The initial value problems for the Korteweg-de Vries (KdV) and modified KdV (mKdV) equations under periodic and decaying boundary conditions are considered. These initial value problems are shown to be globally well-posed in all Sobolev spaces -based where local well-posedness is presently known, apart from the endpoint for mKdV and the endpoint for KdV. The result for KdV relies on a new method for constructing almost conserved quantities using multilinear harmonic analysis and the available local-in-time theory. Miura’s transformation is used to show that global well-posedness of modified KdV is implied by global well-posedness of the standard KdV equation.
The initial value problem for the Korteweg-de Vries (KdV) equation,
has been shown to be locally well-posed (LWP) for Kenig, Ponce and Vega Reference32 extended the local-in-time analysis of Bourgain Reference5, valid for to the range , by constructing the solution of Equation1.1 on a time interval with depending upon Earlier results can be found in .Reference4, Reference28, Reference23, Reference31, Reference12. We prove here that these solutions exist for in an arbitrary time interval thereby establishing global well-posedness (GWP) of Equation1.1 in the full range The corresponding periodic initial value problem for KdV -valued
is known Reference32 to be locally well-posed for These local-in-time solutions are also shown to exist on an arbitrary time interval. Bourgain established .Reference9 global well-posedness of Equation1.2 for initial data having (small) bounded Fourier transform. The argument in Reference9 uses the complete integrability of KdV. Analogous globalizations of the best known local-in-time theory for the focussing and defocussing modified KdV (mKdV) equations ( in Equation1.1, Equation1.2 replaced by and respectively) are also obtained in the periodic , and real line ( settings.
The local-in-time theory globalized here is sharp (at least up to certain endpoints) in the scale of Sobolev spaces -based Indeed, recent examples .Reference33 of Kenig, Ponce and Vega (see also Reference2, Reference3) reveal that focussing mKdV is ill-posed for and that KdV ( -valued is ill-posed for ) (The local theory in .Reference32 adapts easily to the situation.) A similar failure of local well-posedness below the endpoint regularities for the defocusing modified KdV and the -valued KdV has been established -valuedReference13 by Christ, Colliander and Tao. The fundamental bilinear estimate used to prove the local well-posedness result on the line was shown to fail for by Nakanishi, Takaoka and Tsutsumi Reference45. Nevertheless, a conjugation of the local well-posedness theory for defocusing mKdV using the Miura transform established Reference13 a local well-posedness result for KdV at the endpoint Global well-posedness of KdV at the . endpoint and for mKdV in remain open problems.
solutions of KdV satisfy -valued conservation: Consequently, a local well-posedness result with the existence lifetime determined by the size of the initial data in . may be iterated to prove global well-posedness of KdV for data Reference5. What happens to solutions of KdV which evolve from initial data which are less regular than Bourgain observed, in a context ?Reference8 concerning very smooth solutions, that the nonlinear Duhamel term may be smoother than the initial data. This observation was exploited Reference8, using a decomposition of the evolution of the high and low frequency parts of the initial data, to prove polynomial-in-time bounds for global solutions of certain nonlinear Schrödinger (NLS) and nonlinear wave (NLW) equations. In Reference10, Bourgain introduced a general high/low frequency decomposition argument to prove that certain NLS and NLW equations were globally well-posed below the natural regularity associated with the conserved Hamiltonian. Subsequently, Bourgain’s high/low method has been applied to prove global well-posedness below the natural regularity of the conserved quantity in various settings ,Reference20, Reference50, Reference48, Reference34, including KdV Reference18 on the line. A related argument—directly motivated by Bourgain’s work—appeared in Reference29, Reference30 where the presence of derivatives in the nonlinearities leaves a Duhamel term which cannot be shown to be smoother than the initial data. Global rough solutions for these equations are constructed with a slightly different use of the original conservation law (see below).
We summarize the adaptation Reference18 of the high/low method to construct a solution of Equation1.1 for rough initial data. The task is to construct the global solution of Equation1.1 evolving from initial data for with The argument .Reference18 accomplishes this task for initial data in a subset of consisting of functions with relatively small low frequency components. Split the data with where , is a parameter to be determined. The low frequency part of is in (in fact for all with a big norm while the high frequency part ) is the tail of an function and is therefore small (with large in ) for any The low frequencies are evolved according to KdV: . The high frequencies evolve according to a “difference equation” which is selected so that the sum of the resulting high frequency evolution, and the low frequency evolution solves Equation1.1. The key step is to decompose where , is the solution operator of the Airy equation. For the selected class of rough initial data mentioned above, one can then prove that and has a small (depending upon ) norm. Then an iteration of the local-in-time theory advances the solution to a long (depending on time interval. An appropriate choice of ) completes the construction.
