# Regularity criteria for the Kuramoto-Sivashinsky equation in dimensions two
and three

Adam Larios  Department of Mathematics, University of Nebraska–Lincoln,
Lincoln, NE 68588-0130, USA<EMAIL_ADDRESS>,  Mohammad Mahabubur Rahman
Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX,
79409, USA<EMAIL_ADDRESS>and  Kazuo Yamazaki  Department of
Mathematics and Statistics, Texas Tech University, Lubbock, TX, 79409, USA
<EMAIL_ADDRESS>
###### Abstract.

We propose and prove several regularity criteria for the 2D and 3D Kuramoto-
Sivashinsky equation, in both its scalar and vector forms. In particular, we
examine integrability criteria for the regularity of solutions in terms of the
scalar solution $\phi$, the vector solution $u\triangleq\nabla\phi$, as well
as the divergence $\text{div}(u)=\Delta\phi$, and each component of $u$ and
$\nabla u$. We also investigate these criteria computationally in the 2D case,
and we include snapshots of solutions for several quantities of interest that
arise in energy estimates.

###### Key words and phrases:

(Kuramoto-Sivashinsky, Navier-Stokes Equations, Regularity, Global Well-
Posedness.)

MSC 2010 Classification: 35A01, 35K25, 35K51, 35K58, 35B65, 35B10, 65M70.
*Corresponding author

##  1\. Introduction

For those with an interest in nonlinear PDEs, the two-dimensional (2D)
Kuramoto-Sivashinsky equation (KSE) is full of tantalizing possibilities.
There is such strong diffusion in the equation that the nonlinear term seems
incapable of overpowering it, and yet, it is precisely the structure of the
nonlinear term that has so far thwarted all efforts at a general proof of
global well-posedness. It is sometimes thought that this is due to the
backward diffusion term in the equations, but this term serves mainly as the
source of energy and instability, fueling a kind of self-perpetuating chaos.
Instead, the major difficulty is due to the fact that the nonlinear term does
not vanish in any known energy estimates, similar to the vorticity stretching
term in the Navier-Stokes equations (NSE), where an analogous barrier arises.
This paper aims to quantify these difficulties by proving several
integrability-type component based regularity criteria for well-posedness, and
also to shed light on well-posedness issues via numerical simulations.

For those with a more practical eye, the KSE enjoys a wealth of applications.
Originally proposed in the late 1970’s by Kuramoto, Sivashinsky, and Tsuzuki
in investigations of crystal growth [46, 47] and flame-front instabilities
[65] (see also [66]), it has since made many appearances, such as in studies
of inclined planes [67], and has even been shown to be a general feature of
certain unstable behaviors [52]. Although we consider both 2D and 3D cases, we
note that, at least in certain situations such as flame-front propagation, the
2D case is more relevant, as it describes the evolving 2D front of a 3D flame.
However, the 3D case is of interest in terms of making analogies with the 3D
NSE.

For the $N$D case, short-time existence of smooth solutions (specifically,
Gevrey class regularity) was proved in [6]. The triviality of steady states
was studied in [12]. Variations of the 2D KSE have been studied in, e.g., [2,
34, 18, 26, 38, 50, 72], as well as variations on its boundary conditions [30,
49, 57]. The question of the global well-posedness of KSE for $N\geq 2$ in the
periodic case or $\mathbb{R}^{N}$ is still open in general; however, in
dimensions $N=2$ and $3$ for the case of radially symmetric initial data in an
annular domain, global well-posedness was proved in [4], assuming homogeneous
Neumann boundary conditions. It was shown in [1] (see also [2, 18, 26]) that,
in the case where there are no linearly growing modes, global existence holds
for sufficiently small initial data in a certain function space based on the
Wiener algebra. We also mention [61], which studied global existence and
attractors in 2D thin domains (see also [34, 5]).

The two major difficulties in the $N\geq 2$ case are (i) the fact that the
nonlinear term does not vanish in energy estimates (since the solution is not
divergence-free), and hence no $L^{p}$ norm is conserved, and (ii) the fact
that, due to the 4th-order derivatives, no maximum principle is known to hold.
However, we note an interesting recent result in [36] that proves global
regularity (without any smallness condition) for a modified version of the
Michelson-Sivashinsky equation. While lower-order than KSE, this equation
shares many similarities with KSE in that it also does not conserve $L^{p}$
norms, has no known maximum principle, and has no divergence-free condition.

In contrast to the 2D KSE, the 1D KSE is a seemingly limitless playground,
where one has essentially everything one could want. Well-posedness is
straight-forward [53, 69], the large-time dynamics are chaotic (unlike in the
case of the 1D Burgers’ equation, where the large-time dynamics are trivial)
but finite-dimensional [14, 54, 56], and much work on quantities of interest,
such as the existence of the attractor and estimates on its dimension, have
seen excellent progress in recent decades, see, e.g. [14, 13, 17, 16, 27, 28,
29, 31, 32, 33, 35, 37, 55, 60, 69, 70, 43, 73] and the references therein.

