A-priori estimates for generalized
Korteweg–de Vries equations in $H^{-1}(\mathbb{R})$

The Korteweg–de Vries equation

(where $u^{\prime}=\partial_{x}u$) was originally derived as a model for the propagation of waves in a shallow channel of water [Korteweg1895]. However, over the next century, (KdV) has proved to be a fundamental model for nonlinear dispersive waves, with applications to a wide variety of physical systems spanning many fields of science (see, for example, [Crighton1995]).

Mathematically, (KdV) has also played a central role in our understanding of dispersive equations. In particular, the (KdV) equation was the first discovered example of a completely integrable PDE, a feature that has been heavily exploited in some works. This discovery sparked a pursuit into the well-posedness and dispersive behavior for this system over the next 50 years, and generated a long list of cutting-edge techniques  [Bona1976, Kato1975, Saut1976, Temam1969, Tsutsumi1971, Kenig1991, Bourgain1993, Christ2003, Kenig1996, Colliander2003, Guo2009, Kishimoto2009, KT, MR4706572], some of which rely on complete integrability, and some of which do not. On the integrable side this effort culminated in global well-posedness for initial data in $H^{-1}$ on the line and the circle [Killip2019, Kappeler2006], a result that is sharp in the class of $H^{s}$ spaces for both geometries [Molinet2011, Molinet2012].

Although (KdV) lied at the center of this effort, it also served as a testing ground for the introduction of new tools and innovative techniques. After their introduction, many of these methods (both integrable and non-integrable) were later adapted to broader classes of dispersive equations. However, this important step has yet to be achieved for the completely integrable methods used to prove the sharp well-posedness results [Killip2019, Kappeler2006].

One physically important class of examples are the KdV equations with variable bottom. As (KdV) has proved to be an effective model for the propagation of waves in a shallow channel of water, many authors have introduced generalizations to describe an uneven bottom [Dingemans1997, Grimshaw1999, Groesen1993, Johnson1973, Miles1979, Pudjaprasetya1996, Pudjaprasetya1999, Yoon1994, Israwi2010, Lannes2013, Tian2001]. For concreteness, consider the model

which was derived and rigorously justified in [Israwi2010]. Here, $b(x)$ is defined in terms of a function $c(x)$ which describes the bottom of the channel:

When $c(x)$ is a small Schwartz function, the equation (gKdV) closely resembles (KdV). Nevertheless, our understanding of the well-posedness problem is comparatively lacking.

After a change of variables (see (5.15) below), the equation (1.1) can be put in the form

for certain coefficients $a_{j}$. In this work, we will assume that the coefficients $a_{j}(t,x)$ are given functions that are sufficiently regular and localized in space, and study the corresponding solutions $u:[-T,T]\times\mathbb{R}\to\mathbb{R}$.

Our main contribution is the following local-in-time a-priori estimate in $H^{-1}$:

We do not claim that the conditions (1.2)–(1.3) in this theorem are sharp. In fact, our proof will require slightly weaker hypotheses (see (4.1)–(4.4) below for details). The main thrust of this work is to match the sharp regularity of the well-posedness theory for (KdV), and so any hypotheses on the coefficients that work constitutes significant progress. In particular, (1.5) rules out strong forms of instantaneous norm inflation in $H^{-1}$.

Applying our general result to the equation (1.1), we obtain:

We contend that for each of the numerous applications of (KdV), our a-priori estimate (1.5) could also be employed to describe small physical imperfections using similar methods.

Another main way in which (gKdV) arises naturally is in the study of localized perturbations of a background solution to (KdV): if we take our solution $u=q+V$ to be a given background wave $V$ plus a perturbation $q$, then the equation for $q$ often takes the form (gKdV). For example, if $V(t,x)$ solves (KdV) then $q$ solves

and if $V(x)$ is a fixed background profile then a forcing term $-V^{\prime\prime\prime}+6VV^{\prime}$ is added to the RHS above. Through this lens, the special case (1.6) of gKdV equations have been of great interest in the literature.

