Sharp global well-posedness for KdV and modified KdV on $\mathbb{R}$ and $\mathbb{T}$
By J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao
Abstract
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 $L^2$-based Sobolev spaces $H^s$ where local well-posedness is presently known, apart from the $H^{\frac{1}{4}} ({\mathbb{R}})$ endpoint for mKdV and the $H^{-\frac{3}{4}}$ 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.
1. Introduction
The initial value problem for the Korteweg-de Vries (KdV) equation,
has been shown to be locally well-posed (LWP) for $s> - \frac{3}{4}.$ Kenig, Ponce and Vega Reference 32 extended the local-in-time analysis of Bourgain Reference 5, valid for $s \geq 0$, to the range $s > - \frac{3}{4}$ by constructing the solution of Equation 1.1 on a time interval $[0, \delta ]$ with $\delta$ depending upon ${{\| \phi \|}_{H^s ({\mathbb{R}})}}$. Earlier results can be found in Reference 4, Reference 28, Reference 23, Reference 31, Reference 12. We prove here that these solutions exist for $t$ in an arbitrary time interval $[0,T]$ thereby establishing global well-posedness (GWP) of Equation 1.1 in the full range $s > - \frac{3}{4}.$ The corresponding periodic ${\mathbb{R}}$-valued initial value problem for KdV
is known Reference 32 to be locally well-posed for $s \geq - \frac{1}{2}$. These local-in-time solutions are also shown to exist on an arbitrary time interval. Bourgain established Reference 9 global well-posedness of Equation 1.2 for initial data having (small) bounded Fourier transform. The argument in Reference 9 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 ($u^2$ in Equation 1.1, Equation 1.2 replaced by $-u^3$ and $u^3$, respectively) are also obtained in the periodic $(s \geq \frac{1}{2})$ and real line ($s > \frac{1}{4})$ settings.
The local-in-time theory globalized here is sharp (at least up to certain endpoints) in the scale of $L^2$-based Sobolev spaces $H^s$. Indeed, recent examples Reference 33 of Kenig, Ponce and Vega (see also Reference 2, Reference 3) reveal that focussing mKdV is ill-posed for $s < \frac{1}{4}$ and that ${\mathbb{C}}$-valued KdV ($u: {\mathbb{R}}\times [0,T] \longmapsto {\mathbb{C}}$) is ill-posed for $s < -\frac{3}{4}$. (The local theory in Reference 32 adapts easily to the ${\mathbb{C}}$-valued situation.) A similar failure of local well-posedness below the endpoint regularities for the defocusing modified KdV and the ${\mathbb{R}}$-valued KdV has been established Reference 13 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 $s \leq -\frac{3}{4}$ by Nakanishi, Takaoka and Tsutsumi Reference 45. Nevertheless, a conjugation of the $H^{\frac{1}{4}}$ local well-posedness theory for defocusing mKdV using the Miura transform established Reference 13 a local well-posedness result for KdV at the endpoint $H^{-\frac{3}{4}} ({\mathbb{R}})$. Global well-posedness of KdV at the $-\frac{3}{4}$ endpoint and for mKdV in $H^{\frac{1}{4}}$ remain open problems.
1.1. GWP below the conservation law
${\mathbb{R}}$-valued solutions of KdV satisfy $L^2$ conservation: ${{\| u(t) \|}_{L^2}} = {{\| \phi \|}_{L^2}}$. Consequently, a local well-posedness result with the existence lifetime determined by the size of the initial data in $L^2$ may be iterated to prove global well-posedness of KdV for $L^2$ data Reference 5. What happens to solutions of KdV which evolve from initial data which are less regular than $L^2$? Bourgain observed, in a context Reference 8 concerning very smooth solutions, that the nonlinear Duhamel term may be smoother than the initial data. This observation was exploited Reference 8, 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 Reference 10, Bourgain introduced a general high/low frequency decomposition argument to prove that certain NLS and NLW equations were globally well-posed below $H^1$, 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 Reference 20, Reference 50, Reference 48, Reference 34, including KdV Reference 18 on the line. A related argument—directly motivated by Bourgain’s work—appeared in Reference 29, Reference 30 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 Reference 18 of the high/low method to construct a solution of Equation 1.1 for rough initial data. The task is to construct the global solution of Equation 1.1 evolving from initial data $\phi \in H^s ( {\mathbb{R}})$ for $s_0 < s < 0$ with $-3/4 \ll s_0 \lesssim 0$. The argument Reference 18 accomplishes this task for initial data in a subset of $H^s({\mathbb{R}})$ consisting of functions with relatively small low frequency components. Split the data $\phi = \phi _0 + \psi _0$ with ${\widehat{\phi _0}} (k) = \chi _{[-N,N]} (k) {\widehat{\phi }} (k)$, where $N = N(T)$ is a parameter to be determined. The low frequency part $\phi _0$ of $\phi$ is in $L^2 ({\mathbb{R}})$ (in fact $\phi _0 \in H^s$ for all $s$) with a big norm while the high frequency part $\psi _0$ is the tail of an $H^s ({\mathbb{R}})$ function and is therefore small (with large $N$) in $H^\sigma ({\mathbb{R}})$ for any $\sigma < s$. The low frequencies are evolved according to KdV: $\phi _0 \longmapsto u_0 (t).$ The high frequencies evolve according to a “difference equation” which is selected so that the sum of the resulting high frequency evolution, $\psi _0 \longmapsto v_0 (t)$ and the low frequency evolution solves Equation 1.1. The key step is to decompose $v_0 (t) = S(t) \psi _0 + w_0 (t)$, where $S(t)$ is the solution operator of the Airy equation. For the selected class of rough initial data mentioned above, one can then prove that $w_0 \in L^2 ( {\mathbb{R}})$ and has a small (depending upon $N$)$L^2$ norm. Then an iteration of the local-in-time theory advances the solution to a long (depending on $N$) time interval. An appropriate choice of $N$ completes the construction.
The nonlinear Duhamel term for the “difference equation” mentioned above is
The local well-posedness machinery Reference 5, Reference 32 allows us to prove that $w_0 (t) \in L^2 ({\mathbb{R}})$ if we have the extra smoothing bilinear estimate
$$\begin{equation} {{\left\| \partial _x (u v) \right\|}_{{X_{0,b-1}}}} \lesssim {{\left\| u \right\|}_{{X_{s,b}}}} {{\left\| v \right\|}_{{X_{s,b}}}} ,\qquad {s < 0, b= \frac{1}{2}+}, \cssId{extrasmoothing}{\tag{1.3}} \end{equation}$$
with the space ${X_{s,b}}$ defined below (see Equation 1.11). The estimate Equation 1.3 is valid for functions $u,\nobreakspace v$ such that $\widehat{u},\nobreakspace \widehat{v}$ are supported outside $\{ |k | \leq 1 \}$, in the range $-\frac{3}{8} < s$Reference 18, Reference 15. The estimate Equation 1.3 fails for $s < - \frac{3}{8}$ 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, Equation 1.3 fails without some assumptions on the low frequencies of $u$ and $v$, hence the initial data considered in the high/low argument of Reference 18. We showed that the low frequency issue may indeed be circumvented in Reference 15 by proving Equation 1.1 is GWP in $H^s ( {\mathbb{R}}),\nobreakspace s> - \frac{3}{10}$. The approach in Reference 15 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 Equation 1.1 and Equation 1.2.
1.2. The operator $I$ and almost conserved quantities
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 $[0,T]$. 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 $H^s$ is to establish upper bounds on ${{\left\| u(t) \right\|}_{{H^s}}}$ for solutions $u(t)$ which are strong enough to prove that $[0,T]$ 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 $L^2$ conservation property of solutions of KdV.
Consider the following Fourier proofFootnote1 that ${\left\| u(t) \right\|}_{L^ 2 } = {\left\| \phi \right\|}_{L^ 2 }\nobreakspace \forall t \in {\mathbb{R}}$. By Plancherel,
1
This argument was known previously; see a similar argument in Reference 27.
The first expression is symmetric under the interchange of $\xi _1$ and $\xi _2$ so $\xi _1^3$ may be replaced by $\frac{1}{2}( \xi _1^3 + \xi _2^3 )$. Since we are integrating on the set where $\xi _1 + \xi _2 = 0$, the integrand is zero and this term vanishes. Calculating $\widehat{u^2} ( \xi ) = \int \limits _{\xi = \xi _1 + \xi _2 } \widehat{u} ( \xi _1 ) \widehat{u} ( \xi _2 )$, the remaining term may be rewritten
On the set where $\xi _1 + \xi _2 + \xi _3 = 0$,$\xi _1 + \xi _2 = - \xi _3$ which we symmetrize to replace $\xi _1 + \xi _2$ in Equation 1.4 by $-\frac{1}{3} ( \xi _1 + \xi _2 + \xi _3 )$ and this term vanishes as well. Summarizing, we have found that ${\mathbb{R}}$-valued solutions $u(t)$ of KdV satisfy
with an arbitrary ${\mathbb{C}}$-valued multiplier $m$. A formal imitation of the Fourier proof of $L^2$-mass conservation above reveals that for ${\mathbb{R}}$-valued solutions of KdV we have
The term arising from the dispersion cancels since $\xi _1^3 + \xi _2^3 =0$ on the set where $\xi _1 + \xi _2 =0$. The remaining trilinear term can be analyzed under various assumptions on the multiplier $m$ giving insight into the time behavior of ${{\| I u (t) \|}_{L^2}}$. Moreover, the flexibility in our choice of $m$ may allow us to observe how the conserved $L^2$ mass is moved around in frequency space during the KdV evolution.
