diffusion-dispersion limit

In this paper, we consider the controllability of a transport equation perturbed by small diffusion and dispersion terms. We prove that for a sufficiently large time, the cost of the null-controllability tends to zero exponentially as the perturbation vanishes. For small times, on the contrary, we prove that this cost grows exponentially.


Introduction
In this paper, we consider some null-controllability problems for a transport equation perturbed by small di®usion and dispersion e®ects.Precisely, the system under review is the following where T > 0, > 0, (typically non-negative) and M are real numbers.In (1.1), we have denoted y 0 the initial condition and v the control.For 2 R and > 0 ¯xed, the null-controllability of this system is due to Rosier. 14As a matter of fact, Rosier studies the case ¼ 0, but a elementary change of unknown function of the form u ¼ e ax y transforms this particular case into the general one (see Remark 1.3).
Our main problem is the following.Consider the unperturbed transport equation: controlled from the boundary by y jx¼0 if M < 0 and by y jx¼1 if M > 0. This equation is null-controllable provided that T > 1=jMj: it su±ces to take 0 as a control (hence the cost is null in this case).On the contrary, if T < 1=jMj, it is elementary to see that the system is not null-controllable.The question which arises is to determine whether it is possible for times larger than 1=jMj to control (1.1) at a uniform cost as and tend to 0. On the other hand, it is to expect that for times T < 1=jMj, the cost of null-controllability will dramatically increase.These problems have already been treated in the case of vanishing viscosity (see Refs. 4 and 9), and in the case of vanishing dispersion (see Ref. 8).Several statements proved below are new even in the case of pure dispersion ( ¼ 0), see Theorems 1.1 and 1. 4.
The motivation for studying the dissipationÀdispersion mechanism arises from continuum mechanics.In particular, in nonlinear elastodynamics, these terms can model viscosity and capillarity e®ects.These are particularly important in the theory of nonclassical shock waves (see in particular the book of LeFloch 12 ).Nonclassical shock waves are shock waves for conservation laws with nonconvex °ux, which are selected through perturbative terms such as the ones of (1.1); in that case they can di®er from the classical shock waves selected by vanishing viscosity.Hence, although the system which we consider here is linear, one can see the results described below as a ¯rst attempt to control nonlinear conservation laws in a dissipativeÀdispersive limit.Such a study in the purely dissipative limit has been handled by the authors in the case of the Burgers equation. 7A purely dispersive limit in the case of Burgers equation would seem also of interest; see the celebrated paper by Lax and Levermore 11 for the direct problem.
We establish the following results.
Theorem 1.1.There exists C 0 > 0 such that the following holds.For any M > 0, any T !C 0 =M, there are positive constants c and C (depending on T ) such that for any ð; Þ 2 ð0; 1 Â ½0; 1, there exists v 2 L 2 ð0; T Þ driving y 0 2 L 2 ð0; 1Þ to 0 and which can be estimated as follows: exp À cM maxfðMÞ 1=2 ; g jjy 0 jj L 2 : ð1:2Þ Remark 1.1.Theorem 1.1 improves the one in Ref. 8. In fact, for ¼ 0 a similar result was proved in Ref. 8 but with the help of three controls, one on each boundary condition.
The meaning of a solution of system (1.1) will be given in Sec. 2 (see De¯nition 2.1 and Proposition 2.1).In a context of more regular solutions (which are easier to de¯ne via a lifting of boundary conditions), one can state the following.
Corollary 1.1.In the above framework, we can obtain a more regular control v in H 1 ð0; T Þ with the following estimate jjvjj H 1 Cð; MÞ ffiffiffiffiffi ffi M p exp À cM maxfðMÞ 1=2 ; g jjy 0 jj L 2 ; ð1:3Þ where Cð; MÞ behaves at most polynomially in À1 and in M (that is to say, jCð; MÞj Kð À1 þ MÞ r for some K > 0 and some r 2 N).
Let us recall that Theorem 1.1 is valid for the heat equation ( ¼ 0) with Dirichlet boundary conditions regardless of the sign of M, see Ref. 4. Hence it is natural to wonder if it is still valid when > 0 is small with respect to .An answer is given in the next result.
Theorem 1.2.Let 0 < 1.Then there exists C 0 (depending on ), such that for any M < 0, any T !C 0 =jMj, there are positive constants c and C (depending on T and ) such that for any ð; Þ 2 ð0; 1 2 satisfying 2 !jMj; ð1:4Þ one can ¯nd a control driving y 0 to 0 and which can be estimated as follows: In the next result we consider the case of negative .It is somewhat surprising that the dispersive term can overpower a small dissipation term with the wrong sign.
Corollary 1.2.In the framework of Theorem 1.3, it is possible to design a more regular control v in H 1 ð0; T Þ with estimate (1.3) ful¯lled.Now we consider the case where T < 1=jMj.In this case, the transport equation ( ¼ ¼ 0) is no longer controllable.One should hence expect the cost of nullcontrollability to blow up as ; !0. This is shown in the next result.Then there are some constants c > 0 and ' 2 N (independent of 2 ½0; 1 and 2 ð0; 1) and initial states y 0 2 L 2 ð0; 1Þ such that any control v 2 L 2 ð0; T Þ driving y 0 to 0 is estimated from below as follows : 3. Let us recall that in a bounded domain, one can transform Eq. (1.1) in a linear KdV equation without di®usion term through the following transformation.We set z ¼ expðÀxÞy with ¼ 3 : ð1:8Þ Then z satis¯es So the zero-controllability of (1.1) for ¯xed and follows from Rosier's result 14 on the controllability of the linear KdV equation.The exact controllability with two Dirichlet controls also follows from this remark and the results from Ref. 8.However, the estimate of the cost (1.2) cannot be obtained by this transformation and our method for the linear KdV equation from Ref. 8. In particular, the above transformation gives bad estimates in the regime when the di®usion dominates, that is, when 2 ) .It seems natural that di±culties appear in the above transformation when the third-order term is very small.

