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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 578701, 22 pages
Global Analysis for Rough Solutions to the Davey-Stewartson System
1College of Mathematics, Southwest Jiaotong University, Chengdu 610031, China
2College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, China
Received 7 June 2012; Accepted 15 September 2012
Academic Editor: Abdelghani Bellouquid
Copyright © 2012 Han Yang 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.
The global well-posedness of rough solutions to the Cauchy problem for the Davey-Stewartson system is obtained. It reads that if the initial data is in with s > 2/5, then there exists a global solution in time, and the norm of the solution obeys polynomial-in-time bounds. The new ingredient in this paper is an interaction Morawetz estimate, which generates a new space-time estimate for nonlinear equation with the relatively general defocusing power nonlinearity.
The Davey-Stewartson system has their origin in fluid mechanics, where it appears as mathematical models for the evolution of weakly nonlinear water waves having one predominant direction of travel, but in which the wave amplitude is modulated slowly in two horizontal directions (see ). In dimensionless they read as the following system for the (complex) amplitude and the (real) mean velocity potential where , and ; the parameters are real constants. According to signs of the and , these systems may be classified as
In the last two decades, the Cauchy problem for the Davey-Stewartson system (1.1) has focused on intense mathematical research. In 1990, Ghidaglia and Saut  established the local well-posedness for the Cauchy problem of (1.1) in the cases of (1.2)–(1.4). It reads that for , the systems (1.1) have a local solution in time. Hayashi and Hirata  studied the initial value problem to the Davey-Stewartson system for the elliptic-hyperbolic case (1.3) in the usual Sobolev space, they proved local existence and uniqueness for the initial data in whose norm is sufficiently small. Tsutsumi  obtained the -decay estimates of solutions to the systems (1.1) in the elliptic-hyperbolic case (1.3). Hayashi and Saut  and Linares and Ponce  studied some generalized Davey-Stewartson systems in different spaces, and their main tools are estimates of solutions to linear Schrödinger equations. These estimates are usually named generalized Strchartz inequality. Guo and Wang  studied the Cauchy problem for a generalized Davey-Stewartson system in the elliptic-elliptic case (1.2), and they proved the global well-posedness results for initial data in . Recently, Shu and Zhang  and Gan and Zhang  obtained the sharp conditions of global existence for Davey-Stewartson system in the elliptic-elliptic case (1.2) by constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds generated by the evolution flow. Zhang and Zhu  obtained a more precisely sharp criteria of blow-up and global existence. C. Sulem and P. L. Sulem  obtained some numerical observations on blow-up solutions. Richards  showed the mass concentration phenomenon of blow-up solutions. Li et al.  obtained some dynamical properties of blow-up solutions. Recently, Babaoglu and Erbay  proposed a generalized Davey-Stewartson system, which was studied in [15–17].
In this paper, we consider the Cauchy problem of (1.1) in the elliptic-elliptic case (without loss of generality, we may take for simplify), As is well known, the system (1.6)–(1.8) enjoys two useful conservation laws: one is the energy conservation law: and the other is the mass conservation law: For details one can see Ghidaglia and Saut . Moreover, one can easily establishes for (with bounds uniformly in ), and with some additional arguments one can deduce the same claim for . The mass conservation law (1.10) also gives (1.11) for , but unfortunately this does not immediately imply any results for except in the small mass case.
To make the statement more precise, we denote as the solutions of (1.7)–(1.9). It follows from (1.8) that where and denote the Fourier transform and its inversion. For brevity we denote Combining (1.8) and (1.9), (1.6)–(1.8) are changed to
It is conjectured that the system (1.14)-(1.15) is globally well posed in for all and in particular (1.11) holds for all . This conjecture remains open now. In this paper we aim to prove that the Cauchy problem for (1.1) is globally well posed below the energy norms. That is, we will prove the global well-posedness for initial data with sufficiently close to one, then we meet the obstacle that there is no conservation law. Indeed, if the initial data is in then it is bounded in for all time and hence the norm is similarly bounded, but if the initial data is only in then the norm may be infinite, and also the conservation of the Hamiltonian appears to be useless. Conservation of the norm also appears to be unhelpful for this particular problem.
For solutions below the energy threshold the first result was established by Bourgain for nonlinear Schrödinger equation with critical nonlinearity in space dimension two (see ). Bourgain came up with the idea of introducing a large frequency parameter by dividing the solution into the low-frequency portion (when ) and the high frequency portion (when ). The main tool is an extrasmoothing estimate, which shows that if the high frequencies would be merely in for some , then interactions arising from high frequencies were significantly smooth. In fact, they were in the energy class . Moreover, if we denote as the nonlinear flow and is the linear group, Bourgain’s method shows addition that for all time provided , . Thus, he showed that the solution is globally well posed with initial data in for any . Recently, Kenig, Ponce, Fonseca, Ginibre, Molinet, Pecher [19–23], and Miao and Zhang  have developed this methods to study different evolution systems.