Consider now the problem of proving well-posedness of Equation 1.1 or Equation 1.2, with $s< 0$, on an arbitrary time interval $[0,T]$. We define a spatial Fourier multiplier operator $I$ which acts like the identity on low frequencies and like a smoothing operator of order $|s|$ on high frequencies by choosing a smooth monotone multiplier satisfying
The parameter $N$ marks the transition from low to high frequencies. When $N=1$, the operator $I$ is essentially the integration (since $s<0$) operator $D^s$. When $N= \infty$,$I$ acts like the identity operator. Note that ${{\| I \phi \|}_{L^2}}$ is bounded if $\phi \in H^s$. We prove a variant local well-posedness result which shows the length of the local existence interval $[0, \delta ]$ for Equation 1.1 or Equation 1.2 may be bounded from below by ${{\| I \phi \|}_{L^2}^{-\alpha }},\nobreakspace \alpha > 0$, for an appropriate range of the parameter $s$. The basic idea is then to bound the trilinear term in Equation 1.6 to prove, for a particular small $\beta > 0$, that
$$\begin{equation} \sup _{t \in [0, \delta ]} {{\| I u(t) \|}_{L^2}} \leq {{\| I u(0) \|}_{L^2}} + c N^{-\beta } {{\| I u (0 ) \|}_{L^2}^3}. \cssId{basicidea}{\tag{1.7}} \end{equation}$$
If $N$ is huge, Equation 1.7 shows there is at most a tiny increment in ${{\| I u(t) \|}_{L^2}}$ as $t$ evolves from $0$ to $\delta$. An iteration of the local theory under appropriate parameter choices gives global well-posedness in $H^s$ for certain $s < 0$.
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 $N$, the trilinear term in Equation 1.6 may be replaced by a quintilinear term improving Equation 1.7 to
$$\begin{equation} \sup _{t \in [0, \delta ]} {{\| I u(t) \|}_{L^2}} \leq {{\| I u(0) \|}_{L^2}} + c N^{-3 - \frac{3}{4} + \epsilon } {{\| I u (0 ) \|}_{L^2}^5}, \cssId{betteridea}{\tag{1.8}} \end{equation}$$
where $\epsilon$ is tiny. The rescaling argument reduces matters to initial data $\phi$ of fixed size: ${{\| I \phi \|}_{L^2}} \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}\epsilon _0 \ll 1$. In the periodic setting, the rescaling we use forces us to track the dependence upon the spatial period in the local well-posedness theory Reference 5, Reference 32.
The main results obtained here are:
The infinite-dimensional symplectic nonsqueezing machinery developed by S. Kuksin Reference 36 identifies $H^{-\frac{1}{2}} ({\mathbb{T}})$ 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 Reference 6 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 spaces $H^s$ for the polynomial generalized KdV equations. The initial value problem
2
There are results, e.g. Reference 9, in function spaces outside the $L^2$-based Sobolev scale.
The replacement $u \longmapsto -u$ shows that the $\pm$ choice is irrelevant when $k$ is even, but, when $k$ is odd there are two distinct cases in Equation 1.9: $+$ is called focussing and $-$ is called defocussing. The usefulness of the Hamiltonian in controlling the $H^1$ norm can depend upon the $\mp$ choice in Equation 1.10.
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).
Table 1.
${\mathbb{R}}$-Valued Generalized KdV on ${\mathbb{R}}$ Well-posedness Summary Table
Our results here and elsewhere Reference 16, Reference 14, Reference 17 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 $H^{\frac{1}{4}} ( {\mathbb{R}})$ and KdV in $H^{-\frac{3}{4}} ({\mathbb{R}})$ and also extend the GWP intervals in the cases $k \geq 4$. However, our results rely on the fact that we are considering the ${\mathbb{R}}$-valued KdV equation and, due to a lack of conservation laws, we do not know if the local results for the ${\mathbb{C}}$-valued KdV equation may be similarly globalized. An adaptation of techniques from Reference 13 may provide ill-posedness results in the higher power defocussing cases. Blow up in the focussing supercritical ($k \geq 6$ or, more generally, $k \in {\mathbb{R}}$ with $k > 5$) is expected to occur but no rigorous results in this direction have been so far obtained Reference 39.
1.3. Outline
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 ${\mathbb{R}}$ setting, to prove the bulk of Equation 1.8 in Section 5. Section 6 contains the variant local well-posedness result and the proof of global well-posedness for Equation 1.1 in $H^s ( {\mathbb{R}}),\nobreakspace s > -\frac{3}{4}.$ We next consider the periodic initial value problem Equation 1.2 with period $\lambda$. Section 7 extends the local well-posedness theory for Equation 1.2 to the $\lambda$-periodic setting. Section 8 proves global well-posedness of Equation 1.2 in $H^s ( {\mathbb{T}}),\nobreakspace s \geq - \frac{1}{2}$. The last section exploits Miura’s transform to prove the corresponding global well-posedness results for the focussing and defocussing modified KdV equations.
1.4. Notation
We will use $c,C$ to denote various time independent constants, usually depending only upon $s$. In case a constant depends upon other quantities, we will try to make that explicit. We use $A \lesssim B$ to denote an estimate of the form $A \leq C B$. Similarly, we will write $A \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}B$ to mean $A \lesssim B$ and $B \lesssim A$. To avoid an issue involving a logarithm, we depart from standard practice and write $\langle k \rangle = 2 + |k|.$ The notation $a +$ denotes $a+ \epsilon$ for an arbitrarily small $\epsilon$. Similarly, $a-$ denotes $a - \epsilon$. We will make frequent use of the two-parameter spaces $X_{s,b} ({\mathbb{R}}\times {\mathbb{R}})$ with norm
For any time interval $I$, we define the restricted spaces $X_{s,b} (R \times I)$ by the norm
$$\begin{equation*} {{\| u \|}_{X_{s,b} ( {\mathbb{R}}\times I )}} = \inf \{ {{\left\| U \right\|}_{{X_{s,b}}}} : U|_{{\mathbb{R}}\times I} = u \}. \end{equation*}$$
These spaces were first used to systematically study nonlinear dispersive wave problems by Bourgain Reference 5. Klainerman and Machedon Reference 37 used similar ideas in their study of the nonlinear wave equation. The spaces appeared earlier in a different setting in the works Reference 46, Reference 1 of Rauch, Reed, and M. Beals. We will systematically ignore constants involving $\pi$ in the Fourier transform, except in Section 7. Other notation is introduced during the developments that follow.
2. Multilinear forms
In this section, we introduce notation for describing certain multilinear operators; see for example Reference 41, Reference 40. Bilinear versions of these operators will generate a sequence of almost conserved quantities involving higher order multilinear corrections.
The domain of $m$ is ${\mathbb{R}}^k$; however, we will only be interested in $m$ on the hyperplane $\xi _1 + \dots + \xi _k = 0$.
As an example, suppose that $u$ is an ${\mathbb{R}}$-valued function. We calculate ${{\| u \|}_{L^2}^2} = \int \widehat{u} (\xi ) {\overline{\widehat{u}}} ( \xi ) d\xi = \int \limits _{\xi _1 + \xi _2 = 0} \widehat{u} (\xi _1 ) \widehat{u} (\xi _2 ) = \Lambda _2 (1).$
The time derivative of a symmetric $k$-linear functional can be calculated explicitly if we assume that the function $u$ satisfies a particular PDE. The following statement may be directly verified by using the KdV equation.
Note that the second term in Equation 2.3 may be symmetrized.
3. Modified energies
Let $m: {\mathbb{R}}\longmapsto {\mathbb{R}}$ be an arbitrary even ${\mathbb{R}}$-valued 1-multiplier and define the associated operator by
$$\begin{equation*} E_I^2 ( t) = {{\| Iu (t) \|}_{L^2}^2}. \end{equation*}$$
The name “modified energy” is in part justified since in case $m=1,\nobreakspace E^2_I (t) = {{\| u(t) \|}_{L^2}^2}.$ We will show later that for $m$ of a particular form, certain modified energies enjoy an almost conservation property. By Plancherel and the fact that $m$ and $u$ are ${\mathbb{R}}$-valued,
Observe that if $m =1$, the symmetrization results in $M_3 = c (\xi _1 + \xi _2 + \xi _3 )$. This reproduces the Fourier proof of $L^2$-mass conservation from the introduction.
to force the two $\Lambda _3$ terms in Equation 3.4 to cancel. With this choice, the time derivative of $E^3_I (t)$ is a 4-linear expression $\Lambda _4 ( M_4 )$ where
These higher degree corrections to the modified energy $E_I^2$ may be of relevance in studying various qualitative aspects of the KdV evolution. However, for the purpose of showing GWP in $H^s ( {\mathbb{R}})$ down to $s > - \frac{3}{4}$ and in $H^s ({\mathbb{T}})$ down to $s \geq - \frac{1}{2}$, we will see that almost conservation of $E^4_I (t)$ suffices.
The modified energy construction process is illustrated in the case of the Dirichlet energy
$$\begin{equation*} E^2_D (t) = {{\| \partial _x u \|}_{L^2_x}^2} = \Lambda _2 ( (i\xi _1 ) (i \xi _2)). \end{equation*}$$
Define $E^3_D (t) = E^2_D (t) + \Lambda _3 ( \sigma _3 )$, and use Equation 2.3 to see
where $M_4$ is explicitly obtained from $\sigma _3$. Noting that $i (\xi _1 + \xi _2 ) i \xi _3 \{ \xi _1 + \xi _2 \} = - \xi _3^3$ on the set $\xi _1 + \xi _2 + \xi _3 =0$, we know that
Therefore, $E^3_D (t) = \Lambda _2 ( (i \xi _1 ) (i \xi _2 )) + \Lambda _3 ( \frac{1}{3} )$ 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 $L^2$ will similarly lead to the higher conservation laws of KdV.