The nonlinear Duhamel term for the “difference equation” mentioned above is
with the space defined below (see Equation1.11). The estimate Equation1.3 is valid for functions such that are supported outside in the range , Reference18, Reference15. The estimate Equation1.3 fails for and this places an intrinsic limitation on how far the high/low frequency decomposition technique may be used to extend GWP for rough initial data. Also, Equation1.3 fails without some assumptions on the low frequencies of and hence the initial data considered in the high/low argument of ,Reference18. We showed that the low frequency issue may indeed be circumvented in Reference15 by proving Equation1.1 is GWP in The approach in .Reference15 does not rely on showing the nonlinear Duhamel term has regularity at the level of the conservation law. We review this approach now and motivate the nontrivial improvements of that argument leading to sharp global regularity results for Equation1.1 and Equation1.2.
Global well-posedness follows from (an iteration of) local well-posedness (results) provided the successive local-in-time existence intervals cover an arbitrary time interval The length of the local-in-time existence interval is controlled from below by the size of the initial data in an appropriate norm. A natural approach to global well-posedness in . is to establish upper bounds on for solutions which are strong enough to prove that may be covered by iterated local existence intervals. We establish appropriate upper bounds to carry out this general strategy by constructing almost conserved quantities and rescaling. The rescaling exploits the subcritical nature of the KdV initial value problem (but introduces technical issues in the treatment of the periodic problem). The almost conserved quantities are motivated by the following discussion of the conservation property of solutions of KdV.
is the (spatial) Fourier transform. Fourier transform properties imply
Since we are assuming is we may replace -valued, by Hence, .
We apply and we use symmetry and the equation to find ,
The first expression is symmetric under the interchange of and so may be replaced by Since we are integrating on the set where . the integrand is zero and this term vanishes. Calculating , the remaining term may be rewritten ,
On the set where , which we symmetrize to replace in Equation1.4 by and this term vanishes as well. Summarizing, we have found that solutions -valued of KdV satisfy
and both integrands on the right side vanish.
We introduce the (spatial) Fourier multiplier operator defined via
with an arbitrary multiplier -valued A formal imitation of the Fourier proof of . conservation above reveals that for -mass solutions of KdV we have -valued
The term arising from the dispersion cancels since on the set where The remaining trilinear term can be analyzed under various assumptions on the multiplier . giving insight into the time behavior of Moreover, the flexibility in our choice of . may allow us to observe how the conserved mass is moved around in frequency space during the KdV evolution.
Our use of the multiplier to localize the mass in frequency space is analogous to the use of cutoff functions to spatially localize the conserved density in physical space. In that setting, the underlying conservation law is multiplied by a cutoff function. The localized flux term is no longer a perfect derivative and is then estimated, sometimes under an appropriate choice of the cutoff, to obtain bounds on the spatially localized energy.
Consider now the problem of proving well-posedness of Equation1.1 or Equation1.2, with on an arbitrary time interval , We define a spatial Fourier multiplier operator . which acts like the identity on low frequencies and like a smoothing operator of order on high frequencies by choosing a smooth monotone multiplier satisfying
The parameter marks the transition from low to high frequencies. When the operator , is essentially the integration (since operator ) When . , acts like the identity operator. Note that is bounded if We prove a variant local well-posedness result which shows the length of the local existence interval . for Equation1.1 or Equation1.2 may be bounded from below by for an appropriate range of the parameter , The basic idea is then to bound the trilinear term in .Equation1.6 to prove, for a particular small that ,
If is huge, Equation1.7 shows there is at most a tiny increment in as evolves from to An iteration of the local theory under appropriate parameter choices gives global well-posedness in . for certain .
The strategy just described is enhanced with two extra ingredients: a multilinear correction technique and rescaling. The correction technique shows that, up to errors of smaller order in the trilinear term in ,Equation1.6 may be replaced by a quintilinear term improving Equation1.7 to
where is tiny. The rescaling argument reduces matters to initial data of fixed size: In the periodic setting, the rescaling we use forces us to track the dependence upon the spatial period in the local well-posedness theory .Reference5, Reference32.
The main results obtained here are:
The initial value problem -valuedEquation1.1 is globally well-posed for initial data
The periodic initial value problem -valuedEquation1.2 is globally well-posed for initial data .
The initial value problem for modified KdV -valuedEquation9.1 (focussing or defocussing) is globally well-posed for initial data .
The periodic initial value problem for modified KdV (focussing or defocussing) is globally well-posed for initial data -valued.
The infinite-dimensional symplectic nonsqueezing machinery developed by S. Kuksin Reference36 identifies as the Hilbert Darboux (symplectic) phase space for KdV. We anticipate that Theorem 3 will be useful in adapting these ideas to the KdV context. The main remaining issue is an approximation of the KdV flow using finite-dimensional Hamiltonian flows analogous to that obtained by Bourgain Reference6 in the NLS setting. We plan to address this topic in a forthcoming paper.