The first phase of results for (1.6) addressed the construction of solutions using the inverse scattering transform. Two particularly common choices for $V$ are profiles that are periodic [Kuznetsov1974, Ermakova1982, Ermakova1982a, Firsova1988], which describe localized defects to periodic wave trains, and step-like [Buslaev1962, Cohen1984, Cohen1987, Kappeler1986], which arise in the study of bore propagation and rarefaction waves. Later, the authors of [Egorova2009, Egorova2009a] established a general framework that encompasses both of these cases, and even a mixture of the two as $x\to+\infty$ and $x\to-\infty$. Although the main thrust is to prove existence, a key step in many of these works is establishing persistence of regularity for solutions, much like (1.5).

While many of these results work at high regularity, these methods for existence have been adapted to classes of one-sided step-like initial data [Grudsky2014, Rybkin2011, Rybkin2018] and even to one-sided step-like elements of $H^{-1}_{\text{loc}}(\mathbb{R})$ [Grudsky2015]. Despite the lack of assumptions as $x\to-\infty$ (the direction in which radiation propagates), these low-regularity results require rapid decay at $x\to+\infty$ and global boundedness from below.

An alternative approach was introduced in [Menikoff1972], which proves global existence and uniqueness for initial data that satisfies $u(0,x)=o(|x|)$ as $x\to\pm\infty$. Here, the background wave $V$ is evolved according to the inviscid Burger’s equation $\partial_{t}V=-6VV^{\prime}$ using the method of characteristics, and then solutions to the equation for $q$ are constructed using a family of discretized approximate equations. Existence and uniqueness for $q$ is then established in a weighted $H^{3}$ space.

Inspired by the theory of tidal bores and rarefaction waves (as well as kink solutions to KdV with higher-power nonlinearities), authors began turning their attention to the well-posedness problem for step-like initial data. The first phase of results employed BBM and parabolic regularizations [Bona1994, Iorio1998, Zhidkov2001], which involve adding a term to the LHS of (1.6) that eases the proof of well-posedness ($-\varepsilon\partial_{t}\partial^{2}_{x}q$ and $-\varepsilon\partial^{2}_{x}q$ for $\varepsilon>0$, respectively) and then sending $\varepsilon\to 0$. The former was introduced by Bona and Smith [Bona1975] in the case $V\equiv 0$ and leads to the Benjamin–Bona–Mahony equation, for which the method is named. These approaches culminated in local well-posedness for initial data $q\in H^{s}$ for $s>\frac{3}{2}$, and was later advanced to $s>1$ in [Gallo2005] through the incorporation of Strichartz estimates.

Recently, the work [Palacios2023] extends local well-posedness to the range $s>\frac{1}{2}$ using a synthesis of much more modern tools for well-posedness. In addition to the cases of step-like and periodic background waves $V$, this result also applies to more general choices of $V$ with bounded asymptotics as $x\to\pm\infty$, and even more general nonlinearities in (KdV). Nevertheless, it only applies to (gKdV) in the special case where $a_{1}\equiv 0$ and $a_{2}$ is constant.

Around the same time, the second author proved that (KdV) is globally well-posed for initial perturbations $q\in H^{-1}$ in [Laurens2023], provided that the background wave $V$ is a suitable solution to (KdV). These conditions on $V$ do include regularity but do not impose any assumptions on spatial asymptotics. In particular, this result applies to the important cases of smooth periodic [Laurens2023] and step-like [Laurens2022] initial data.

All of these works for (1.6) focus on the coefficients $a_{3}$ and $a_{4}$ in (gKdV). The introduction of the coefficient $a_{1}$ already significantly changes the analysis. The local well-posedness of this equation for initial data in $H^{s}$ with $s>\frac{3}{2}$ and suitable coefficients was demonstrated in [Israwi2013] using energy methods. Recently, local well-posedness was extended to the range $s>\frac{1}{2}$ in [Molinet2023].

In the most general case, the optimal well-posedness results that cover the $a_{2}$ term in (gKdV) are [Craig1992, Akhunov2019, Ben] to the best of our knowledge. These works establish well-posedness for initial data at a considerably high regularity, but they apply to very general families of nonlinear equations with a KdV-type dispersion.