Cauchy Problem
Let us brie°y discuss the Cauchy problem for Eq.(1.1).For recent references concerning the initial boundary value problem for the KortewegÀde Vries equation, let us cite 2,3,10 and references therein.
The solutions of system (1.1) are to be understood in the sense of transposition.
Of course, any regular solution of (1.1) is a solution in the above sense, as easily shown by integration by parts.Proposition 2.1.For M 2 R, 2 ð0; 1, and either 2 ½0; 1Þ or < 0 and À < (for some ¯xed < 3 2 ), T > 0, y 0 2 H À1 ð0; 1Þ and v 2 L 2 ð0; T Þ, there exists a unique solution of transposition of (1.1).Moreover, there exists C > 0 independent of and such that jjyjj L 2 ðð0;T ÞÂð0;1ÞÞ\C 0 ð½0;T ;H À1 ð0;1ÞÞ ; 1ÞÞ, we have w 2 C 0 ð½0; T ; H 1 0 ð0; 1ÞÞ and w xxjx¼0 2 L 2 ð0; T Þ, together with an estimate on these quantities in terms of f.For further purposes, we will consider w 0 not necessarily 0, although this is not needed to establish Proposition 2.1.Let us remark that the existence of solutions of (2.1) for regular data follows for instance from the transformation (1.8)À(1.9).
To do this, we perform several di®erent energy inequalities.As we will see, this is valid in both cases when 2 ½0; 1Þ and when < 0 but À < 3 2 .Consider a regular solution w of (2.1).The general case follows from a regularization procedure.
First inequality.We multiply (2.1) by w: this yields wf dx: ð2:4Þ Second inequality.We multiply (2.1) by ð1 À xÞw: this yields When adding (2.4) and (2.5) we arrive at In the ¯rst case, we integrate between t and T, take the supremum over t and use Young's inequality.In the second case, we use Young's inequality.In both cases, we deduce an a priori estimate on w in L 1 ð½0; T ; L 2 ð0; 1ÞÞ \ L 2 ð½0; T ; H 1 ð0; 1ÞÞ in terms of f and w 0 : where the constant C depends on M. Note that, due to the fact that < 3=2, (2.6) provides an estimate of w in L 2 ð½0; T ; H 1 ð0; 1ÞÞ in terms of w 0 and f, with a coe±cient which grows as 1=.
Higher-order estimates.Now let us denote Together with estimate (2.13), this implies that w xxjx¼0 2 L 2 ð0; T Þ, with the estimate 3. Proof of Theorem 1.1