4. Pointwise multiplier bounds
This section presents a detailed analysis of the multipliers $M_3,\nobreakspace M_4,\nobreakspace M_5$ 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.
4.1. Arithmetic and calculus facts
The following arithmetic facts may be easily verified:
A related observation for the circle was exploited by C. Fefferman Reference 19 and by Carleson and Sjölin Reference 11 for curves with nonzero curvature. These properties were also observed by Rosales Reference 47 and Equation 4.1 was used by Bourgain in Reference 5.
With this notion, we can state the following forms of the mean value theorem.
We will sometimes refer to our use of Equation 4.4 as applying the double mean value theorem.
4.2. $M_3$ bound
The multiplier $M_3$ was defined in Equation 3.3. In this section, we will generally be considering an arbitrary even ${\mathbb{R}}$-valued 1-multiplier $m$. We will specialize to the situation when $m$ is of the form Equation 4.7 below. Recalling that $\xi _1 + \xi _2 + \xi _3 =0$ and that $m$ is even allows us to re-express Equation 3.3 as
The multiplier $M_5$ was defined in Equation 3.9, with $\sigma _4 = - \frac{M_4}{\alpha _4}.$ Our work on $M_4$ above showed that $M_4$ vanishes whenever $\alpha _4$ vanishes so there is no denominator singularity in $M_5$. Moreover, we have the following upper bound on $M_5$ in the particular case when $m$ is of the form Equation 4.7.
5. Quintilinear estimate on ${\mathbb{R}}$
The $M_5$ upper bound contained in Lemma 4.6 and the local well-posedness machinery Reference 31, Reference 5, Reference 32 are applied to prove an almost conservation property of the modified energy $E^4_I$. The almost conservation of $E^4_I$ is the key ingredient in our proof of global well-posedness of the initial value problem for KdV with rough initial data.
Recall that $X_{s,b}^\delta$ denotes the Bourgain space Reference 5 associated to the cubic $\{ \tau = \xi ^3 \}$ on the time interval $[0,\delta ]$. We begin with a quintilinear estimate.
Lemma 5.1 is combined with the $M_5$ upper bound of Lemma 4.6 in the next result.
A glance back at Equation 3.8 shows that for solutions of KdV, we can now control the increment of the modified energy $E^4_I$.
6. Global well-posedness of KdV on ${\mathbb{R}}$
The goal of this section is to construct the solution of the initial value problem Equation 1.1 on an arbitrary fixed time interval $[0,T]$. We first state a variant of the local well-posedness result of Reference 32. Next, we perform a rescaling under which the variant local result has an existence interval of size 1 and the initial data is small. This rescaling is possible because the scaling invariant Sobolev index for $KdV$ is $- \frac{3}{2}$ which is much less than $-\frac{3}{4}$. Under the rescaling, we show that Equation 3.8 and Equation 5.6 allow us to iterate the local result many times with an existence interval of size 1, thereby extending the local-in-time result to a global one. This will prove Theorem 1.
6.1. A variant local well-posedness result
The expression ${\left\| Iu(t) \right\|}_{L^ 2 }$, where ${\widehat{Iu(t)}}( \xi ) = m(\xi ) {\widehat{u(t)}}(\xi )$ and $m$ is of the form Equation 4.7, is closely related to the $H^s ( {\mathbb{R}})$ norm of $u$. Recall that the definition of $m$ in Equation 4.7 depends upon $s$. An adaptation of the local well-posedness result in Reference 32, along the lines of Lemma 5.2 in Reference 16 and Section 12 in Reference 17, establishes the following result.
We briefly describe why this result follows from the arguments in Reference 32. The norm ${{\| Iu \|}_{L^2}}$ is connected to the norm ${{\| u \|}_{H^s}}$ by the identity ${{\| Iu \|}_{L^2}} = {{\| Du \|}_{H^s}}$ where $D$ is the Fourier multiplier operator with symbol
Since $-\frac{3}{4} < s < 0,\nobreakspace d$ is essentially nondecreasing and $d(\xi ) \gtrsim 1$ so $D$ acts like a differential operator. The crucial bilinear estimate required to prove Proposition 2 is
$$\begin{equation*} {{\| I(uv)_x \|}_{X_{0, -\frac{1}{2}+}}} \lesssim {{\| I u \|}_{X_{0,\frac{1}{2}+}}} {{\| I v \|}_{X_{0, \frac{1}{2}+}}}, \end{equation*}$$
which is equivalent to showing
$$\begin{equation} {{\| D (uv)_x \|}_{X_{s, -\frac{1}{2}+}}} \lesssim {{\| D u \|}_{X_{s,\frac{1}{2}+}}} {{\| D v \|}_{X_{s,\frac{1}{2}+}}}. \cssId{nearlydone}{\tag{6.3}} \end{equation}$$
Since $d(\xi _1 + \xi _2) \lesssim d(\xi _1 ) + d(\xi _2 )$, the operator $D$ may be moved onto the higher frequency factor inside the parenthesis in the left side of Equation 6.3 and the bilinear estimate of Reference 32 then proves Equation 6.3.
6.2. Rescaling
Our goal is to construct the solution of Equation 1.1 on an arbitrary fixed time interval $[0, T]$. We rescale the solution by writing $u_\lambda ( x, t) = \lambda ^{-2} u ( \frac{x}{\lambda } , \frac{t}{\lambda ^3 } )$. We achieve the goal if we construct $u_\lambda$ on the time interval $[0 , \lambda ^3 T ]$. A calculation shows that
$$\begin{equation*} {\left\| I \phi _\lambda \right\|}_{L^ 2 } \lesssim \lambda ^{- \frac{3}{2} - s } N^{-s} {{\left\| \phi \right\|}_{{H^s}}}. \end{equation*}$$
The choice of the parameter $N= N(T)$ will be made later but we select $\lambda$ now by requiring
and the task is to construct the solution of Equation 1.1 on the time interval $[0, \lambda ^3 T]$.
6.3. Almost conservation
Recall the modified energy $E^2_I (0) = {{\| I \phi \|}_{L^2}^2} = \Lambda _2 ( m(\xi _1 ) m(\xi _2 )) (0).$ This subsection shows that the modified energy $E^2_I (t)$ of our rescaled local-in-time solution $u$ is comparable to the modified energy $E^4_I (t)$. Next, as forecasted in Section 5, we use Equation 3.8 and the bound Equation 5.6 to show $E^4_I (t)$ is almost conserved, implying almost conservation of $E^2_I (t) = {{\| I u(t) \|}_{L^2}^2}$. Since the lifetime of the local result Equation 6.1 is controlled by ${{\| I \phi \|}^{2}_{L^2}}$, this conservation property permits us to iterate the local result with the same sized existence interval.
Since our rescaled solution satisfies ${{\| I \phi \|}_{L^2}^2} = \epsilon _0^2 < 1$, we are certain that
whenever ${{\| I u(t) \|}_{L^2}^2} = E^2_I (t) < 2 \epsilon _0 .$ Using the estimate Equation 5.6 in Equation 3.8, the rescaled solution is seen to satisfy
Consequently, using Equation 6.17, we see that the rescaled solution has
$$\begin{equation*} {{\| I u (1) \|}_{L^2}^2} = \epsilon _0^2 + O (\epsilon _0^3) + C \epsilon _0^5 N^{-3 - \frac{3}{4}+} < 4 \epsilon _0^2. \end{equation*}$$
6.4. Iteration
We may now consider the initial value problem for KdV with initial data $u(1)$ and, in light of the preceding bound, the local result will advance the solution to time $t=2$. We iterate this process $M$ times and, in place of Equation 6.18, we have
$$\begin{equation*} E^4_I (t) \leq E^4_I (0) + M C \epsilon _0^5 N^{- 3 - \frac{3}{4}+} \quad \text{for all $t \in [0, M+1]$}. \end{equation*}$$
As long as $M N^{-3 - \frac{3}{4}+} \lesssim 1$, we will have the bound
$$\begin{equation*} {{Iu(M)}_{L^2}^2} = \epsilon _0^2 + O (\epsilon ^3 ) + M C \epsilon _0^5 N^{-3 - \frac{3}{4}+} < 4 \epsilon _0^2, \end{equation*}$$
and the lifetime of the local results remains uniformly of size 1. We take $M \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}N^{3 + \frac{3}{4}-}.$ This process extends the local solution to the time interval $[0, N^{3 + \frac{3}{4}-} ]$. We choose $N = N(T)$ so that
which may certainly be done for $s > - \frac{3}{4}.$ This completes the proof of global well-posedness for $KdV$ in $H^s ({\mathbb{R}}),\nobreakspace s> - \frac{3}{4}$.
We make two observations regarding the rescalings of our global-in-time KdV solution:
In fact, the selection of $N$ is polynomial in the parameter $T$ so Equation 6.22 gives a polynomial-in-time upper bound on ${{\left\| u(t) \right\|}_{{H^s}}}$.