Let us now turn to our methods for proving the priori estimate (1.5). At the center of our analysis lies the renormalized perturbation determinant $\alpha(\kappa,u)$ (see (2.9) below for details). The perturbation determinant is a conserved quantity for (KdV) originating from scattering theory, and is rooted in the complete integrability of this system. However, the authors of [Killip2018] (and independently [Rybkin2010]) discovered a renormalization $\alpha$ of this quantity that satisfies

and is still conserved, and used this to prove a global-in-time a-priori estimate for solutions to (KdV) in $H^{s}$ spaces for $-1\leq s<1$. At the same time, conserved energies in $H^{s}$ spaces were independently constructed in [KT] for a full range of Sobolev exponents $s\geq-1$. As it turned out, these energies are also connected to the perturbation determinant, but with a different choice of renormalization. See also the earlier work [B] where similar bounds were proved at slightly higher regularity $s>-\frac{4}{5}$.

The same quantity $\alpha$ was then used as a starting point in [Killip2019] to prove that (KdV) is well-posed in $H^{-1}$ using the method of commuting flows.

Of course, introducing the coefficients $a_{j}$ in (gKdV) breaks the conservation of $\alpha$, along with all of the other conservation laws of (KdV). In general, the value of $\alpha$ can grow in time, and consequently we cannot expect a global-in-time estimate. Instead, in order to establish (1.5), we prove that $\alpha$ must remain finite for a short period of time using a bootstrap/continuity argument.

However, we immediately encounter an obstacle. As an illustrative example, consider the $L^{2}$ norm; if $u$ is a Schwartz solution of (KdV), then the $L^{2}$ norm of $u(t)$ is constant in time. On the other hand, if $u$ solves (gKdV), then the $L^{2}$ norm evolves according to

How can we bound the right-hand side solely in terms of $\left\lVert u\right\rVert_{L^{2}}$ in order to close the argument?

Our strategy in this paper is to use local smoothing: due to the dispersive nature of the equation, we expect a gain in the regularity of solutions locally in space on average in time. For (KdV), this effect was discovered by Kato [Kato1983], who proved that $L^{2}$ solutions are in fact in $H^{1}$ locally in space at almost every time. In particular, this gives us a way to make sense of the RHS of (1.8) even for $L^{2}$ solutions $u$.

For the a-priori estimate (1.5), we encounter an analogous loss of derivatives phenomenon in computing the time evolution of $\alpha$. The local smoothing effect attendant to this problem is to show that an $H^{-1}$ solution of (gKdV) is in $L^{2}$ locally in space and time:

We will prove (1.5) and (1.10) simultaneously, using a bootstrap argument in terms of both $\alpha$ and a local smoothing norm.

In the case where the coefficients $a_{j}\equiv 0$ vanish, an analogous local smoothing estimate was first proved in [Buckmaster2015]; see also the earlier work in [B], where similar local smoothing estimates are proved for $H^{s}$ solutions with $s>-\frac{4}{5}$. However, our proof is more closely related to the alternative approach in [Killip2019]*Th. 1.2 using the renormalized perturbation determinant. This argument is made possible by the discovery of a density $\rho(x;\kappa,u)$ for $\alpha$ (see (2.9) below) not known during the earlier work [Killip2018]. Moreover, if $u$ solves (KdV), then $\rho$ satisfies the density-flux equation (or ‘microscopic conservation law’)

for a certain current $j(x;\kappa,u)$ (see (2.12)). Integrating this equation in space yields the conservation of $\alpha$. Alternatively, first multiplying by a smooth step function and then integrating in space leads to a local smoothing estimate; this is basis of the short proof presented in [Killip2019].

Turning our attention back to (gKdV), we again encounter obstacles. The presence of the coefficients $a_{j}$ breaks the conservation law (1.11), and instead we have

(Here, $d\rho|_{u}$ denotes the functional derivative of $\rho$ at $u$; see (2.14) for details.) All of terms on the RHS of (1.12) are new, and must that must be controlled in terms of $\alpha$ and our local smoothing norm.

Many of our estimates are in the spirit of [Bringmann2021], which proves that the fifth-order KdV equation is globally well-posed in $H^{-1}$. In order to prove their result, the authors had to prove an analogous local smoothing estimate for $H^{-1}$ solutions of their system and a family of approximate equations. Together, the estimates in [Killip2019, Bringmann2021] provide upper bounds on the density $\rho$ and current $j$ in terms of the local smoothing and $H^{-1}$ norms, the latter of which can then be controlled by $\alpha$ using (1.7). For example, when $\kappa\geq 1$ we may bound

and use this to construct $\alpha$ for all $\left\lVert u\right\rVert^{2}_{H^{-1}}\ll 1$. For the (KdV) and fifth-order KdV equations, this small-data assumption does not pose a problem; one can use the scaling symmetry to make the initial data arbitrarily small, and then the exact conservation of $\alpha$ implies that solutions remain small globally in time.