Carleman inequality
Let us consider the following backwards (in time) problem, which is the adjoint system associated to (1.1): The objective of this section is to state a Carleman inequality for the solutions of this system.In order to state this estimate, let us set ðt; xÞ ¼ ðxÞ t ðT À tÞ ; ðt; xÞ 2 Q; ð3:2Þ Uniform Controllability of a Transport Equation 1573for some 2 ½1=2; 1.Here, is a strictly positive, strictly increasing, and concave polynomial of degree 2. Weight functions of this kind were ¯rst introduced by Fursikov and Imanuvilov; we refer to Ref. 6 for a systematic use of them.
Observe that the function satis¯es where C; C 0 and C 1 are positive constants independent of T.
Proposition 3.1.There exists a positive constant C independent of T, > 0, !0 and M 2 R such that, for any ' T 2 L 2 ð0; 1Þ, we have ÞÞ, where ' is the solution of (3.1).
Remark 3.1.Note that in the dispersive regime (that is, when & 2 ), one could deduce Proposition 3.1 from the Carleman estimate of Ref. 8, by putting the di®usion term ' xx on the right-hand side.In passing, when this term is put in the right-hand side, the sign of does no longer matter.See Proposition 4.2 for a precise statement with a negative .
Since the proof of Proposition 3.1 is very technical, we postpone it to an Appendix, at the end of the paper.

Exponential dissipation result
Let us consider > 0, !0, M > 0. It follows from (2.4) that the solution of the adjoint system (3.1)satis¯es In this paragraph we will prove that, whenever the time passed t 2 À t 1 is larger than 1=M, the constant K can be dramatically improved: typically, it behaves like The precise result is stated in the next proposition: We have the following decay properties for the solution of (3.1): . If 2 !3ðM À 1=ðt 2 À t 1 ÞÞ, then Proof of Proposition 3.2.This is inspired by Ref. 5. Let us multiply (3.1) by expfrðMðT À tÞ À xÞg', where r is a positive constant which will be chosen below.Then, integrating in ð0; 1Þ and integrating by parts with respect to x, we deduce Integrating again by parts, we obtain with Now, we choose r > 0 to minimize K, that is to say, we take With this choice of r, we have that K (given by (3.12)) coincides with We ¯rst use that K written in the form expfÀy=zg ðy; z > 0Þ is an increasing function of z and we get Similarly as in the previous case, we ¯rst have that Finally, we take into account that ( ) This establishes (3.7) and (3.8).
To do this we must distinguish two regimes: the \dispersive regime" where M & 2 and the di®usive regime where M . 2 .We consider ' a regular solution of (3.1) and use Proposition 3.1 for a time T . First regime: M ! 2 .We ¯x ¼ 1=2.We consider s ful¯lling the assumptions of Proposition 3.1: Observe in particular that T 1 =s .1. From (3.5), we infer for some C > 0 independent of ; , and M. From the de¯nition of (see (3.2)), this yields for some C 2 ; C 3 > 0.Then, we use here the following energy inequality In particular, using T 1 =s .1, this allows us to deduce the following observability inequality from (3.14): for some C 4 > 0 independent of M, < 1 and < 1.Now, we use ffiffiffiffiffiffiffi ffi M p and deduce an estimate on the observability constant . Second regime: M . 2 .We choose ¼ 1.Here we deduce from (3.5) that for s as in Proposition 3.1 (observe now that T 2 1 =s .1): for some C > 0 independent of ; , and M. From the de¯nition of (see (3.2)), this yields Uniform Controllability of a Transport Equation 1577for some C 6 ; C 7 > 0. Proceeding as previously we obtain (3.15) with C Ã estimated by 2. Now, given T !C 0 =M (with C 0 > 2 to be chosen large enough later), we use the above observability inequality (3.15) between times T À 1=M and T; we deduce During the time interval ½0; T À 1=M, we use Proposition 3.2 to compare jj'ðT À 1=MÞjj 2 L 2 to jj'ð0Þjj 2 L 2 (that is, we take t 1 ¼ 0 and t 2 ¼ T À 1=M).We ¯nally obtain with Kð0; T À 1=MÞ and C Ã given by (3.6)À(3.8)and (3.16) in the ¯rst regime and by (3.6), (3.7), and (3.17) in the second one.It is then clear that by taking C 0 large enough (independently of the parameters , , M ), we can bound the observability constant in the following way: in the first regime; in the second regime: Now, from these observability inequalities for the solutions of (3.1), it is classical to prove that for any y 0 2 L 2 ð0; 1Þ, there exists a control v 2 L 2 ð0; T Þ such that the solution y 2 L 2 ð0; T ; H 1 ð0; 1ÞÞ of (1.1) satis¯es yðT ; xÞ ¼ 0 for x 2 ð0; 1Þ with v 1 estimated by Let us emphasize that the factor comes from writing the duality relation between (1.1) and (3.1) and applying the standard H.U.M. procedure (see Ref. 13).Then, one can estimate the factor C obs = 2 in the following way: in the first regime; and hence one can obtain the form (1.2) in both cases (slightly reducing c 2 ).This concludes the proof of Theorem 1.1.