We conclude this subsection with a discussion culminating in a table which summarizes the well-posedness theory in SobolevFootnote2 There are results, e.g. Reference9, in function spaces outside the Sobolev scale. -based✖ spaces for the polynomial generalized KdV equations. The initial value problem
has the associated Hamiltonian
The replacement shows that the choice is irrelevant when is even, but, when is odd there are two distinct cases in Equation1.9: is called focussing and is called defocussing. The usefulness of the Hamiltonian in controlling the norm can depend upon the choice in Equation1.10.
We now summarize the well-posedness theory for the generalized KdV equations. The notation D and F in Table 1 refers to the defocussing and focussing cases. We highlight with the notation ?? some issues which are not yet resolved (as far as we are aware).
Our results here and elsewhere Reference16, Reference14, Reference17 suggest that local well-posedness implies global well-posedness in subcritical dispersive initial value problems. In particular, we believe our methods will extend to prove GWP of mKdV in and KdV in and also extend the GWP intervals in the cases However, our results rely on the fact that we are considering the . KdV equation and, due to a lack of conservation laws, we do not know if the local results for the -valued KdV equation may be similarly globalized. An adaptation of techniques from -valuedReference13 may provide ill-posedness results in the higher power defocussing cases. Blow up in the focussing supercritical ( or, more generally, with is expected to occur but no rigorous results in this direction have been so far obtained )Reference39.
Sections 2 and 3 describe the multilinear correction technique which generates modified energies. Section 4 establishes useful pointwise upper bounds on certain multipliers arising in the multilinear correction procedure. These upper bounds are combined with a quintilinear estimate, in the setting, to prove the bulk of Equation1.8 in Section 5. Section 6 contains the variant local well-posedness result and the proof of global well-posedness for Equation1.1 in We next consider the periodic initial value problem Equation1.2 with period Section 7 extends the local well-posedness theory for .Equation1.2 to the setting. Section 8 proves global well-posedness of -periodicEquation1.2 in The last section exploits Miura’s transform to prove the corresponding global well-posedness results for the focussing and defocussing modified KdV equations. .
We will use to denote various time independent constants, usually depending only upon In case a constant depends upon other quantities, we will try to make that explicit. We use . to denote an estimate of the form Similarly, we will write . to mean and To avoid an issue involving a logarithm, we depart from standard practice and write . The notation denotes for an arbitrarily small Similarly, . denotes We will make frequent use of the two-parameter spaces . with norm
For any time interval we define the restricted spaces , by the norm
These spaces were first used to systematically study nonlinear dispersive wave problems by Bourgain Reference5. Klainerman and Machedon Reference37 used similar ideas in their study of the nonlinear wave equation. The spaces appeared earlier in a different setting in the works Reference46, Reference1 of Rauch, Reed, and M. Beals. We will systematically ignore constants involving in the Fourier transform, except in Section 7. Other notation is introduced during the developments that follow.
In this section, we introduce notation for describing certain multilinear operators; see for example Reference41, Reference40. Bilinear versions of these operators will generate a sequence of almost conserved quantities involving higher order multilinear corrections.
A k-multiplier is a function A . is symmetric if -multiplier for all the group of all permutations on , objects. The symmetrization of a -multiplier is the multiplier
The domain of is however, we will only be interested in ; on the hyperplane .
A generates a k-linear functional or k-form acting on -multiplier functions ,
We will often apply to copies of the same function in which case the dependence upon may be suppressed in the notation: may simply be written .
If is symmetric, then is a symmetric functional. -linear
As an example, suppose that is an function. We calculate -valued
The time derivative of a symmetric functional can be calculated explicitly if we assume that the function -linear satisfies a particular PDE. The following statement may be directly verified by using the KdV equation.
Suppose satisfies the KdV equation Equation1.1 and that is a symmetric Then -multiplier.
Note that the second term in Equation2.3 may be symmetrized.
Let be an arbitrary even 1-multiplier and define the associated operator by -valued
We define the modified energy by
The name “modified energy” is in part justified since in case We will show later that for of a particular form, certain modified energies enjoy an almost conservation property. By Plancherel and the fact that and are -valued,
Using Equation2.3, we have
The first term vanishes. We symmetrize the remaining term to get
Note that the time derivative of is a 3-linear expression. Let us denote
Observe that if the symmetrization results in , This reproduces the Fourier proof of . conservation from the introduction. -mass
Form the new modified energy
where the symmetric 3-multiplier will be chosen momentarily to achieve a cancellation. Applying Equation2.3 gives
to force the two terms in Equation3.4 to cancel. With this choice, the time derivative of is a 4-linear expression where
This process can clearly be iterated to generate satisfying
These higher degree corrections to the modified energy may be of relevance in studying various qualitative aspects of the KdV evolution. However, for the purpose of showing GWP in down to and in down to we will see that almost conservation of , suffices.