The choice of $\lambda$. The parameter $\lambda$ was chosen above so that
Since, from Equation 6.20, ${{\| I \phi _\lambda \|}_{L^2}} \lesssim N^{-s} \lambda ^{-\frac{3}{2} -s } {{\left\| \phi \right\|}_{{H^s}}}$, we see that Equation 6.23 holds provided we choose
The choice of $N$. The parameter $N$ is chosen so that
$$\begin{equation} N^\beta > \lambda ^3 T \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}c_{{{\left\| \phi \right\|}_{{H^s}}}, \epsilon _0 } N^{-\frac{6s}{3+2s}} T , \cssId{polyseven}{\tag{6.25}} \end{equation}$$
where $\beta$ is the exponent appearing in Equation 5.6 (in the ${\mathbb{R}}$-case just presented, $\beta = 3 + \frac{3}{4} -$). This unravels to give a sufficient choice of $N$:
In the range $-\frac{3}{2} < s$, the numerator of the exponent on $T$ is positive. The denominator is positive provided $\beta > - \frac{6}{3+2s}$. For $s = -\frac{3}{4},\nobreakspace - \frac{6}{3+2s} = 3$ so we require better than third order decay with $N$ in the local-in-time increment Equation 5.6. With $s = -\frac{3}{4}+,\nobreakspace \beta = 3 + \frac{3}{4} -$, calculating $\gamma (s)$ and inserting the resulting expression for $N$ in terms of $T$ into Equation 6.22 reveals that, for our global-in-time solutions of Equation 1.1, we have
Observe that the polynomial exponent $1+$ in Equation 6.27 does not explode as we approach the critical regularity value $- \frac{3}{4}$. This is due to the fact that Equation 5.6 gave us much more decay than required for iterating the local result. In principle, the decay rate in Equation 5.6 could be improved by going further along the sequence $\{ E^n_I \}$ of modified energies.
7. Local well-posedness of KdV on ${\mathbb{T}}$
This section revisits the local-in-time theory for periodic KdV developed by Kenig, Ponce and Vega Reference 32 and Bourgain Reference 5. Our presentation provides details left unexposed in Reference 32 and Reference 5 and quantifies the dependence of various implied constants on the length of the spatial period. This quantification is necessary for the adaptation of the rescaling argument used in Section 6 to the periodic setting.
7.1. The $\lambda$-periodic initial value problem for KdV
We consider the $\lambda$-periodic initial value problem for KdV:
$$\begin{equation} \left\{ \begin{matrix} \partial _t u + \partial _x^3 u + \frac{1}{2}\partial _x u^2 =0,& x \in [0, \lambda ], \\ u(x, 0) = \phi (x). \end{matrix} \right. \cssId{lamkdv}{\tag{7.1}} \end{equation}$$
We first want to build a representation formula for the solution of the linearization of Equation 7.1 about the zero solution. So, we wish to solve the linear homogeneous $\lambda$-periodic initial value problem
$$\begin{equation} \left\{ \begin{matrix} \partial _t w + \partial _x^3 w =0,& x \in [0, \lambda ], \\ w(x, 0) = \phi (x). \end{matrix} \right. \cssId{lamhomog}{\tag{7.2}} \end{equation}$$
Define $(dk)_\lambda$ to be normalized counting measure on ${\mathbb{Z}}/ \lambda$:
and so on. If we apply $\partial _x^m,\nobreakspace m \in {\mathbb{N}}$, to Equation 7.5, we obtain
$$\begin{equation*} \partial _x^m f(x) = \int e^{2 \pi i k x } {{(2 \pi i k )}^m} {\widehat{f}} (k ) (dk )_\lambda . \end{equation*}$$
This, together with Equation 7.6, motivates us to define the Sobolev space$H^s ( 0 , \lambda )$ with the norm
$$\begin{equation} {{\| f \|}_{H^s ( 0 , \lambda )}} = {{\| {\widehat{f}} ( k ) \langle k \rangle ^s \|}_{L^2 ( (dk)_\lambda )}}. \cssId{Hs}{\tag{7.9}} \end{equation}$$
We will often denote this space by $H^s$ for simplicity. Note that there are about $\lambda$low frequencies in the range $|k| \lesssim 1$ where the $H^s$ norm consists of the $L^2$ norm.
The Fourier inversion formula Equation 7.5 allows us to write down the solution of Equation 7.2:
$$\begin{equation} w(x,t) = S_\lambda (t) \phi (x) = \int e^{2 \pi i k x} e^{- {{(2 \pi i k)}^3} t} {\widehat{\phi }} (k) (dk )_\lambda . \cssId{Slam}{\tag{7.10}} \end{equation}$$
For a function $v = v(x,t)$ which is $\lambda$-periodic with respect to the $x$ variable and with the time variable $t \in {\mathbb{R}}$, we define the space-time Fourier transform${\widehat{v}} = {\widehat{v}}(k , \tau )$ for $k \in {\mathbb{Z}}/ \lambda$ and $\tau \in {\mathbb{R}}$ by
$$\begin{equation} {\widehat{v}} (k, \tau ) = \int \int _0^\lambda e^{-2 \pi i k x } e^{- 2 \pi i \tau t } v(x,t) dx dt. \cssId{spacetimeFT}{\tag{7.11}} \end{equation}$$
This transform is inverted by
$$\begin{equation} v(x,t) = \int \int e^{2 \pi i k x } e^{2 \pi i \tau t } {\widehat{v}} ( k , \tau ) (dk )_\lambda d\tau . \cssId{spacetimeinvFT}{\tag{7.12}} \end{equation}$$
The expression Equation 7.10 may be rewritten as a space-time inverse Fourier transform,
where $\delta ({ \eta } )$ represents a 1-dimensional Dirac mass at $\eta = 0$. This recasting shows that $S_\lambda ( \cdot ) \phi$ has its space-time Fourier transform supported precisely on the cubic $\tau = 4 \pi ^2 k^3$ in ${\mathbb{Z}}/ \lambda \times {\mathbb{R}}$.
We next find a representation for the solution of the linear inhomogeneous $\lambda$-periodic initial value problem
$$\begin{equation} \left\{ \begin{matrix} \partial _t v + \partial _x^3 v = f, & x \in [0, \lambda ], \\ v(x, 0 ) = 0, \end{matrix} \right. \cssId{laminhomog}{\tag{7.14}} \end{equation}$$
with $f = f(x,t)$ a given time-dependent $\lambda$-periodic (in $x$) function. By Duhamel’s principle,
The spatial mean $\int _{{\mathbb{T}}} u(x,t) dx$ is conserved during the evolution Equation 7.1. We may assume that the initial data $\phi$ satisfies a mean-zero assumption $\int _{{\mathbb{T}}} \phi (x) dx$ since otherwise we can replace the dependent variable $u$ by $v = u - \int _{{\mathbb{T}}} \phi$ at the expense of a harmless linear first order term. This observation was used by Bourgain in Reference 5. The mean-zero assumption is crucial for some of the analysis that follows.
7.2. Spaces of functions of space-time
The integral equation Equation 7.17 will be solved using the contraction principle in spaces introduced in this subsection. We also introduce some other spaces of functions of space-time which will be useful in our analysis of Equation 7.17.
We define the ${X_{s,b}}$ spaces for $\lambda$-periodic KdV via the norm
(We will suppress reference to the spatial period $\lambda$ in the notation for the space-time function spaces $X_{s,b}$ and the related spaces below.)
The study of periodic KdV in Reference 32, Reference 5 has been based around iteration in the spaces $X_{s, \frac{1}{2}}$. This space barely fails to control the $L^\infty _t H^s_x$ norm. To ensure continuity of the time flow of the solution we construct, we introduce the slightly smaller space $Y^s$ defined via the norm
$$\begin{equation} {{\| u \|}_{Y^s}} = {{\left\| u \right\|}_{{X_{s,\frac{1}{2}}}}} + {{\|\langle k \rangle ^s \widehat{u} (k , \tau ) \|}_{ L^2 ( (dk)_\lambda ) L^1 (d \tau )}}. \cssId{Ys}{\tag{7.19}} \end{equation}$$
If $u \in Y^s$, then $u \in L^\infty _t H^s_x$. We will construct the solution of Equation 7.17 by proving a contraction estimate in the space $Y^s$. The mapping properties of Equation 7.15 motivate the introduction of the companion spaces $Z^s$ defined via the norm
Let $\eta \in C_0^\infty ({\mathbb{R}})$ be a nice bump function supported on $[-2,2]$ with $\eta =1$ on $[-1,1]$. It is easy to see that multiplication by $\eta (t)$ is a bounded operation on the spaces $Y^s$,$Z^s$, and ${X_{s,b}}$.
By applying a smooth cutoff, we may assume that $F$ is supported on ${\mathbb{T}}\times [-3, 3]$. Let $a(t) = \operatorname {sgn} (t) {\tilde{\eta }} ( t )$, where ${\tilde{\eta }}$ is a smooth bump function supported on $[-10, 10]$ which equals 1 on $[-5,5]$. The identity
Since the Fourier transform of $\int a(t') S( - t') F(t') dt'$ evaluated at $\xi$ is given by $\int \widehat{a} ( \tau - \xi ^3 ) \widehat{F} ( \xi , \tau ) d\tau$ and one can easily verify that $|\widehat{a} ( \tau ) | = O ( \langle \tau \rangle ^{-1} )$, the claimed estimate follows using the definition Equation 7.20.
For Equation 7.25, we discard the cutoff $\eta (t)$ and note that the space-time Fourier transform of $\int a( t-t') S(t-t' ) F (t') dt'$ evaluated at $(\xi , \tau )$ is equal to $\widehat{a} ( \tau - 4 \pi ^2 \xi ^3 ) \widehat{F} ( \xi , \tau )$. The claimed estimate then follows from the definitions Equation 7.20, Equation 7.19 and the decay estimate for $\widehat{a}$ used above.