Faced with an obstacle in proving global well-posedness in case $N=2$ or 3, a
natural strategy is to investigate its criterion following analogous works for
the 3D NSE such as (2.3) from [25, 42, 59, 62]. In this endeavor, we first
obtained criterion that seemed to be natural extensions from those of the NSE
(see (2.11) and Remark 2.3). To our surprise, subsequently we were able to
improve such results significantly (see Theorem 3.1 and Remark 3.2). This
motivated us to pursue another direction of research that has caught much
attention in the past few decades, specifically component reduction of such
classical regularity criterion (see (2.12)). Despite a large amount of work
dedicated to such results on various systems including the NSE,
magnetohydrodynamics (MHD) system and surface quasi-geostrophic (SQG)
equations, divergence-free property of velocity field was crucial in all of
their results; consequently, we are not aware of any example of a PDE that
does not involve divergence-free velocity field and yet admit component
reduction results. Unexpectedly, we were able to obtain such results by making
use of special structure of the KSE, which seems to be a unique property that
is absent in the NSE or Burgers’ equation (see Remark 3.5).

Now let us write $\partial_{t}\triangleq\frac{\partial}{\partial t}$ and
introduce several equations of our main concern. First, the $N$D KSE in vector
form is given by

(1.1) $\displaystyle\partial_{t}u+(u\cdot\nabla)u+\lambda\Delta
u+\Delta^{2}u=0,$

with appropriate boundary conditions (here taken to be periodic) and initial
data $u^{in}$. Here, $\lambda>0$ is a constant. We note that one often
considers the KSE with $\lambda=1$ and a domain with size determined by a
parameter $L>0$, such as $[-\tfrac{L}{2},\tfrac{L}{2}]^{N}$. We choose to work
with a fixed domain $\mathbb{T}^{N}\triangleq[-\pi,\pi]^{N}$ and a general
parameter $\lambda$; these formulations are equivalent under the rescaling
$u^{\prime}(x,t)\mapsto(L/2\pi)^{-3}u(\tfrac{x}{L/2\pi},\tfrac{t}{(L/2\pi)^{4}})$
after setting $\lambda=(L/2\pi)^{2}$.

The scalar form of (1.1) is formally given by

(1.2)
$\displaystyle\partial_{t}\phi+\tfrac{1}{2}|\nabla\phi|^{2}+\lambda\Delta\phi+\Delta^{2}\phi$
$\displaystyle=0,$

By setting $u=\nabla\phi$ in (1.2), one formally obtains (1.1). Our analytical
results throughout this manuscript go through independently of the fact that
$u=\nabla\phi$; i.e., we will obtain results on (1.1) and (1.2) separately and
independently of each other. For comparison, let us recall the NSE:

(1.3) $\partial_{t}u+(u\cdot\nabla)u+\nabla\Pi=\Delta u,\hskip
8.53581pt\nabla\cdot u=0,$

where $\Pi$ represents the pressure. Due to the condition $\nabla\cdot u=0$,
an $L^{2}(\mathbb{T}^{N})$-inner product of $(u\cdot\nabla)u$ with $u$
vanishes for the NSE, which is crucial in energy estimates. However, this does
not happen for the KSE, nor for the $N$D Burgers’ equation

(1.4) $\partial_{t}u+(u\cdot\nabla)u=\Delta u$

with $N\geq 2$, although Burgers’ equation enjoys a maximum principle allowing
for a proof of global well-posedness; see, e.g., [68, 58] for details.
Currently, no such maximum principle is known to hold for the KSE, and hence,
global well-posedness for arbitrary smooth initial data remains an open
problem. The case $\Pi\equiv 0$ and $\nabla\cdot u=0$ condition being dropped
reduces the NSE to Burgers’ equation.

######  Remark 1.1.

A major interest of ours is the Navier-Stokes equations (1.3), where, not
counting the pressure, the nonlinearity is (at least notationally) the same as
in equation (1.1). Hence, our we aim to prove as much as possible without
using the assumption that $u=\nabla\phi$. However, all of our results can be
translated to results about equation (1.2) by thinking of $u$ as merely a
notation for $\nabla\phi$; for instance, $u_{1}=\partial_{1}\phi$,
$\nabla\cdot u=\Delta u$, and so on.

Note that if one does not assume that $u=\nabla\phi$, then the vector and
scalar forms of the KSE might not be equivalent. For instance, if $\phi$ is a
given smooth solution to (1.2) with initial data $\phi^{in}$, then
$u\triangleq\nabla\phi$ automatically solves (1.1) with initial data
$u^{in}\triangleq\nabla\phi^{in}$. However, given a solution $u$ to the vector
form (1.1), it may not be possible to find a solution $\phi$ to the scalar
form such that $u=\nabla\phi$, since $u$ might not be a pure gradient; for
instance, when the initial data $u_{0}$ of (1.1) is not a pure gradient.

Figure 1.1.  (Log-linear plot) The growth of the $L^{2}$-norm of the
divergence-free part $v=P_{\sigma}u$ of a vector solution $u$ of (1.1)
($\lambda=8.1$; i.e., 24 unstable modes), with initial data
$u^{in}=\nabla\phi^{in}$, where $\phi^{in}$ is given by (4.1). Since
$P_{\sigma}u^{in}\equiv 0$, one expects that $P_{\sigma}u(t)\approx 0$ for
$t>0$, but this is not what is observed computationally.