By comparison, (gKdV) does not posses a scaling symmetry in general, and we expect that the $H^{-1}$ norm of solutions can be growing in time without bound. This in turn forces us to choose $\kappa\gg\left\lVert u\right\rVert^{2}_{H^{-1}}$ large, which introduces a large implicit constant in (1.13). Consequently, the estimates in [Killip2019, Bringmann2021] are insufficient, even in estimating the LHS of (1.12). In order to close our bootstrap argument, we must establish new estimates for the density $\rho$ and current $j$ using the $H^{-1}_{\kappa}$ norm, rather than $H^{-1}$, as this is the quantity whose growth we can efficiently estimate using (1.7). In comparison to the estimate (1.13) used throughout [Killip2019, Bringmann2021], this requires a much more careful estimation of high frequency contributions.

This paper is organized as follows. We begin in Section 2 by reviewing the material developed in [Killip2019, Bringmann2021] that we will need, including the key players $\alpha$, $\rho$, and $j$ mentioned previously as well as the estimates they satisfy. We then proceed in Section 3 by introducing our local smoothing norm and developing the new estimates required for our bootstrap argument.

We employ these ingredients in Sections 4 and 5 to prove more precise versions of our local smoothing and a-priori estimates, Theorem 4.1 and Proposition 5.1, respectively. We then combine these two estimates with a bootstrap argument, and present the full result in Theorem 5.2. Finally, we conclude Section 5 by describing how the statements of Theorems 1.1 and 1.5 and Corollary 1.4 represent special cases of Theorem 5.2.

Here we recall some of the key objects from [Killip2019] which are used in order to prove that (KdV) is well-posed in $H^{-1}$, and which will serve us well in obtaining the a-priori bounds for our nonintegrable model (gKdV) displayed in Theorem 1.1 and Theorem 1.5. Our starting point is represented by the self-adjoint Lax operator associated to the KdV flow,

Formally, the spectrum of $L_{u}$ is conserved along the KdV flow. More precisely, $L_{u(t)}$ and $L_{u(s)}$ are unitarily equivalent operators. In general $L_{u}$ is not invertible, which is one reason it is convenient to replace it with $L_{u}+\kappa^{2}$; given $u\in H^{-1}$ the operator $L_{u}+\kappa^{2}$ is invertible if $\kappa>0$ is large enough. One may construct conserved quantities for KdV by looking at the trace of functions of $L_{u}+\kappa^{2}$, and in particular the trace of its renormalized logarithm, which can be also viewed as a function of $\kappa$. As shown in [Killip2019], this trace is closely related to the diagonal of the kernel of $L_{u}+\kappa^{2}$.

The first step in the analysis is to construct the diagonal of the integral kernel $G(x,y)$ for the resolvent

of the Lax operator associated to a state $u$. With this operator in mind, it will be convenient to work with the norms

where our convention for the Fourier transform is

In the definition of $H^{s}_{\kappa}$, the constant $4$ is simply included in order to make (2.4) an equality. One should see these spaces as versions of the traditional inhomogeneous Sobolev spaces $H^{s}$ but adapted to the frequency scale $\kappa$ rather than the unit frequency. Topologically these are the same as the classical Sobolev spaces $H^{s}$, but with the implicit constants in the norm equivalence depending on $\kappa$. In particular, we have the elementary estimates

uniformly for $\kappa\geq 1$. Notice that the results for $H^{-1}_{\kappa}$ follow from those for $H^{1}_{\kappa}$ by duality. We will also frequently use the embedding

which implies the algebra bound

The resolvent $R_{0}$ associated to $u\equiv 0$,

plays an important role in what follows, and in particular has the mapping property

This property is also shared by $R_{u}$ for $u\in H^{-1}$ for $\kappa$ large enough, but only for restricted range of $s$. One can formally use $R_{0}$ to obtain an expansion for $R_{u}$:

which is justified for $u\in H^{-1}$ for $\kappa$ large enough.

