support.

Endnotes

1. In (6), (11), and other similar expressions in the text, it is assumed that .

2. The constants in (6) and (7) are not optimal; minimal values are not known.

3. In (6), (11), and other similar expressions in the text, it is assumed that .

4. In (6), (11), and other similar expressions in the text, it is assumed that .

5. For , any satisfies property (b); compare (7) in Section 1.

6. In case we have for all , , the answer is given in (2) above.

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