■Proposition 3.
Let $\phi$ be a $\lambda$-periodic function whose Fourier transform is supported on $\{ k : |k| \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}N\}$. Then
Proposition 3 (and the related Proposition 4 below) will not be used in our proof of local and global well-posedness of KdV on ${\mathbb{T}}$ but may be of relevance in studying other properties of the long period limit of the KdV equation.
In case $N \leq 1$, the cardinality of the set is $O(\lambda )$ so $C(N,\lambda ) \lesssim 1$ for $N \leq 1$. Assume now that $N > 1$ and rename $k_1 = x$. The task is to estimate
This set is largest when the parabola is the flattest, i.e., when $x \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}\frac{k}{2}$. We find that
If $v=v(x,t)$ is a $\lambda$-periodic function of $x$ and the spatial Fourier transform of $v$ is supported on $\{ k: |k| \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}N\}$, then
$$\begin{equation} {{\| \eta (t) v \|}_{L^4_{x,t}}} \lesssim C(N, \lambda ) {{\left\| v \right\|}_{{X_{0,\frac{1}{2}+}}}}, \cssId{lamBouStrichartz}{\tag{7.30}} \end{equation}$$
where $C(N, \lambda )$ is as it appears in Equation 7.27.
This follows easily by stacking up cubic level sets on which Equation 7.26 holds.
Lemma 7.3.
If $v=v(x,t)$ is a $\lambda$-periodic function of $x$, then
$$\begin{equation} {{\|\eta (t) v \|}_{L^4_{x,t}}} \lesssim {{\left\| v \right\|}_{{X_{0,\frac{1}{3}}}}} . \cssId{rescaledBourgain}{\tag{7.31}} \end{equation}$$
The estimate Equation 7.31 is a rescaling of the $\lambda =1$ case proven by Bourgain Reference 5.
$$\begin{equation} {{\| \eta (t) v \|}_{L^4_{x,t}}} \lesssim \{C(N, \lambda )\}^{1-} {{\left\| v \right\|}_{{X_{0,\frac{1}{2}}}}} . \cssId{atahalf}{\tag{7.32}} \end{equation}$$
7.4. Bilinear estimate
Proposition 5.
If $u$ and $v$ are $\lambda$-periodic functions of $x$, also depending upon $t$ having zero $x$-mean for all $t$, then
$$\begin{equation} {{\| \eta (t) \partial _x ( u v) \| }_{Z^{-\frac{1}{2}}}} \lesssim \lambda ^{0+} {{\| u \|}_{X_{-\frac{1}{2}, \frac{1}{2}}}} {{\| v \|}_{X_{-\frac{1}{2}, \frac{1}{2}}}}. \cssId{bilinearestimate}{\tag{7.33}} \end{equation}$$
Note that Equation 7.33 implies ${{\| \eta (t) \partial _x ( u v) \| }_{Z^{-\frac{1}{2}}}} \lesssim \lambda ^{0+} {{\| u \|}_{Y^{-\frac{1}{2}}}} {{\| v \|}_{Y^{-\frac{1}{2}}}}.$ We will relax the notation by dispensing with various constants involving $\pi$ with the recognition that some of the formulas which follow may require adjusting the constants to be strictly correct.
Proof.
The norms involved allow us to assume that $\widehat{u}$ and $\widehat{v}$ are nonnegative. There are two contributions to the $Z^{-\frac{1}{2}}$ norm we must control. We begin with the $X_{-\frac{1}{2}, -\frac{1}{2}}$ contribution. Duality and an integration by parts shows that it suffices to prove
Using Hölder, we split the left side into $L^4_{x,t} L^4_{x, t} L^2_{x,t}$ and apply Equation 7.31 to finish this case. (In fact, we control the left side of Equation 7.37 with $\lambda ^{0+} {{\left\| v_1 \right\|}_{{X_{0,\frac{1}{3}}}}} {{\left\| v_2 \right\|}_{{X_{0,\frac{1}{3}}}}} {{\left\| v_3 \right\|}_{{X_{0,0}}}} .$)
Case 2.$|k_1|, |k_2|, |k_3| \lesssim 1.$
Derivatives are cheap in this frequency setting. We use Hölder to estimate Equation 7.35 in $L^4 L^4 L^2$, and then we apply Equation 7.31 to control the $L^4$ norms. Finally, we use the Case 2 defining conditions and Sobolev to move $X_{0,\frac{1}{3}}$ to $X_{-\frac{1}{2}, \frac{1}{2}}$ on two factors and ${X_{0,0}}$ to $X_{-\frac{1}{2}, \frac{1}{2}}$ on the remaining factor. (Again, we have encountered $X_{0, \frac{1}{3}}$ on two factors.)
Since $k_1 + k_2 + k_3 = 0$, the only remaining case to consider is when one of the frequencies is small and the other two are big. Symmetry permits us to focus on
Case 3.$0 < |k_3| \lesssim 1 \lesssim |k_1|, |k_2|.$
Since we are in the $\lambda$-periodic setting and our functions have zero $x$-mean, we have $|k_3 | \gtrsim \frac{1}{\lambda }.$ We analyze this expression in two cases: when the $j=3$ term dominates the $j=1, j=2$ terms and when the $j=1$ or $j=2$ term dominates the $j=3$ term. In case $j=3$ dominates, it suffices to prove
(Strictly speaking, each $\eta$ that appears in Equation 7.39 should be replaced by $\eta ^{\frac{1}{3}}$ but we abuse the notation with the understanding that all smooth cutoff functions are essentially the same within this analysis.) The left side of Equation 7.39 is estimated via Hölder by
It remains to control the weighted $L^2_{k} L^1_{\tau }$ portion of the $Z^{-\frac{1}{2}}$ norm to complete the proof of Equation 7.33. Since $|\langle k\rangle ^{-\frac{1}{2}} {\widehat{\partial _x (uv) }} ( k, \tau )| \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}\langle k\rangle ^\frac{1}{2}| {\widehat{uv}} ( k, \tau )|$, it suffices to prove that
Upon rewriting the left side using duality, we see that an $L^4_{xt} L^2_{xt} L^4_{xt}$ Hölder application using Equation 7.31 finishes off this case. The situation when $\langle \tau _2 - k_2^3 \rangle$ is the maximum is symmetric so we are reduced to considering the case when $\langle \tau - k^3 \rangle$ is the maximum in Equation 7.46.
we get a little help from the 1-denominator in Equation 7.44. We cancel $\langle \tau _1 - k_1^3 \rangle ^{\frac{1}{6}}$ leaving $\langle \tau _1 - k_1^3 \rangle ^\frac{1}{3}$ in the denominator and ${{(\langle k\rangle \langle k_1 \rangle \langle k_2 \rangle )}^{ \frac{1}{2}-}}$ in the numerator. After the natural cancellation using Equation 7.46, we collapse to needing to prove
uniformly in the parameter $k$. Note that familiar arguments complete the proof of Equation 7.49 (and, hence, Equation 7.42) provided we show Equation 7.50.
Remark 7.4.
The condition Equation 7.48 restricts the functions $\widehat{u_i}$ essentially to the dispersive curve $\{(k,k^3): k\in {\mathbb{Z}}/ \lambda \}$. Suppose for the moment that $\lambda = 1$ and we restrict our attention to those $k$ satisfying $|k| \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}N$. Observe that the projection of the point set $S_N = \{ (k, k^3 ) \in {\mathbb{Z}}_k\times {\mathbb{R}}_{\tau }: k\in {\mathbb{Z}}, |k| \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}N \}$ onto the $\tau$-axis is a set of $N$ points which are $N^2$-separated. Therefore, if we “vertically thicken” these points $O(N^{\alpha })$ for $\alpha \ll 2$, the projected set remains rather sparse on the $\tau$-axis. The intuition underlying the proof of Equation 7.50 is that a vertical thickening of the set $S_{N_1} + {S_{N_2}}$ also projects onto a thin set on the $\tau$ axis.
Lemma 7.4.
Fix $k\in {\mathbb{Z}}\backslash \{ 0 \}$. For $k_1 , k_2 \in {\mathbb{Z}}\backslash \{0\},$ we have for all dyadic $M \geq 1$ that
The hypotheses are symmetric in $k_1 , k_2$ so we may assume $|k_1 | \geq |k_2 |.$ We first consider the situation when $|k| \geq |k_1 |$. The expression
allows us to conclude that $|k| \lesssim |\mu | \lesssim |k|^3$ since $k_1, k_2 \in {\mathbb{Z}}\backslash \{0 \}$ and $|kk_1 k_2 | \lesssim |k|^3$. Suppose $|\mu | \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}M$ (dyadic) and $|k| \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}N$ (dyadic). We have, for some $p \in [1,3]$, that $M \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}N^p$. For $\mu$ to satisfy Equation 7.52, $|k_1 k_2 | \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}M^{1 - \frac{1}{p}}$. We make the crude observation that there are at most $M^{1-\frac{1}{p}}$ multiples of $M^{\frac{1}{p}}$ in the dyadic block $\{ |\mu | \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}M\}$. Hence, the set of possible $\mu$ satisfying Equation 7.52 must lie inside a union of $M^{1-\frac{1}{p}}$ intervals of size $M^{\frac{1}{100}}$, each of which contains an integer multiple of $k$. We have then that
In case $|k| \leq |k_1 |$, we must have $|k_1 | \lesssim |\mu | \lesssim |k_1 |^3$ so, if $|k_1 | \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}N_1$ (dyadic), we must have $M \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}N_1^p$ for some $p \in [1,3]$ and we can repeat the argument presented above.