Indeed, denoting the Helmholtz-Hodge decomposition of a solution $u$ to (1.1)
by $u=\nabla q+v$ where $\nabla\cdot v=0$, a straight-forward calculation
shows, at least formally, that  (1.5)
$\displaystyle\frac{1}{2}\frac{d}{dt}\|v\|_{L^{2}}^{2}+\|\Delta
v\|_{L^{2}}^{2}$ $\displaystyle=$ $\displaystyle\lambda\|\nabla
v\|_{L^{2}}^{2}-2(\nabla q\cdot S,v),$

where $S\triangleq\tfrac{1}{2}(\nabla v-(\nabla v)^{T})$ is the anti-symmetric
part of the gradient of $v$. If $(\nabla q\cdot S,v)\not\equiv 0$, there is
potential for the gradient part of $u$ to “push” the solution off of the
gradient manifold. Moreover, even if one could show analytically111We denote
the Leray-Helmholtz projection $P_{\sigma}u=v$. See, e.g., [15, 71] for
details. that $P_{\sigma}u=0$ is preserved by (1.1), in computations, small
errors could be amplified due to the destabilizing term in (1.1), and the non-
gradient part of the solution could grow rapidly. To test this
computationally, we simulated the vector equation (1.1) starting from a pure-
gradient initial condition. The results are shown in Figure 1.1, where we
observe that the deviation from the gradient manifold grows exponentially fast
in time (see Figure 1.1).

Therefore, we choose to study both (1.1) and (1.2) independently, although if
one assumes that $u=\nabla\phi$, then all the results we obtain for (1.1),
specifically Theorems 3.1, 3.6, 3.10, 3.12, 3.13, and 3.16, immediately imply
equivalent results for $\phi$ that solves (1.2). Informally, we give a
comprehensive list our results of regularity criteria here for clarity:

  1. (1)

in terms of $u$ in 2D and 3D (Theorem 3.1);

  2. (2)

in terms of $u_{1}$ in 2D (Theorem 3.6);

  3. (3)

in terms of $u_{1},u_{2}$ in 3D (Theorem 3.10);

  4. (4)

in terms of $\nabla u$ in 2D and 3D (Theorem 3.12);

  5. (5)

in terms of $\nabla\cdot u$ in 2D and 3D (Theorem 3.13);

  6. (6)

in terms of $\partial_{2}u_{2}$ in 2D (Theorem 3.16);

  7. (7)

in terms of $\phi$ in 2D and 3D (Theorem 3.18);

  8. (8)

in terms of $\partial_{12}\phi$ in 2D (Theorem 3.20).

######  Remark 1.2.

We note that the proofs of the results on individual components are non-
standard, and do not follow as in, e.g., the Navier-Stokes case; a major
difference being the loss of the divergence-free condition, which is what
makes the multi-dimensional KSE difficult in the first place.

This paper is organized as follows. In Section 2, we lay out notation and
mention some preliminary results. In Section 3, we state and prove our main
results. In Section 4, we present and discuss our computational results.

##  2\. Preliminaries

For simplicity we write $\partial_{j}\triangleq\frac{\partial}{\partial
x_{j}}$ for $j\in\\{1,\ldots,N\\}$ and $\int
F\triangleq\int_{\mathbb{T}^{N}}F(x)dx$; for further brevity we may write
$f_{x}\triangleq\partial_{x}f$, etc., when $f$ is a scalar field. We write
$A\lesssim_{a,b}B$ when there exists a constant $C=C(a,b)\geq 0$ such that
$A\leq CB$. We recall that we may write

$f(x)=\sum_{k\in\mathbb{Z}^{N}}\hat{f}(k)e^{ik\cdot x},$

and the inhomogeneous and homogeneous Sobolev norms

$\lVert f\rVert_{H^{s}}\triangleq\Big{(}\sum_{k\in\mathbb{Z}^{N}}(1+\lvert
k\rvert^{2s})\lvert\hat{f}(k)\rvert^{2}\Big{)}^{\frac{1}{2}},\hskip
8.53581pt\lVert
f\rVert_{\dot{H}^{s}}\triangleq\Big{(}\sum_{k\in\mathbb{Z}^{N}}\lvert
k\rvert^{2s}\lvert\hat{f}(k)\rvert^{2}\Big{)}^{\frac{1}{2}},$

respectively. We denote by $(\cdot,\cdot)$ the standard
$L^{2}(\mathbb{T}^{N})$ inner-product, and by $\left<\cdot,\cdot\right>$ the
$H^{-1}$-$H^{1}$ duality paring.

Let us formally write down the definition of a strong solution to the KSE
equation (1.1).