The modified energy construction process is illustrated in the case of the Dirichlet energy
Define and use ,Equation2.3 to see
where is explicitly obtained from Noting that . on the set we know that ,
The choice of results in a cancellation of the terms and
Therefore, is an exactly conserved quantity. The modified energy construction applied to the Dirichlet energy led us to the Hamiltonian for KdV. Applying the construction to higher order derivatives in will similarly lead to the higher conservation laws of KdV.
This section presents a detailed analysis of the multipliers which were introduced in the iteration process of the previous section. The analysis identifies cancellations resulting in pointwise upper bounds on these multipliers depending upon the relative sizes of the multiplier’s arguments. These bounds are applied to prove an almost conservation property in the next section. We begin by recording some arithmetic and calculus facts.
The following arithmetic facts may be easily verified:
A related observation for the circle was exploited by C. Fefferman Reference19 and by Carleson and Sjölin Reference11 for curves with nonzero curvature. These properties were also observed by Rosales Reference47 and Equation4.1 was used by Bourgain in Reference5.
Let and be smooth functions of the real variable We say that . is controlled by if is nonnegative and satisfies for and
for all nonzero .
With this notion, we can state the following forms of the mean value theorem.
If is controlled by and then ,
If is controlled by and then ,
We will sometimes refer to our use of Equation4.4 as applying the double mean value theorem.
The multiplier was defined in Equation3.3. In this section, we will generally be considering an arbitrary even 1-multiplier -valued We will specialize to the situation when . is of the form Equation4.7 below. Recalling that and that is even allows us to re-express Equation3.3 as
If is even and -valued is controlled by itself, then, on the set (dyadic),
Symmetry allows us to assume In case . the claimed estimate is equivalent to showing ,
In the particular case when the multiplier is smooth, monotone, and of the form
This subsection establishes the following pointwise upper bound on the multiplier .
Assume is of the form Equation4.7. In the region where for dyadic,
We begin by deriving two explicit representations of in terms of These identities are then analyzed in cases to prove .Equation4.9.
and We shall ignore the irrelevant constant in .Equation4.10. Therefore,
Recall also from Equation4.2 that
We can now rewrite the first term in Equation4.12
The second term in Equation4.12 is rewritten, using and the fact the , is even,
We record two identities for .
If is even and the following two identities for -valued, are valid:
Proof of Lemma Equation4.4.
The proof consists of a case-by-case analysis pivoting on the relative sizes of Symmetry properties of . permit us to assume that Consequently, we assume . Since for a glance at ,Equation4.12 shows that vanishes when We may therefore assume that Since . we must also have ,.
From Equation4.13, we know that we can replace on the right side of Equation4.9 by Suppose . Then, and so Thus, at least one of . must be at least of size comparable to The right side of .Equation4.9 may be re-expressed as
Case 1. .
Term in Equation4.16 is bounded by and therefore, after cancelling , with one of the satisfies ,Equation4.9. Term is treated next. In case ,Equation4.18 is an upper bound of and the triangle inequality gives since is a decreasing function. If and we rewrite ,
Applying the mean value theorem and using gives since and so this subcase is fine. If , and the double mean value theorem ,Equation4.4 applied to term gives the bound
Our assumptions on give the bound which is smaller than Equation4.18.
The remaining subcases have either precisely one element of the set much smaller than or precisely two elements much smaller than In the case of just one small . we apply the mean value theorem as above. When there are two small , we apply the double mean value theorem as above. ,
Certainly, in this region. It is not possible for both and in this region. Indeed, we find then that and which with implies but while We need to show ..
Case 2A. .
Replacing by and by we identify three differences poised for the mean value theorem. We find this term equals ,
with for so This expression is also ..
Case 2B. .
Since and we must have , We have . and here so our desired upper bound is We recall .Equation4.16 and evaluate when we can to find
The last term is dangerous so we isolate a piece of the first term to cancel it out. Expanding we see that ,
The first piece cancels with in Equation4.19 and the second piece is of size which is fine. It remains to control ,
by Expand using Equation4.13 to rewrite this expression as
(The second term in Equation4.20 cancelled with part of the first.) The second and third terms in Equation4.21 are and may therefore be ignored. We rewrite the first term in Equation4.21 using the fact that is even as
Since we can apply the double mean value theorem to obtain ,
with Therefore, this term is bounded by .
which is smaller than as claimed.
Case 2C. .
This case follows from a modification of Case 2A.
This case does not occur because but is very small which forces to also be small, which is a contradiction.
The multiplier was defined in Equation3.9, with Our work on above showed that vanishes whenever vanishes so there is no denominator singularity in Moreover, we have the following upper bound on . in the particular case when is of the form Equation4.7.
If is of the form Equation4.7, then