■Remark 7.5.
If we change the setting of the lemma to the case where $k, k_1, k_2 \in {\mathbb{Z}}/ \lambda \backslash \{0 \}$, we have to adjust the conclusion to read
Consider the $\lambda$-periodic initial value problem Equation 7.1 with periodic initial data $\phi \in H^s (0, \lambda ),\nobreakspace s \geq - \frac{1}{2}.$ We show first that, for arbitrary $\lambda$, this problem is well-posed on a time interval of size $\mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}1$ provided ${{\| \phi \|}_{H^{-\frac{1}{2}} (0, \lambda )}}$ is sufficiently small. Then we show by a rescaling argument that Equation 7.1 is locally well-posed for arbitrary initial data $\phi \in H^s (0, \lambda )$.
As mentioned before in Remark 7.1, we restrict our attention to initial data having zero $x$-mean.
Fix $\phi \in H^s (0,\lambda ),\nobreakspace s \geq - \frac{1}{2}$ and for $w \in Z^{-\frac{1}{2}}$ define
is defined on $Y^{-\frac{1}{2}}$. Observe that $\Gamma (u) = u$ is equivalent, at least for $t \in [-1,1]$, to Equation 7.17, which is equivalent to Equation 7.1.
The preceding discussion establishes well-posedness of Equation 7.1 on a $O(1)$-sized time interval for any initial data satisfying Equation 7.54 .
Finally, consider Equation 7.1 with $\lambda = \lambda _0$ fixed and $\phi \in H^s ( 0, \lambda _0 ),\nobreakspace s \geq - \frac{1}{2}$. This problem is well-posed on a small time interval $[0, \delta ]$ if and only if the $\sigma$-rescaled problem
provided $\sigma = \sigma ( \lambda _0 , {{\| \phi \|}_{H^{-\frac{1}{2}} (0 , \lambda _0 )}})$ is taken to be sufficiently large. This verifies Equation 7.54 for the problem Equation 7.55 proving well-posedness of Equation 7.55 on the time interval, say $[0, 1]$. Hence, Equation 7.1 is locally well-posed for $t \in [0, \sigma ^{-3} ]$.
The preceding discussion reproves the local well-posedness result for periodic KdV in Reference 32. We record the following simple variant which will be used in proving the global result for Equation 1.2. See Section 11 of Reference 17 for a general interpolation lemma related to this proposition.
Proposition 6.
If $s \geq - \frac{1}{2}$, the initial value problem Equation 1.2 is locally well-posed for data $\phi$ satisfying $I \phi \in L^2 ( {\mathbb{T}})$. Moreover, the solution exists on a time interval $[0, \delta ]$ with the lifetime
$$\begin{equation*} \delta \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}{{\| I \phi \|}_{L^2}^{-\alpha }}, \end{equation*}$$
and the solution satisfies the estimate
$$\begin{equation*} {{\| I u \|}_{Y^0}} \lesssim {{\| I \phi \|}_{L^2}}. \end{equation*}$$
8. Almost conservation and global well-posedness of KdV on ${\mathbb{T}}$
This section proves that the 1-periodic initial value problem Equation 7.1 for KdV is globally well-posed for initial data $\phi \in H^s ({\mathbb{T}})$ provided $s \geq -\frac{1}{2}$. In particular, we prove Theorem 2. The proof is an adaptation of the argument presented for the real line to the periodic setting.
8.1. Quintilinear estimate
The following quintilinear space-time estimate controls the increment of the modified energy $E^4_I$ during the lifetime of the local well-posedness result.
Lemma 8.1.
Let $w_i = w_i ( x,t)$ be $\lambda$-periodic function in $x$ also depending upon $t$. Let $\mathbb{P}$ denote the orthogonal projection onto mean zero functions, $\mathbb{P}u (x) = u(x) - \int _0^\lambda u(y) dy.$ Assume that $\int _0^\lambda w_i (x,t) dx = 0$ for all $t$. Then
The multilinear estimate Equation 8.3 is proved in the forthcoming paper Reference 17. Here we indicate the proof for the $k=3$ case of Equation 8.3, namely Equation 8.2, when $s \in (\frac{1}{2}, 1]$. The proof for $s=\frac{1}{2}$ in Reference 17 supplements the discussion presented below with some elementary number theory. The reader willing to accept Equation 8.2 may proceed to Lemma 8.2.
The Fourier transform of $\prod _{i=1}^3 {\widehat{u_i}}(x,t)$ equals
where $\int _*$ denotes an integration over the set where $k= k_1 + k_2 + k_3 ,\nobreakspace \tau = \tau _1 + \tau _2 + \tau _3$. We make a case-by-case analysis by decomposing the left side of Equation 8.2 into various regions. We may assume that $\widehat{u_i}\nobreakspace ,i = 1,2,3$, are nonnegative ${\mathbb{R}}$-valued functions.
Finally, using Sobolev again and the embedding $Y^s \subset L^\infty _t H^s_x$, we conclude that Equation 8.5 holds. Since $\langle \tau _j - k_j^3 \rangle \gtrsim \langle \tau - k^3 \rangle$ for $j = 2,3$ is symmetric with the Case 1 defining condition, we may assume that we are in Case 2.
Case 2.$\langle \tau _i - k_i^3 \rangle \ll \langle \tau - k^3 \rangle$ for $i = 1,2,3.$
The convolution constraints $k= k_1 + k_2 + k_3,\nobreakspace \tau = \tau _1 + \tau _2 + \tau _3$ in this case imply that
Symmetry allows us to assume $|k_1 | \geq |k_2 | \geq |k_3 |$ and we must have $|k| \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}|k_1 |$. Since $k= k_1 + k_2 + k_3$, we also have $|k^3 - (k_1^3 + k_2^3 + k_3^3 )| \lesssim |kk_1 k_2 |$ (see Equation 4.2). Therefore, in this case, the left side of Equation 8.7 is bounded by
We recall from Reference 5 that $X_{\delta , \frac{1}{2}} \subset L^6_{x,t}$ for any $\delta > 0$. Therefore, we validate Equation 8.8 using a Hölder application in $L^4_{x,t} L^6_{x,t} L^{12}_{x,t}$, with the required $L^{12}_{x,t}$ estimate given by Sobolev and the $L^4_{x,t}$ estimate from Equation 7.31.
Case 2B.$|k| \ll |k_i |$ for $i = 1,2,3.$
We bound $|k^3 - (k_1^3 + k_2^3 + k_3^3 ) | \lesssim |k_1 k_2 k_3 |$ in this region and control the left side of Equation 8.7 by
If $s \in (\frac{1}{2}, 1]$, we may ignore $\frac{1}{\langle k\rangle ^{1-s}}$ and finish things off with an $L^2_{x,t} L^6_{x,t} L^6_{x,t} L^6_{x,t}$ Hölder argument using $X_{\delta , \frac{1}{2}} \subset L^6_{x,t}$ for any $\delta >0$.
This completes the proof of Equation 8.2 for $s \in (\frac{1}{2}, 1]$.
■Remark 8.1.
Bourgain has conjectured Reference 5 that $X_{0 , \frac{1}{2}} \subset L^6_{x,t}$. If this estimate were known, the previous discussion could be substantially simplified. Our proof of the $s=\frac{1}{2}$ case in Reference 17 is partly motivated by an effort to prove this embedding estimate.
Lemma 8.2.
If $m$ is of the form Equation 4.7 with $s = - \frac{1}{2}$, then
The proof is a simple modification of the proof of Lemma 5.2 with Equation 8.1 playing the role of Equation 5.1. The projection appearing in the left side of Equation 8.1 causes no trouble in this application, since, as shown in Lemma 4.6, $M_5$ vanishes when $k_4 + k_5 = 0$ so it also vanishes when $k_1 + k_2 + k_3 = 0.$ Note also that $- \frac{3}{4}+$ and $\frac{1}{4}-$ are systematically replaced by $-\frac{1}{2}$ and $\frac{1}{2}$ throughout the argument.
8.2. Rescaling
Our task is to construct the solution of the 1-periodic Equation 7.1 on an arbitrary fixed time interval $[0,T]$. This is equivalent to showing the $\lambda$-rescaled problem with corresponding solution $u_\lambda (x,t) = \lambda ^{-2} u( \frac{x}{\lambda } , \frac{t}{\lambda ^3} )$ has a solution which exists on $[0, \lambda ^3 T]$. The lifetime of the variant local result is controlled by ${{\| I \phi \|}_{L^2}}$ and
Recall that $\lambda = \lambda (N)$ so we may ignore $\lambda ^{0+}$ by slightly adjusting $-\frac{5}{2}+$. Therefore, since by a (modification of) Equation 6.6$E^4_I (t) \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}{{\| I \phi _\lambda (t) \|}_{L^2}^2},$ we have that
$$\begin{equation*} {{\| I \phi _\lambda (1) \|}_{L^2}^2} = \epsilon _0 + C \epsilon _0^5 N^{-\frac{5}{2}+} + O (\epsilon _0^3 ). \end{equation*}$$
For small $\epsilon _0$ and large $N$ we see then that ${{\| I \phi _\lambda (1) \|}_{L^2}}$ is also of size $\epsilon _0$. We may iterate the local result $M$ times until, say, $E^4_I ( M )$ first exceeds $2 E^4_I (0)$, that is, until
$$\begin{equation*} M N^{-\frac{5}{2}+} \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}\epsilon _0 \implies M \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}N^{\frac{5}{2}-}. \end{equation*}$$
The solution of the $\lambda (N)$-periodicEquation 7.1 is thus extended to the interval $[0, N^{\frac{5}{2}-} ]$. We now choose $N = N(T)$ such that $N^{\frac{5}{2}-} > [\lambda (N)]^3 T \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}N^\frac{3}{2} T.$ This completes the proof that Equation 1.2 is globally well-posed in $H^{-\frac{1}{2}} ( {\mathbb{T}})$. Comments similar to those presented in Equation 6.19-Equation 6.26 apply to the periodic case showing that for our solution of Equation 1.2 we have
The results obtained for KdV are combined with some properties of the Miura transform Reference 42 (see also the survey Reference 43, Reference 44) to prove global well-posedness results for modified KdV (mKdV). This section contains the proofs of Theorems 3 and 4. The initial value problem for ${\mathbb{R}}$-valued mKdV on the line is
The choice of sign distinguishes between the focussing ($+$) and defocussing ($-$) cases. This problem is known Reference 31 to be locally well-posed in $H^s$ for $s \geq \frac{1}{4}$. The regularity requirement $s \geq \frac{1}{4}$ is sharp Reference 33, Reference 13. We establish global well-posedness of Equation 9.1 in the range $s > \frac{1}{4}$ improving the work of Fonseca, Linares and Ponce Reference 20.