Proof of Corollary 1.1
In this proof, Cð; MÞ will denote a generic positive constant depending on À1 and M at most polynomially.The construction consists of two steps: ¯rst, we let the control be zero in (1.1), and prove that this regularizes the state of the system.Next, we ¯nd a convenient control for more regular initial data.
First step.We consider some t Ã 2 ð0; T Þ.Let us prove that setting the control to zero yields a state yðt Ã Þ in H 3 ð0; 1Þ.
We recall that we already have that y 2 L 2 ðð0; T Þ Â ð0 Second step.Hence, considering yðt Ã Þ as our new initial condition, we can consider that y 0 2 ðH 3 \ H 2 0 Þð0; 1Þ.We now reduce Corollary 1.1 to an internal regularity property for system (1.1).In order to do this, we extend y 0 into ỹ0 2 ðH 3 \ H 2 0 Þ Â ðÀ1; 1Þ (in a continuous manner) and we consider system (1.1) in ½À1; 1 rather than in ½0; 1: Due to Theorem 1.1, there exists a control ṽ 2 L 2 ð0; T Þ driving ỹ0 to 0 at time T. Now for system (3.23),we establish in the Appendix the following internal regularity result.
Now it is clear that y ¼ ỹj½0;1 ful¯lls the requirements of Corollary 1.1, since ỹ has a trace at x ¼ 0 belonging to H 1 ð0; T Þ and satisfying estimate (5.10).

Proof of Theorem 1.2
The only part of the proof of Theorem 1.1 which needs to be changed is the dissipation inequality.For that, we start from (3.9), but here, due to the sign of M, we consider r < 0. Provided that r !À 2 3 ; ð3:25Þ the integral concerning ' x has a positive coe±cient.It follows that (3.11) is still valid in this situation, with K here given by instead of (3.12).Now we consider Observe that, since 2 !jMj À 1 t 2 À t 1 (see condition (1.4)) r Ã indeed satis¯es (3.25).Now injecting r Ã in (3.26) and neglecting the ¯rst term in (3.26) yields which, recalling that 1, proves the exponential decay property.Now the rest of the proof follows the lines of Theorem 1.1.

Proof of Theorem 1.3
First, let us recall that the Cauchy problem for such was investigated in Sec. 2. That the required Carleman inequality holds in the context of this result (viz.when is negative but small in absolute value) was explained in Remark 3.1; see Proposition 4.2 for a precise statement.Also, we need to extend the validity of the dissipation estimate (Proposition 3.2) to our context.