######  Definition 2.1.

We call $u$ a strong solution to the KSE (1.1) over a time interval $[0,T]$ if
for any $\psi\in C^{\infty}(\mathbb{T}^{N})$,

(2.1) $\left<\partial_{t}u,\psi\right>+(u\cdot\nabla u,\psi)+(\Delta
u,\Delta\psi)=\lambda(\nabla u,\nabla\psi)$

for almost all $t\in[0,T]$, and

(2.2a) $\displaystyle u\in L^{\infty}([0,T];H^{1}(\mathbb{T}^{N})),u\in
L^{2}([0,T];H^{3}(\mathbb{T}^{N})),$ (2.2b) $\displaystyle u\in
C([0,T];H^{s}(\mathbb{T}^{N}))\hskip 2.84526pt\forall\hskip
2.84526pts\in[0,1),\partial_{t}u\in L^{2}([0,T];H^{-1}(\mathbb{T}^{N})).$

and moreover, the initial data $u_{0}\in H^{1}(\mathbb{T}^{N}))$ is satisfied
in the sense that $(u(t),\psi)\rightarrow(u_{0},\psi)$ as $t\downarrow 0$.

Standard arguments show (as in, e.g., [50]) that the KSE is locally well-posed
in $H^{1}(\mathbb{T}^{N})$ for both $N\in\\{2,3\\}$:

######  Theorem 2.2.

In case $N\in\\{2,3\\}$, given any initial data $u^{in}\in
H^{1}(\mathbb{T}^{N})$, there exists an interval $[0,T)$ where $T=T(u^{in})>0$
over which the KSE (1.1) has a unique strong solution starting from $u^{in}$.

While the proof is standard and follows from the works within [50], because we
could not locate its proof in the literature, especially the case $N=3$, for
completeness we sketch its outline in the Appendix. On the other hand, by [4,
Theorem 1], we know that (1.2) is locally well-posed in
$L^{2}(\mathbb{T}^{N})$ for both $N\in\\{2,3\\}$.

Let us recall two prominent regularity criteria for the 3D NSE (1.3):

(2.3a) $\displaystyle u\in L_{T}^{r}L_{x}^{p}\text{ where
}\frac{3}{p}+\frac{2}{r}\leq 1,p\in[3,\infty],$ (2.3b) $\displaystyle\nabla
u\in L_{T}^{r}L_{x}^{p}\text{ where }\frac{3}{p}+\frac{2}{r}\leq
2,p\in[\tfrac{9}{4},\infty]$

due to [19, 48, 59, 62]. Let us follow the proof of (2.3a) on NSE (1.3) to
examine what type of criteria we may expect for the $N$D KSE. Taking
$L^{2}(\mathbb{T}^{N})$-inner products on (1.1) with $-\Delta u$ gives us

(2.4) $\frac{1}{2}\frac{d}{dt}\lVert\nabla
u\rVert_{L^{2}}^{2}+\lVert\Delta\nabla
u\rVert_{L^{2}}^{2}=\int(u\cdot\nabla)u\cdot\Delta u+\lambda\lVert\Delta
u\rVert_{L^{2}}^{2}.$

Let us already mention that in the case of the KSE (1.1), we need to
additionally estimate $L^{2}(\mathbb{T}^{N})$-norm so that the sum with (2.4)
gives an $H^{1}(\mathbb{T}^{N})$-estimate because the solution to the KSE
(1.1) does not preserve mean-zero property in sharp contrast to the NSE as
$\int(u\cdot\nabla)u\neq 0$ (this is certainly not a problem if we assume
$u=\nabla\phi$). As we will see in (3.4)-(3.5), this will not give us much
trouble, and because we only want to illustrate heuristics here, let us omit
these details. Now, for clarity, we consider $p\in(N,\infty)$ first, and the
case $p=\infty$ afterward. We estimate by H$\ddot{\mathrm{o}}$lder’s
inequality

(2.5) $\int(u\cdot\nabla)u\cdot\Delta u\leq\lVert u\rVert_{L^{p}}\lVert\nabla
u\rVert_{L^{2}}\lVert\Delta u\rVert_{L^{\frac{2p}{p-2}}}.$

We can continue by Gagliardo-Nirenberg inequality of

(2.6) $\displaystyle\lVert\nabla f\rVert_{L^{\frac{2p}{p-2}}}\lesssim$
$\displaystyle\lVert f\rVert_{L^{2}}^{\frac{p-N}{2p}}\lVert
f\rVert_{H^{2}}^{\frac{p+N}{2p}}$ $\displaystyle\text{ for
}p\in\begin{cases}(N,\infty]&\text{ if }N=2,\\\ [N,\infty]&\text{ if
}N=3\end{cases}$

to bound by

$\displaystyle\int(u\cdot\nabla)u\cdot\Delta u\lesssim$ $\displaystyle\lVert
u\rVert_{L^{p}}\lVert\nabla u\rVert_{L^{2}}\lVert\nabla
u\rVert_{L^{2}}^{\frac{p-N}{2p}}\lVert\nabla u\rVert_{H^{2}}^{\frac{p+N}{2p}}$
(2.7) $\displaystyle\leq$ $\displaystyle\frac{1}{4}\lVert\Delta\nabla
u\rVert_{L^{2}}^{2}+C\Big{(}\lVert u\rVert_{L^{p}}^{\frac{4p}{3p-N}}+\lVert
u\rVert_{L^{p}}\Big{)}\lVert\nabla u\rVert_{L^{2}}^{2}$

by Young’s inequality. This estimate is not valid for the case $N=p=2$ because
it is false that $\lVert\Delta u\rVert_{L^{\infty}}$ can be bounded by a
constant multiple of $\lVert\Delta u\rVert_{H^{1}}$ when $N=2$. In case
$p=+\infty$, we easily use the interpolation of