Another improvement was given by Colliander et al. in [25, 26], where the authors used the “I-method” that we state below. If , then the energy is infinite and one cannot compare the norm of the solution with the energy. In order to overcome this difficulty, we introduce the following multiplier : where , is a smooth, radial, nonincreasing in such that: ; , and is a dyadic number playing the role of a parameter to be chosen. Then we plug this multiplier into the energy which generates to so-called modified energy: Note that if then . Note also that as goes to infinity, the multiplier “approaches” the identity operator. Therefore the variant of this smoothed energy is expected to be slow as goes to infinity. This is the “I-method”, originally invented by Colliander et al.  to prove global existence for semilinear Schrödinger equations with rough data.
In this paper we design it for the Davey-Stewartson system. The main purpose of this paper is to study that we can lower the value of to what extent which can also grantees the global existence. In this paper we will prove the following.
Remark 1.3. We improve the results obtained by Shen and Guo , in which they demonstrated the global existence for for the Cauchy problem (1.14)-(1.15). The technique in their proof mainly depends on the Fourier restriction norm method of Bourgain by showing a generalized estimates of Strichartz type and splitting the data into low- and high-frequency parts. The new ingredient in our proof is a priori interaction Morawetz-type estimate, which generates a new space-time estimates for the “approximate solution” to the nonlinear equation with the relatively general defocusing power nonlinearity, and this technique is motivated by the work given by Colliander et al. in .
Remark 1.4. It is worth to remark that Dodson [28–30] proves a frequency-localized interaction Morawetz estimate similar to the estimate made in  for considering an -critical initial value problem for cubic nonlinear Schrodinger equation. The major difference between the cubic nonlinear Schrodinger equation and the elliptic-elliptic Davey-Stewartson system (1.14) is the singular integral operator in (1.14), which may result in some new difficulties to establish the corresponding frequency localized interaction Morawetz estimate. We hope to solve this problem in a forthcoming paper from the arguments derived by Dodson.
2. Notations and Preliminaries
In this paper, we will often use the notation whenever there exist some constants such that . Similarly, we will use if . We use if for some small constant . We use ± to denote the real number for any sufficiently small . and are the real part and imaginary part of the complex number , respectively.
We use to denote the Schwartz space and to denote its topological dual space. We use to denote the usual Lebesgue space of functions : whose norm is finite, with the usual modification in the case . We also define the space-time space by for any space-time slab , with the usual modification when either or are infinity. When we abbreviate by .
Definition 2.1. A pair of exponent is called admissible in if
We recall the known Strichartz estimates  (and the reference therein).
Proposition 2.2. Let and be any two admissible pairs Then one has the estimate with the prime exponents denoting Hölder dual exponents.
We also define the fractional differential operator for any real by and analogously where . We then define the homogeneous Sobolev space and the inhomogeneous Sobolev space by
We also need some Littlewood-Paley theory. Specifically, let be a smooth bump supported in and equalling one on . For each number we define the Littlewood-Paley operators: Similarly, we can define , and , whenever and are dyadic numbers. We will frequently write for and similarly for the other operators. We recall the following standard Bernstein and Sobolev type inequalities.
Lemma 2.3. For any and , one has
We collect the basic properties of into the following.
Lemma 2.4. Let and . Then
Proof. For the proof one can see Colliander et al. .
Now we define the Strichartz norm of functions Then we introduce the following bilinear smoothing property due to Bourgain .
Lemma 2.5. Let such that Then, for , the following inequality holds: That is to say, suppose solves (1.15)–(1.18) on the time interval . Let , for with . Then The estimate (2.17) will be also valid if is replaced by .
We also have the local well posedness result.
3. Almost Conservation Laws
In this section we prove the almost conservation of the modified energy .
Proof. In light of (2.19), it suffices to control the energy increment for in terms of . Applying the operator to the system (1.14)-(1.15):
From now on, we abbreviate as for simplicity. An elementary calculation shows that is controlled by the sum of the space-time integrals:
Here we used the properties of operator for : (i), where denotes the space of bounded linear operators from to ; (ii)then ,.
We estimate first. We use to denote . When is dyadically localized to and we will write by . The analysis will not rely upon the complex conjugate structure in the left side of (3.4). Thus, there is symmetry under the interchange of the indices , and We may assume that .
Case I. . Since the convolution hypersurface is , we have as well. Hence we get , and the bound holds trivially.
Case II. , for , we have . Applying the mean value theorem, we deduce that Moreover, is controlled by
Case III: . We have the bound on the symbol: If , then we bounded by renormalizing the derivatives and multiplier, paring and and using Lemma 2.5: We write this bound as Since is bounded from above by 1 and for is nondecreasing and bounded from above by 1, for , we bound here we used the fact that for . For , by using the definition of : Using the facts and , Hence
If , then paring and and using Lemma 2.5 again, a similar analysis leads to the bound:
Now we turn to give the bound for the term . it required 6-linear estimate for (3.5). We write to denote and use to denote the size of . By symmetry, we may assume . We carry out a case by case analysis.