Dissipation and Carleman estimates
The dissipation result which we use here is the following.
Proposition 4.1.Consider T > 0, < 0, and > 0 satisfying then for every 0 t 1 < t 2 T such that t 2 À t 1 !ð1 þ c 0 Þ=M, the solutions of (3.1) satisfy the decay property (3.6) with the constant K estimated by: Proof of Proposition 4.1.The computations that led to (3.9) are still valid.But due to the negative sign of , estimate (3.10) could possibly no longer occur.However, if we choose r properly and prove that r !À 2 3 ; ð4:3Þ then the sum of the two terms of the second line of (3.9) is non-negative, so that (3.10) holds.
Hence it remains to choose r satisfying (4.3), in order to make our constant K given in (3.12) satisfy (4.2).First, we remark that due to the sign of , we have: Uniform Controllability of a Transport Equation 1581We choose In particular, (4.3) is a direct consequence of (4.1).Due to hence we deduce easily The Carleman estimate that we use here is the following proposition.
There exists a positive constant C independent of the previous quantities such that, for any ' 0 2 L 2 ð0; 1Þ, we have where ' is the solution of (3.1).
Proof of Proposition 4.2.We modify the proof of Proposition 3.1, by taking ¼ 1=2 and placing the ' xx term in (3.1) on the right-hand side.This yields We proceed as in the proof of Theorem 1.1 and get an observability constant which is the product of C Ã and K.If we take a su±ciently long amount of time for the dissipation, we see that we can absorb the constant from (4.7) by the constant from (4.2) (used during the time interval ½0; T À 1=M).This yields an observability constant satisfying: Recall that the constant giving the cost of the control is C 1=2 obs =.This yields the conclusion.

Proof of Corollary 1.2
One can reproduce the proof of Corollary 1.1.Concerning the ¯rst step (using a null control regularizes the data), inequality (3.22) is still true, since we did not use the sign of (apart from the fact that 3=2 þ > ð3=2 À Þ).Concerning the second step (the interior regularity result), we can see in Sec.7 that we only use 3 2 þ > 0.

Proof of Theorem 1.4
We ¯rst consider the case where M > 0. At the end, we will describe the modi¯cations needed in the case where M < 0. We introduce R > 0 such that Uniform Controllability of a Transport Equation 1583We consider the corresponding solution ' of (3.1).Now the proof is twofold.First we show that the mass of ' is essentially conserved in the sense that Z 1 0 j 'ð0; xÞj 2 dx !c > 0; ð5:3Þ for some constant c > 0. Next, we prove that 'xxjx¼0 decays exponentially as First step.We introduce ðt; xÞ as the solution of Now, using We multiply (3.1) by ðx À MðT À tÞÞ expfrðMðT À tÞ À xÞg ', for some r !0. In Proposition 3.2 we multiplied (3.1) with expfrðMðT À tÞ À xÞg ', which led to (3.10).
Here there are additional terms due to the presence of ðx À MðT À tÞÞ; we put them in the right-hand side.Denoting where we have used that ðx À MðT À tÞÞ ¼ 1 for ðt; xÞ 2 ½0; T Â ½0; R=4.Finally, we get Now we choose r as follows Uniform Controllability of a Transport Equation 1585Using, r 1=ðRT Þ 1=2 , this yields (5.7) and (5.8) with C at most polynomial in 1=.Now from (5.7) and (5.8), we are going to deduce an estimate of the type jj 'jj 2 L 2 ð0;T ;H 3 ð0;R=16ÞÞ for some constant CðÞ whose growth in 1= is at most polynomial.
To do so, ¯rst, we consider Eq. (3.1) in ½0; R=4 and multiply by ðR=4 À xÞ 3 ' and get as for (2.5): This yields (using (5.2))When M < 0, we de¯ne R essentially as in (5.1) and replace the condition on the support in (5.2) by Then the step 1 in the above analysis is easily adapted in this situation.Concerning the second step, we rede¯ne 2 C 1 ðRÞ by Then the goal is again to establish (5.7) and (5.8).The same computation as previously gives (5.11)where the limits of the time-integral in the right-hand side have to be replaced with 1 À 3R þ MðT À tÞ and 1 À 2R þ MðT À tÞ.Next, (5.12) still holds with a di®erent coe±cient (recall that E was de¯ned in (5.10)) As previously we integrate in time; here with the new de¯nition of we again have ðx À MðT À tÞÞ ¼ 1 for ðt; xÞ 2 ½0; T Â ½0; R=4, and we get Then one may conclude as previously (note in particular that due to (5.21), (5.15) is valid), which ends the proof of Theorem 1.4.