9.1. Defocussing case
Consider the defocussing case of Equation 9.1. The Miura transform of a solution $u$ is the function $v$ defined by
$$\begin{equation} v = M[u]= \partial _x u + u^2. \cssId{defocusMiura}{\tag{9.2}} \end{equation}$$
A calculation shows that $v$ solves
$$\begin{equation} \left\{ \begin{matrix} \partial _t v + \partial _x^3 v - 6 v \partial _x v =0,& v: {\mathbb{R}}\times [0,T] \longmapsto {\mathbb{R}}, \\ v( 0) = v_0. \end{matrix} \right. \cssId{MiuraKdVivp}{\tag{9.3}} \end{equation}$$Remark 9.1.
Note that the uniqueness of the mKdV evolution is known Reference 31 in a subspace of $C([0,T], H^s),\nobreakspace s>\frac{1}{4}$, obtained by intersection in spaces associated with the maximal function and smoothing effect norms, while the KdV uniqueness holds in $X_{s-1, \frac{1}{2}+}$, a subspace of $C([0,T]; H^{s-1})$ which is not naturally identified within the image of $C([0,T]; H^s)$ under the Miura transform. Nevertheless, the Miura image of the mKdV evolution coincides with the KdV evolution from an element of the Miura image. Let $S_{mKdV}(t) (u_0)$ denote the nonlinear solution flow map for the defocussing initial value problem Equation 9.1 and let $S_{KdV} (t) (v_0)$ denote the flow map of Equation 9.3. For smooth enough $v_0$, we have the intertwining relationship
Since the KdV evolution is uniquely determined in $X_{s-1,\frac{1}{2}+}$ for data in $H^{s-1} ({\mathbb{R}})$, the next lemma provides the regularity to show the Miura image of the mKdV evolution from $H^s,\nobreakspace s>\frac{1}{4}$, data is the unique solution of KdV. A similar remark applies to the focussing case.
Suppose the initial data $u_0$ for Equation 9.1 is in $H^s,\nobreakspace \frac{1}{4} < s < 1$. We show that $v_0$ is in $H^{s-1}$.
Lemma 9.1.
If $u_0 \in H^s,\nobreakspace \frac{1}{4} < s < 1$, then $v_0 = \partial _x u_0 + u_0^2 \in H^{s-1}$.
The lemma verifies that the initial data $v_0$ for the KdV equation Equation 9.3 is in $H^{s-1}$ and $-\frac{3}{4} < {s-1}$ since $\frac{1}{4} < s$. Therefore, the global well-posedness result for KdV just established applies to Equation 9.3 and we know that the solution $v$ exists for all time and satisfies
for some constant $C$. We exploit this polynomial-in-time bound for KdV solutions to control ${{\left\| u(t) \right\|}_{{H^s}}}$ using the Miura transform.
Since our mKdV solution satisfies $L^2$-mass conservation, ${\left\| u(t) \right\|}_{L^ 2 } = {\left\| u_0 \right\|}_{L^ 2 }$, it suffices to control ${{\left\| \partial _x u(t) \right\|}_{{H^{s-1}}}}$ to control ${{\left\| u(t) \right\|}_{{H^s}}}$. By Equation 9.2,
Assuming the lemma for a moment, observe that combining Equation 9.7 and Equation 9.6 implies a polynomial-in-time upper bound on ${{\left\| u(t) \right\|}_{{H^s}}}$ giving global well-posedness of defocussing mKdV. We now turn to the proof of Equation 9.7.
Proof.
We first consider the case when $\frac{1}{2}+ \frac{1}{1000} < s < 1$. The Sobolev estimate in one dimension
$$\begin{equation} {\left\| w \right\|}_{L^ q } \lesssim {\left\| D^\sigma w \right\|}_{L^ p }, \qquad \frac{1}{q} = \frac{1}{p} - \frac{\sigma }{1} \cssId{GagNir}{\tag{9.8}} \end{equation}$$
is applied with $w = D^{s-1} (u^2),\nobreakspace \sigma = (1-s),\nobreakspace q=2$, yielding for
$$\begin{equation} \frac{1}{p} = \frac{3}{2} - s \cssId{pcondition}{\tag{9.9}} \end{equation}$$
We continue the estimate by writing ${{\| u^2 \|}_{L^p}} = {{\| u \|}_{L^{2p}}}$ and using Sobolev to get
$$\begin{equation} \leq {{\| u \|}_{L^{2p}}^2} \leq {{\| u \|}^2_{H^{\sigma (p)}}}, \qquad \sigma (p ) = \frac{1}{2}- \frac{1}{2p}. \cssId{continue}{\tag{9.10}} \end{equation}$$
Finally, we interpolate $H^{\sigma (p)}$ between $H^0 = L^2$ and $H^s$ to obtain
$$\begin{equation*} {{\| D^{s-1} (u^2) \|}_{L^2}} \lesssim {{\| u \|}_{L^2}^{2 (1-\theta )}} {{\| u \|}_{H^s}^{2 \theta }} \end{equation*}$$
where $\theta = \frac{1}{s} \sigma ( p )$. Using Equation 9.9 and Equation 9.10, we can simplify to find $\sigma (p) = \frac{2s -1}{4}$ and $2 \theta = 1- \frac{1}{2s}$. Since ${\left\| u \right\|}_{L^ 2 } \lesssim 1$, we observe that Equation 9.7 holds in case $\frac{1}{2}+ \frac{1}{1000} < s < 1.$
In case $\frac{1}{4} < s \leq \frac{1}{2}+ \frac{1}{1000}$, we begin with a crude step by writing
It is then clear that for $s \in (\frac{1}{4}, \frac{1}{2}+ \frac{1}{1000}]$, we have $2 \theta = 1 - \frac{1}{6s} = 1 - \epsilon$ for an appropriate $\epsilon > 0$ as claimed.
■
9.2. Focussing case
In the focussing case of ${\mathbb{R}}$-valued modified KdV, the Miura transform has a different form:
$$\begin{equation} v = \partial _x u + i u^2 . \cssId{focusMiura}{\tag{9.11}} \end{equation}$$
The function $v$ solves the complex KdV initial value problem
$$\begin{equation} \left\{ \begin{matrix} \partial _t v + \partial _x^3 v -i 6 v \partial _x v =0,& v: {\mathbb{R}}\times [0,T] \longmapsto {\mathbb{C}}, \\ v( 0) = v_0. \end{matrix} \right. \cssId{CKdVivp}{\tag{9.12}} \end{equation}$$
Since the solution $u(t)$ of focussing modified KdV is ${\mathbb{R}}$-valued and derivatives are more costly than squaring in one dimension, we take the perspective that $v$ is “nearly ${\mathbb{R}}$-valued”. The variant local result for Equation 9.12 has an existence interval determined by ${{(\int |I v_0 |^2 dx )}^\frac{1}{2}}$. However, Equation 9.12 does not conserve ${{(\int |I v (t) |^2 dx )}^\frac{1}{2}}$ but instead (almost) conserves ${{|\int (Iv(t))^2 dx |}^\frac{1}{2}}$. An iteration argument showing global well-posedness may proceed if we show that
The equivalence Equation 9.16 links the quantity determining the length of the local existence interval to an almost conserved quantity. Consequently, Equation 9.12 is GWP and ${{| \int (I v(t))^2 dx|}^\frac{1}{2}}$, is polynomially bounded in $t$. Since ${\left\| u(t) \right\|}_{L^ 2 } \leq C$ for solutions of focussing modified KdV, the equivalence Equation 9.16 implies ${{\| u(t) \|}_{H^s }}, s > \frac{1}{4}$, is polynomially bounded in $t$. Therefore, focussing mKdV is globally well-posed in $H^s ({\mathbb{R}}),\nobreakspace s> \frac{1}{4},$provided we prove the lemma above.