(2.8) $\displaystyle\lVert
f\rVert_{\dot{H}^{2}}^{2}\leq\Big{(}\int\lvert\lvert\xi\rvert^{3}\lvert\hat{f}(\xi)\rvert\rvert^{2}d\xi\Big{)}^{\frac{1}{2}}\Big{(}\int\lvert\lvert\xi\rvert\lvert\hat{f}(\xi)\rvert\rvert^{2}d\xi\Big{)}^{\frac{1}{2}}=\lVert
f\rVert_{\dot{H}^{3}}\lVert f\rVert_{\dot{H}^{1}}$

and estimate

$\displaystyle\int(u\cdot\nabla)u\cdot\Delta u\leq$ $\displaystyle\lVert
u\rVert_{L^{\infty}}\lVert\nabla u\rVert_{L^{2}}\lVert\Delta u\rVert_{L^{2}}$
(2.9) $\displaystyle\leq$ $\displaystyle\frac{1}{4}\lVert\Delta\nabla
u\rVert_{L^{2}}^{2}+C\lVert u\rVert_{L^{\infty}}^{\frac{4}{3}}\lVert\nabla
u\rVert_{L^{2}}^{2}$

by Young’s inequality. Using (2.8) we can also estimate the linear term

(2.10) $\lambda\lVert\Delta u\rVert_{L^{2}}^{2}\leq\lambda\lVert\Delta\nabla
u\rVert_{L^{2}}\lVert\nabla u\rVert_{L^{2}}\leq\frac{1}{4}\lVert\Delta\nabla
u\rVert_{L^{2}}^{2}+C\lVert\nabla u\rVert_{L^{2}}^{2}.$

Applying (2.7)-(2.10) to (2.4) leads to

$\frac{1}{2}\frac{d}{dt}\lVert\nabla
u\rVert_{L^{2}}^{2}+\frac{1}{2}\lVert\Delta\nabla
u\rVert_{L^{2}}^{2}\lesssim\lVert\nabla u\rVert_{L^{2}}^{2}(\lVert
u\rVert_{L^{p}}^{\frac{4p}{3p-N}}+\lVert u\rVert_{L^{p}}+1)$

with $\frac{4p}{3p-N}=\frac{4}{3}$ in case $p=\infty$. This leads to a
regularity criteria of

$\displaystyle u\in L_{T}^{r}L_{x}^{p}$ $\displaystyle\text{ where
}\begin{cases}p\in(N,\infty],r\in[\frac{4}{3},2)&\text{ if }N=2,\\\
p\in[N,\infty],r\in[\frac{4}{3},2]&\text{ if }N=3,\end{cases}$ (2.11)
$\displaystyle\text{ satisfy
}\frac{N}{p}+\frac{2}{r}=\frac{3}{2}+\frac{N}{2p}.$

######  Remark 2.3.

If we take $N=2$ and then $p>2$ arbitrarily close to 2 in (2.11), then we see
that the right hand side of (2.11) is almost 2, very reminiscent of (2.3b) in
terms of $\nabla u$ for the 3D NSE (1.3). In fact, (2.11) may be heuristically
explained via scaling argument as follows. Informally assuming that
$\lambda=0$ in (1.1), we realize that if $u(t,x)$ is its solution, then so is
$u_{\beta}(t,x)\triangleq\beta^{3}u(\beta^{4}t,\beta x)$ and $\lVert
u_{\beta}\rVert_{L_{T}^{r}L_{x}^{p}}=\lVert
u\rVert_{L_{\beta^{4}T}^{r}L_{x}^{p}}$ if and only if
$\frac{N}{p}+\frac{2}{r}=\frac{3}{2}+\frac{N}{2p}$. We are indebted to one of
the referees for pointing this out.

As we emphasized after (2.7), we cannot obtain the case $N=p=2$ due to the
fact that $H^{1}(\mathbb{T}^{2})\hookrightarrow L^{\infty}(\mathbb{T}^{2})$ is
false in general. For the NSE, there are typically div-curl lemma or Brezis-
Wainger type inequality arguments to work around this issue and improve $p>N$
to $p=N$. Unfortunately, both methods faced technical difficulty for the KSE.
In fact, we will actually see in Theorem 3.1 that it is possible to improve
(2.11) significantly by rather an unconventional approach that seems unique to
the KSE.