Appendix A. Proof of the Carleman inequality
Let :¼ e Às ', where is given by (3.2) and ' ful¯lls system (3.1).We deduce that with ðA:3Þ Then, we have In the following lines, we will compute the double product term.For the sake of simplicity, let us denote by ðL i Þ j (1 i 2, 1 j 7) the j th term in the expression of L i .To identify the signs of the following integrals, we recall that > 0, x > 0 and xx < 0.
² First, integrating by parts with respect to x, we have Uniform Controllability of a Transport Equation 1587 Here, we have used that jx¼0;1 ¼ xjx¼0 ¼ 0, (3.3) and the fact that xxx ¼ 0.
For the second term, we do very similar computations and we obtain xjx¼1 j xjx¼1 j 2 dt: ðA:6Þ For the third term, we integrate by parts with respect to the x variable and we obtain Z T 0 jx¼0 j xxjx¼0 j 2 dt: ðA:7Þ We consider now the fourth term of L 2 and we readily get The next term gives Furthermore, since xxx ¼ 0 and jx¼0 ¼ xjx¼0 ¼ 0, we have jx¼1 j xjx¼1 j 2 dt: ðA:10Þ Observe that for the last integral of the ¯rst line of (A.10) we have used CauchyÀ Schwarz inequality and the estimate 2 xx = x C T 4 3 for some constant C > 0. The last term in the second line of (A.10) yields a positive term when combined with the last term in the ¯rst line of (A.7) thanks to x > 0.

ðA:12Þ
Uniform Controllability of a Transport Equation 1589² Concerning the second term of L 1 , we ¯rst integrate by parts with respect to t: Similar computations give the following for the second term: For the third one we use that xjt¼0 ¼ xjt¼T ¼ 0 and (3.4) and we get Then, we readily see that ððL 1 Þ 2 ; ðL 2 Þ 4 Þ L 2 ðQÞ ¼ 0. Again using jt¼0 ¼ jt¼T ¼ 0 and (3.4), we deduce This term cancels with the last term in (A.15).Finally, the last product of the second term of L 1 provides Putting together all the computations concerning the second term of L 1 ((A.13)À(A.18)),we obtain þ sjMjT ðþ1Þ= Þj j 2 dx dt: ðA:19Þ ² We consider now the products concerning the third term of L 1 .First, we have Thirdly, For the fourth term, we have Using (3.4), we obtain the following for the ¯fth term: Uniform Controllability of a Transport Equation 1591Furthermore, x xx j x j 2 dx dt: ðA:25Þ Finally, Consequently, we get the following for the third term of L 1 ((A.20)À(A.26)): ² Now, we compute the fourth term.First, we have: x xx j j 2 dx dt: ðA:28Þ Similar computations give x xx j j 2 dx dt: ðA:29Þ For the third term, we get The ¯fth term gives Direct computations for the sixth term provides This term cancels with the ¯rst integral in the right-hand side of (A.30).At last, All these computations ((A.28)À(A.34))give ² Concerning the ¯fth and last term of L 1 , we have: Then, Uniform Controllability of a Transport Equation 1593Now, we compute the third one: Then, Additionally, integrating by parts again with respect to x, we have This term can be combined with the ¯rst integral in the second line of (A.38).Finally, We introduce R k ðxÞ :¼ R x À1 k .The estimates are done in four steps.In what follows, C denotes a constant independent of ỹ, whose growth in 1= and M is at most polynomial, and which can change from one line to another.Consider m ! 5.