where $m( \xi ) \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}1$ when $|\xi | \lesssim N$ and $m( \xi ) \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}N^{-(s-1)} |\xi |^{s-1}$ when $|\xi | > N$ and we have relaxed to the bilinear situation. The function $f$ is introduced to calculate the norm using duality so ${\left\| f \right\|}_{L^ 2 } \leq 1$. We may assume that $\widehat{u_j}$ is nonnegative. Symmetry allows us to assume $|\xi _1 | \geq |\xi _2 |.$
Case 1.$|\xi _1 | \lesssim N \implies m( \xi _1 + \xi _2 ) \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}1.$
In this case, $I$ acts like the identity operator and the task is to control ${{\| u \|}_{L^4}^2}$. By Sobolev and interpolation,
$$\begin{equation*} {{\| u \|}_{L^4}^2} \leq {{\| u \|}_{L^2}} {{\| u \|}_{{\dot{H}}^\frac{1}{2}}} \leq {{\| u \|}_{L^2}^{\frac{3}{2}}} {{\| u \|}_{{\dot{H}}^1}^{\frac{1}{2}}} \end{equation*}$$
and we observe that, in this case,
$$\begin{equation*} {{\| I( u^2 ) \|}_{L^2}} \leq C + \epsilon {{\| I (\partial _x u) \|}_{L^2}}. \end{equation*}$$
In the Case 2A region, $m( \xi _1 ) \xi _1 \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}N_1^s N^{-(s-1)}$. We make this substitution and apply Cauchy-Schwarz in $\xi _2$ to observe
Multiplying through by $1 = \frac{N^{-(s-1)}}{N^{-(s-1)}}$ leads to
$$\begin{equation*} \begin{split} &\lesssim {\left\| f \right\|}_{L^ 2 } {\left\| \widehat{u_{N_2}} \right\|}_{L^ 2 } \frac{N^{s-1}}{N_1^{s-\frac{1}{2}}} {\left\| I \partial _x u_{N_1} \right\|}_{L^ 2 }\\ & \ll {\left\| I \partial _x u_{N_1} \right\|}_{L^ 2 },\nobreakspace {\text{provided}}\nobreakspace \frac{1}{2}< s < 1, \end{split} \end{equation*}$$
since $N_1 > N \gg 1.$ Of course we can sum over the dyadic scales and retain the claim.
It remains to establish the claim for the Case 2A region when $\frac{1}{4} < s \leq \frac{1}{2}$. We rewrite the expression Equation 9.17 differently as
and the prefactor vanishes as $N \rightarrow \infty$, proving the claimed estimate.
Case 2B.$|\xi _1 + \xi _2 | \gg N$.
We multiply Equation 9.17 through by $m(\xi _1) \xi _1$ in numerator and denominator. We observe that $\frac{m(\xi _1 + \xi _2 ) }{m( \xi _1 )} \lesssim 1$ and use the argument passing through $L^1_{\xi _1}$ in Case 2A to complete the proof.
■
9.3. Modified KdV on ${\mathbb{T}}$
Lemmas 9.2 and 9.3 naturally extend to the $\lambda$-periodic setting. These results link the polynomial-in-time upper bound Equation 8.10 for solutions of the $\lambda$-periodic initial value problem Equation 7.1 for KdV to a polynomial-in-time upper bound on ${{\| u(t) \|}_{H^\frac{1}{2}({\mathbb{T}})}}$ for solutions of the $\lambda$-periodic initial value problem for mKdV, implying Theorem 4.
If $m$ is even ${\mathbb{R}}$-valued and $m^2$ is controlled by itself, then, on the set $\xi _1 + \xi _2 + \xi _3 =0,\nobreakspace |\xi _i | \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}N_i$ (dyadic),
Assume $m$ is of the form Equation 4.7. In the region where $|\xi _ i | \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}N_i ,\nobreakspace |\xi _j + \xi _k | \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}N_{jk}$ for $N_i , N_{jk}$ dyadic,
If $s> - \frac{3}{4}$, the initial value problem Equation 1.1 is locally well-posed for data $\phi$ satisfying $I \phi \in L^2 ({\mathbb{R}})$. Moreover, the solution exists on a time interval $[0, \delta ]$ with the lifetime
$$\begin{equation} {{\| Iu \|}_{X^\delta _{0, \frac{1}{2}+}}} \lesssim {{\| I \phi \|}_{L^2}}. \cssId{spacetimebound}{\tag{6.2}} \end{equation}$$
Equation (6.3)
$$\begin{equation} {{\| D (uv)_x \|}_{X_{s, -\frac{1}{2}+}}} \lesssim {{\| D u \|}_{X_{s,\frac{1}{2}+}}} {{\| D v \|}_{X_{s,\frac{1}{2}+}}}. \cssId{nearlydone}{\tag{6.3}} \end{equation}$$
Lemma 6.1.
Let $I$ be defined with the multiplier $m$ of the form Equation 4.7 and $s = - \frac{3}{4}+.$ Then
$$\begin{equation} w(x,t) = S_\lambda (t) \phi (x) = \int e^{2 \pi i k x} e^{- {{(2 \pi i k)}^3} t} {\widehat{\phi }} (k) (dk )_\lambda . \cssId{Slam}{\tag{7.10}} \end{equation}$$
Equation (7.12)
$$\begin{equation} v(x,t) = \int \int e^{2 \pi i k x } e^{2 \pi i \tau t } {\widehat{v}} ( k , \tau ) (dk )_\lambda d\tau . \cssId{spacetimeinvFT}{\tag{7.12}} \end{equation}$$
The spatial mean $\int _{{\mathbb{T}}} u(x,t) dx$ is conserved during the evolution Equation 7.1. We may assume that the initial data $\phi$ satisfies a mean-zero assumption $\int _{{\mathbb{T}}} \phi (x) dx$ since otherwise we can replace the dependent variable $u$ by $v = u - \int _{{\mathbb{T}}} \phi$ at the expense of a harmless linear first order term. This observation was used by Bourgain in Reference 5. The mean-zero assumption is crucial for some of the analysis that follows.
Equation (7.19)
$$\begin{equation} {{\| u \|}_{Y^s}} = {{\left\| u \right\|}_{{X_{s,\frac{1}{2}}}}} + {{\|\langle k \rangle ^s \widehat{u} (k , \tau ) \|}_{ L^2 ( (dk)_\lambda ) L^1 (d \tau )}}. \cssId{Ys}{\tag{7.19}} \end{equation}$$
Let $\phi$ be a $\lambda$-periodic function whose Fourier transform is supported on $\{ k : |k| \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}N\}$. Then
If $v=v(x,t)$ is a $\lambda$-periodic function of $x$ and the spatial Fourier transform of $v$ is supported on $\{ k: |k| \mathrel{\mathchoice{\vcenter{\img[][8pt][3pt][{$\displaystyle\thicksim$}]{Images/imgbc36ca76378f648d5360a7e94ac1fd32.svg}}}{\vcenter{\img[][8pt][3pt][{$\textstyle\thicksim$}]{Images/img847c94f94e3604269ef1f666d7e9e1a5.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptstyle\thicksim$}]{Images/img0d69f157b0aaf28c0f5b0cd8bcd34bdf.svg}}}{\vcenter{\img[][6pt][3pt][{$\scriptscriptstyle\thicksim$}]{Images/imgcd6ef9b49f495ec0cf0d9803181a7ab9.svg}}}}N\}$, then
$$\begin{equation} {{\| \eta (t) v \|}_{L^4_{x,t}}} \lesssim C(N, \lambda ) {{\left\| v \right\|}_{{X_{0,\frac{1}{2}+}}}}, \cssId{lamBouStrichartz}{\tag{7.30}} \end{equation}$$
where $C(N, \lambda )$ is as it appears in Equation 7.27.
Lemma 7.3.
If $v=v(x,t)$ is a $\lambda$-periodic function of $x$, then
$$\begin{equation} {{\|\eta (t) v \|}_{L^4_{x,t}}} \lesssim {{\left\| v \right\|}_{{X_{0,\frac{1}{3}}}}} . \cssId{rescaledBourgain}{\tag{7.31}} \end{equation}$$
Proposition 5.
If $u$ and $v$ are $\lambda$-periodic functions of $x$, also depending upon $t$ having zero $x$-mean for all $t$, then
$$\begin{equation} {{\| \eta (t) \partial _x ( u v) \| }_{Z^{-\frac{1}{2}}}} \lesssim \lambda ^{0+} {{\| u \|}_{X_{-\frac{1}{2}, \frac{1}{2}}}} {{\| v \|}_{X_{-\frac{1}{2}, \frac{1}{2}}}}. \cssId{bilinearestimate}{\tag{7.33}} \end{equation}$$
If we change the setting of the lemma to the case where $k, k_1, k_2 \in {\mathbb{Z}}/ \lambda \backslash \{0 \}$, we have to adjust the conclusion to read
Let $w_i = w_i ( x,t)$ be $\lambda$-periodic function in $x$ also depending upon $t$. Let $\mathbb{P}$ denote the orthogonal projection onto mean zero functions, $\mathbb{P}u (x) = u(x) - \int _0^\lambda u(y) dy.$ Assume that $\int _0^\lambda w_i (x,t) dx = 0$ for all $t$. Then
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Department of Mathematics, University of Toronto, Toronto, ON Canada, M5S 3G3
M. Keel
School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455
G. Staffilani
Department of Mathematics, Stanford University, Stanford, California 94305-2125
Address at time of publication: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02138
Show rawAMSref\bib{1969209}{article}{
author={Colliander, J.},
author={Keel, M.},
author={Staffilani, G.},
author={Takaoka, H.},
author={Tao, T.},
title={Sharp global well-posedness for KdV and modified KdV on $\mathbb R$ and $\mathbb T$},
journal={J. Amer. Math. Soc.},
volume={16},
number={3},
date={2003-07},
pages={705-749},
issn={0894-0347},
review={1969209},
doi={10.1090/S0894-0347-03-00421-1},
}
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