Our first bound involving $R_{0}$ is as follows:

Here, $\mathfrak{I}_{2}$ denotes the class of Hilbert–Schmidt operators on $L^{2}(\mathbb{R})$. Such operators are automatically continuous due to the inequality

where $\mathfrak{I}_{\infty}$ denotes the operator norm. Moreover, the space of such operators forms a two-sided ideal in the space of bounded operators:

Lastly, the product of two Hilbert–Schmidt operators is trace class, and we have

Here, $\mathfrak{I}_{1}$ denotes the trace class; this consists of operators whose singular values are $\ell^{1}$-summable. In this way, the second inequality above is simply an application of the Cauchy–Schwarz inequality.

The Green’s function is the integral kernel for the resolvent of the Lax operator $R_{u}$, which may be expressed as

As explained earlier, its restriction to the diagonal plays a key role in our analysis.

As a consequence of the expansion of $R_{u}$ in (2.3), for $g$ we have a similar expansion

In particular, using the integral kernel $\langle\delta_{x},R_{0}(\kappa)\delta_{y}\rangle=\tfrac{1}{2\kappa}e^{-\kappa|x-y|}$ for the free resolvent, we find that

As it turns out, it is the function $1/g$ that is linearly related to the logarithmic renormalized trace of the resolvent. A consequence of the proposition is that the integral of $1/g$ diverges. Even after removing the constant, its integral still turns out to diverge for $u$ in any $H^{s}$ space. To rectify this, one also needs to remove the linear term in $u$, which formally integrates to a multiple of $\int u\,dx$; this is a conserved quantity for the KdV flow, and thus harmless. A similar renormalization was also implemented in [KT] for the logarithm of the transmission coefficient. Taking these renormalizations into account, we can define the conserved quantity which controls the $H^{-1}$ norm of $u$:

The formula for $\alpha$ is the trace of the integral kernel $-1/2G(x,y;\kappa,u)$ with the first two terms of its Taylor series about $u\equiv 0$ removed.

The authors of [Killip2019] also proved that under the (KdV) flow, we have the following density-flux equation

where $\rho=\rho(x;\kappa,u)$ is defined in (2.9) and

We see in particular by integrating (2.11) in space that $\alpha$ is conserved under the (KdV) flow.

By comparison, the coefficients $a_{j}$ in (gKdV) break the density-flux equation (2.11). Instead, we obtain an additional term as follows:

where

is the functional derivative of $\rho$ at $u$, and

The identity (2.13) lies at the heart of both of the proofs of our a-priori and local smoothing estimates.

In order to leverage local smoothing, we will frequently need to commute slowly varying weights past $R_{0}$, as in the following estimates.

The proof is elementary; see, for example, [Bringmann2021]*Lem. 2.7.

We will use this lemma in particular for the functions $\psi^{\ell}$, $\ell=1,2,3$, where

The constant 6 is simply included to ensure that $\psi^{\ell}$ satisfies (2.16) for $\ell=1,2,3$. (In turn, the constant 2 in (2.16) is based on the integral kernel $\langle\delta_{x},R_{0}\delta_{y}\rangle=\tfrac{1}{2\kappa}e^{-\kappa|x-y|}$ and the condition $\kappa\geq 1$.)

Over the course of our analysis, we will also encounter the ‘double commutator’

In order to bound this operator, the naive estimate (2.17) unfortunately does not yield enough decay as $\kappa\to\infty$. To this end, we record the following operator estimates, which capture the double commutator structure:

In comparison to (2.17), the proofs of (2.18) and (2.19) exhibit extra cancellation by computing the double commutator in a symmetric way. For example, (2.18) follows from [Bringmann2021]*Lem. 2.10, and (2.19) can be verified using a similar argument.

Here, we will record some useful estimates for the following local smoothing norm:

where $T>0$ and

We begin with the following elementary observations:

The operator estimate (3.7) below will play an important role in our analysis. In comparison to (2.4), we now bound the operator norm (rather than Hilbert–Schmidt) and only require a local norm of $u$ to do so.

Our next lemma provides an estimate for the diagonal Green’s function analogous to (2.6) that captures the gain of regularity we expect from our local smoothing norm.