Case I. . On , this forces as well. In this case, , and the bound holds trivially.
Case II. . On , in this case. By the mean value theorem, Applying the Cauchy-Schwartz inequality and the above multiplier bound to (3.5), we deduce that By Hölder inequality and Lemma 2.5, we control the above expression by By the Sobolev’s inequality, we have It follows from Colliander et al. in  that
We use (3.7)–(3.15) to complete the Case II analysis. is bounded by
Case III . We have the bound on the symbol Similar steps leads to the bound Combine the estimates for and , we can complete the proof of Proposition 3.1.
Remark 3.2. One can see that the proof of Proposition 3.1 closely follows the proof from Colliander . However, the proof in this paper provides some clarity to the final stages of the proof in  and the necessary restrictions on .
4. The Interaction Morawetz Inequality
We first recall the generalized viriel identity .
Proof. Since is convex and real valued and , by the fundamental theorem of calculus we can easily deduce the result. In the case of a solution to an equation with a nonlinearity which is not associated to a defocusing potential, the following corollary holds.
Corollary 4.2. Let be convex and be a smooth solution to the equation: Then, the following inequality holds: where is the Morawetz action corresponding to and is the momentum bracket defined by
Now we give the interaction Morawetz inequality, although the results presented here are well known to experts, it seems to us that simple, self-contained proofs are often difficult to locate, so we present them for the convenience of the reader.
Proof. The proof of the Proposition 4.3 is similar to that in Colliander et al. . Now we choose if ; if ; then, is smooth and convex for all . We apply the generalized viriel identity with the weight and the tensor product , where are solutions with to (1.14)-(1.15). It is not hard to see that the tensor product satisfies the equation:
and is the Laplace in .
Then we conclude that where Note that the definition of implies Thus It follows from the Fubuni’s theorem that On the other hand, Picking , we get
Remark 4.4. For the common Morawetz inequality, the nonlinear term (the second term in (4.1)) has played the central role in the scattering theory for the nonlinear Schrödinger equation, and the first term in (4.1) did not play a big role in these works. But now by taking advantage of the first term, we can obtain a global prior estimate for defocusing nonlinearity, and we mention that the heart of the matter is that This idea first appeared in .
Now, we consider the solution of If does not solve (4.7) but the following equation: then the calculations that we have done above would reveal that of course this is not the case. We may rewrite (4.17) as For what follows we abbreviate where is the solution of We aim to prove the following theorem.
Proof. According to Corollary 4.2,
If solves (4.3) for , then solves (4.3) for , with right-hand side given by
Now we decompose as good part and bad part. The good part creates a positive term that we ignore. The bad term produces the error term. Now we have the bound:
where we have used the fact that . Remark that is a real valued, thus
and that . We now compute the dot product under the integral in (4.20), that is,
Using the definition of and the fact that acts only on , we have
Analogously, we can see that the second part is given by We have Here we used the fact that the pair is admissible and . By a similar way we can deduce that Hence, we only need to estimate . Observe that , and Using the fact that and the properties of operator , we have There is symmetry under interchange of the indices 1, 2, 3. We may assume that Let We carry out a case by case analysis for (4.28).
Case I. , this force , then there no contribution to (4.25).
Case II. , we have where we have used the fact that and by mean value theorem that Therefore,
Case III . We also have
Case IV . We estimate as follows: where we have used the estimate Finally, since the pair (3, 6) is admissible, we can get Combining (4.22), (4.24), and (4.40), we complete the proof Theorem 4.5.
5. The Proof of Theorem 1.1
The idea is followed from [26, 34]. The first observation is the fact that if is a solution of the Cauchy problem (1.14)-(1.15), we can scale it and obtain a new solution, namely, the scale function satisfies the same equation with initial data . This scaling preserves the norms of as well as the space-time norm. From estimate (2.13), we have We can choose the parameter in the manner so that .
Then, for any (arbitrarily large), define where is a constant to be chosen later. We claim that is the whole interval . Indeed, if is not the whole interval , then using the fact that is a continuous function of time, there exist some with the properties, Now we divide the time interval into subintervals , in such a way that where is as the same as in Proposition 2.6. This is possible because of (5.7). Then, the number of the slices, which we will call , is most like
According to Propositions 2.6 and 3.1, we have that, for any for our choice of , . Noting that , we can apply the Proposition 3.1. In order to guarantee that for all , we require Since , this is fulfilled as long as note (5.3), that is to say we have If , we have that is arbitrarily large if we send to infinity.
We use Theorem 4.5 to show that (5.6) is not true. Recall the estimates (4.22)–(4.24), we have which implies that This estimate contradicts to (5.6) for suitable choice of (namely, we choose ). Therefore , and can be chosen arbitrarily large. In addition, Since is arbitrarily large, the priori bound on the norm concludes the global well-posedness of the Cauchy Problem (1.14)-(1.15).
This work is supported by National Natural Science Foundation of China (11071177) and Fundamental Research Funds for the Central Universities (SWJTU12CX061 and SWJTU12ZT13).
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