Very recently, the research direction of component reduction from the criteria
in (2.3) for the 3D NSE has received much attention. While we acknowledge that
there have been many extensions and improvements, let us list the following
examples as a reference: starting from an initial data that is sufficiently
smooth, there exists a unique solution to the 3D NSE for all $t>0$ if for any
$j,k\in\\{1,2,3\\}$,

(2.12a) $\displaystyle u_{j}\in L_{T}^{r}L_{x}^{p}\text{ where
}\frac{3}{p}+\frac{2}{r}\leq\frac{5}{8},\hskip
2.84526ptp\in[\tfrac{24}{5},\infty]$ (2.12b)
$\displaystyle\partial_{j}u_{j}\in L_{T}^{r}L_{x}^{p}\text{ where
}\frac{3}{p}+\frac{2}{r}<\frac{3(p+2)}{4p},\hskip 2.84526ptp>2,\hskip
2.84526ptr\in[1,\infty),$ (2.12c) $\displaystyle\partial_{k}u_{j}\in
L_{T}^{r}L_{x}^{p}\text{ for }k\neq j\text{ where
}\frac{3}{p}+\frac{2}{r}<\frac{p+3}{2p},\hskip 2.84526ptp>3,\hskip
2.84526ptr\in[1,\infty),$

due to [8, 44] (see also [7, 45, 76]). We point out that while the dependence
on $u=(u_{1},u_{2},u_{3})$ or $\nabla u=(\partial_{j}u_{k})_{1\leq j,k\leq 3}$
was remarkably reduced down respectively to $u_{j}$, $\partial_{j}u_{j}$, or
$\partial_{k}u_{j}$, the integrability condition of 1 and 2 in (2.3)
deteriorated respectively to $\frac{5}{8}$, $\frac{3(p+2)}{4p}$, or
$\frac{p+3}{2p}$ in (2.12). This is the price one must pay for reducing
components; to the best of our knowledge, despite much effort by many
mathematicians, it remains unknown if we can improve $\frac{5}{8}$ up to 1 or
$\frac{3(p+2)}{4p},\frac{p+3}{2p}$ up to 2, as in (2.3). All the proofs of
such component reduction results rely crucially on the divergence-free
property of $u$ in (1.3), and some of such results have been extended to other
systems of PDEs that involve divergence-free velocity field (e.g., [9, 75] on
MHD system; [74] on SQG equations). In short, the proof of (2.12a) relies on
the fact that upon
$\lVert(\partial_{1}u,\partial_{2}u)\rVert_{L^{2}}^{2}$-estimate of the 3D NSE
(1.3), one can separate $u_{3}$ within the nonlinear terms as follows:

$\displaystyle\int(u\cdot\nabla)u\cdot(\partial_{1}^{2}+\partial_{2}^{2})u=$
$\displaystyle-\sum_{i,j=1}^{3}\sum_{k=1}^{2}\int\partial_{k}u_{i}\partial_{i}u_{j}\partial_{k}u_{j}$
$\displaystyle\lesssim$ $\displaystyle\int\lvert u_{3}\rvert\lvert\nabla
u\rvert\lvert\nabla(\partial_{1},\partial_{2})u\rvert.$

E.g., we can estimate by summing the terms when $(i,j,k)=(1,1,2)$ and
$(i,j,k)=(2,1,2)$ to bound

$\displaystyle\int\partial_{2}u_{1}\partial_{1}u_{1}\partial_{2}u_{1}+\partial_{2}u_{2}\partial_{2}u_{1}\partial_{2}u_{1}=$
$\displaystyle-\int(\partial_{2}u_{1})^{2}\partial_{3}u_{3}$
$\displaystyle\lesssim$ $\displaystyle\int\lvert u_{3}\rvert\lvert\nabla
u\rvert\lvert\nabla(\partial_{1},\partial_{2})u\rvert$

where the divergence-free property played an important role. Despite the large
number of papers devoted to this direction of research, we are not aware of
any example of a PDE that admitted such component reduction of regularity
criteria without a divergence-free property. Remarkably, we have been able to
prove such results for the KSE (see Theorems 3.6, 3.10, 3.16, and 3.20).

##  3\. Regularity criteria for $N$d KSE and their proofs

The following computations hold on $\mathbb{R}^{N}$ or $\mathbb{T}^{N}$ for
$N\in\\{2,3\\}$, but we choose to focus on the latter, and also elaborate more
in the case $N=2$ as our main concern. Results in Subsections 3.1-3.4 focus on
(1.1) independently of (1.2) while Subsection 3.5 focuses on (1.2)
independently of (1.1).

###  3.1. Result on $u$

Let us continue the $\dot{H}^{1}(\mathbb{T}^{N})$-estimate from (2.4). For the
sake of generality, let us also aim to attain a criterion in $W_{x}^{m,p}$ for
$m\in[0,1)$ so that taking $m=0$ reduces to the standard criterion in
$L_{x}^{p}$. We proceed as follows:

(3.1) $\displaystyle\int(u\cdot\nabla)u\cdot\Delta u\leq\lVert
u\rVert_{L^{\frac{Np}{N-mp}}}\lVert\nabla
u\rVert_{L^{\frac{2Np}{Np+mp-N}}}\lVert\Delta
u\rVert_{L^{\frac{2Np}{Np+mp-N}}}$

by H$\ddot{\mathrm{o}}$lder’s inequality for $p$ in an appropriate range to be
specified subsequently. Now we use two Gagliardo-Nirenberg inequalities:

(3.2a) $\displaystyle\lVert f\rVert_{L^{\frac{2Np}{Np+mp-N}}}\lesssim$
$\displaystyle\lVert f\rVert_{L^{2}}^{\frac{(4+m)p-N}{4p}}\lVert
f\rVert_{H^{2}}^{\frac{N-mp}{4p}}$ $\displaystyle\text{ for any
}N\in\\{2,3\\},p\in\begin{cases}[1,\frac{N}{m}]&\text{ if }m\in(0,1),\\\
[1,\infty]&\text{ if }m=0,\\\ \end{cases}$ (3.2b) $\displaystyle\lVert\nabla
f\rVert_{L^{\frac{2Np}{Np+mp-N}}}\lesssim$ $\displaystyle\lVert
f\rVert_{L^{2}}^{\frac{(2+m)p-N}{4p}}\lVert
f\rVert_{H^{2}}^{\frac{(2-m)p+N}{4p}}$
$\displaystyle\begin{cases}p\in[1,\infty]&\text{ if }N=2,m\in(0,1),\\\
p\in[\frac{3}{m+2},\infty]&\text{ if }N=3,m\in(0,1),\\\ p\in(1,\infty]&\text{
if }N=2,m=0,\\\ p\in[\frac{3}{2},\infty]&\text{ if }N=3,m=0.\end{cases}$

Now if $m\in(0,1)$, then we apply Sobolev embedding
$W^{m,p}(\mathbb{T}^{N})\hookrightarrow L^{\frac{Np}{N-mp}}(\mathbb{T}^{N})$
assuming $p<\frac{N}{m}$ as part of hypothesis in case $m\in(0,1)$ and see
that

$\displaystyle\int(u\cdot\nabla)u\cdot\Delta u\lesssim$ $\displaystyle\lVert
u\rVert_{W^{m,p}}\lVert\nabla
u\rVert_{L^{2}}^{\frac{(m+3)p-N}{2p}}\lVert\nabla
u\rVert_{H^{2}}^{\frac{(1-m)p+N}{2p}}$ (3.3) $\displaystyle\leq$
$\displaystyle\frac{1}{8}\lVert\Delta\nabla u\rVert_{L^{2}}^{2}+C(\lVert
u\rVert_{W^{m,p}}^{\frac{4p}{(3+m)p-N}}+1)\lVert\nabla u\rVert_{L^{2}}^{2}$

by Young’s inequality. As mentioned, we also need an $L^{2}(\mathbb{T}^{N})$
estimate; thus, taking $L^{2}(\mathbb{T}^{N})$ inner-products on (1.1) with
$u$ we obtain

(3.4) $\frac{1}{2}\frac{d}{dt}\lVert u\rVert_{L^{2}}^{2}+\lVert\Delta
u\rVert_{L^{2}}^{2}=\lambda\lVert\nabla
u\rVert_{L^{2}}^{2}-\int(u\cdot\nabla)u\cdot u$

where we estimate

$\displaystyle\int(u\cdot\nabla)u\cdot u$ $\displaystyle\leq$
$\displaystyle\lVert u\rVert_{L^{\frac{Np}{N-mp}}}\lVert
u\rVert_{L^{\frac{2Np}{Np+mp-N}}}\lVert\nabla
u\rVert_{L^{\frac{2Np}{Np+mp-N}}}$ $\displaystyle\lesssim$
$\displaystyle\lVert u\rVert_{W^{m,p}}\lVert
u\rVert_{L^{2}}^{\frac{(m+3)p-N}{2p}}\lVert
u\rVert_{H^{2}}^{\frac{(1-m)p+N}{2p}}$ $\displaystyle\lesssim$
$\displaystyle\lVert u\rVert_{W^{m,p}}\Big{(}\lVert
u\rVert_{L^{2}}^{\frac{(m+3)p-N}{2p}}\lVert\nabla
u\rVert_{L^{2}}^{\frac{(1-m)p+N}{4p}}\lVert\Delta\nabla
u\rVert_{L^{2}}^{\frac{(1-m)p+N}{4p}}+\lVert u\rVert_{L^{2}}^{2}\Big{)}$ (3.5)
$\displaystyle\leq$ $\displaystyle\frac{1}{8}\lVert\Delta\nabla
u\rVert_{L^{2}}^{2}+C\Big{(}1+\lVert u\rVert_{W^{m,p}}+\lVert
u\rVert_{W^{m,p}}^{\frac{4p}{(3+m)p-N}}\Big{)}\lVert u\rVert_{H^{1}}^{2}$

by H$\ddot{\mathrm{o}}$lder’s inequality, Sobolev embedding
$W^{m,p}(\mathbb{T}^{N})\hookrightarrow L^{\frac{Np}{N-mp}}(\mathbb{T}^{N})$
assuming $p<\frac{N}{m}$ in case $m\in(0,1)$ as part of hypothesis,
(3.2a)-(3.2b), and Young’s inequality. Applying (2.10) and (3.3) to (2.4), and
(3.5) to (3.4), we showed the following criteria:

######  Theorem 3.1.

Suppose $u\in L_{T}^{r}W_{x}^{m,p}$ where $m\in[0,1)$, and

(3.6) $\begin{cases}p\in[1,\frac{N}{m}),r\in(\frac{4}{3},\frac{4}{1+m}]&\text{
if }N=2,m\in(0,1),\\\
p\in[\frac{N}{m+2},\frac{N}{m}),r\in(\frac{4}{3},4]&\text{ if
}N=3,m\in(0,1),\end{cases}$

and

(3.7) $\begin{cases}p\in(\frac{N}{2},\infty],r\in[\frac{4}{3},4)&\text{ if
}N=2,m=0,\\\ p\in[\frac{N}{2},\infty],r\in[\frac{4}{3},4]&\text{ if
}N=3,m=0,\end{cases}$

satisfy

(3.8) $\displaystyle\frac{N}{p}+\frac{2}{r}=\frac{3+m}{2}+\frac{N}{2p}.$

Then the $N$D KSE is globally well-posed in $H^{1}(\mathbb{T}^{N})$.

######  Remark 3.2.

In comparison to Remark 2.3, because we improved the range of $p$ in (2.11) to
(3.7) in Theorem 3.1, the right hand side of (3.8) with $m=0$ can achieve
$\frac{3}{2}+\frac{N}{2p}>2$ for $p\in(\frac{N}{2},N)$; in fact, taking
$p\approx\frac{N}{2}$ gives $\frac{3}{2}+\frac{N}{2p}$ to be almost
$\frac{5}{2}$. We recall from Remark 2.3 that
$\frac{N}{p}+\frac{2}{r}=\frac{3}{2}+\frac{N}{2p}$ can be explained
heuristically via scaling argument. We also emphasize that this new approach
in the proof of Theorem 3.1 is impossible for the NSE. Indeed, if we choose
$\lVert\Delta u\rVert_{L^{\frac{2Np}{Np+mp-N}}}$ in the first
H$\ddot{\mathrm{o}}$lder’s inequality of (3.1), then this term already cannot
be handled by the diffusion in the NSE, specifically $\lVert\Delta
u\rVert_{L^{2}}^{2}$ in $H^{1}$-estimate. It is the fourth-order diffusion in
the KSE that is allowing us to proceed with this new approach.

###  3.2. Result on $u_{1}$

Next, we obtain a criterion on only one component $u_{1}$ in the 2D case while
on $(u_{1},u_{2})$ in the 3D case.

######  Proposition 3.3.

Suppose $u_{1}\in L_{T}^{r}W_{x}^{m,p}$ where $m\in[0,1)$ and

$\begin{cases}p\in[1,\frac{2}{m}),r\in(\frac{4}{3},\frac{4}{1+m}]&\text{ if
}m\in(0,1),\\\ p\in(1,\infty],r\in[\frac{4}{3},4)&\text{ if }m=0,\end{cases}$

satisfy

$\displaystyle\frac{2}{p}+\frac{2}{r}=\frac{1}{p}+\frac{3+m}{2}.$

Then $u_{2}\in L_{T}^{\infty}L_{x}^{2}\cap L_{T}^{2}H_{x}^{2}$.

###### Proof.

Taking $L^{2}(\mathbb{T}^{2})$-inner products with $u_{2}$ on the second
component of (1.1) gives

(3.9) $\frac{1}{2}\frac{d}{dt}\lVert u_{2}\rVert_{L^{2}}^{2}+\lVert\Delta
u_{2}\rVert_{L^{2}}^{2}=-\int(u\cdot\nabla)u_{2}u_{2}+\lambda\lVert\nabla
u_{2}\rVert_{L^{2}}^{2}$

where

(3.10) $\displaystyle\lambda\lVert\nabla
u_{2}\rVert_{L^{2}}^{2}\leq\lambda\lVert u_{2}\rVert_{L^{2}}\lVert\Delta
u_{2}\rVert_{L^{2}}\leq\frac{1}{4}\lVert\Delta u_{2}\rVert_{L^{2}}^{2}+C\lVert
u_{2}\rVert_{L^{2}}^{2},$

and

$\displaystyle-\int(u\cdot\nabla)u_{2}u_{2}=$ $\displaystyle-\int
u_{1}(\partial_{1}u_{2})u_{2}-\int u_{2}\partial_{2}u_{2}u_{2}$
$\displaystyle=$ $\displaystyle-\int
u_{1}(\partial_{1}u_{2})u_{2}-\int\frac{1}{3}\partial_{2}(u_{2})^{3}$
$\displaystyle\leq$ $\displaystyle\lVert
u_{1}\rVert_{L^{\frac{2p}{2-mp}}}\lVert
u_{2}\rVert_{L^{\frac{4p}{2p+mp-2}}}\lVert\nabla
u_{2}\rVert_{L^{\frac{4p}{2p+mp-2}}}$ $\displaystyle\lesssim$
$\displaystyle\lVert u_{1}\rVert_{W^{m,p}}\lVert
u_{2}\rVert_{L^{2}}^{\frac{(3+m)p-2}{2p}}\lVert
u_{2}\rVert_{H^{2}}^{\frac{(1-m)p+2}{2p}}$ (3.11) $\displaystyle\leq$
$\displaystyle\frac{1}{4}\lVert\Delta u_{2}\rVert_{L^{2}}^{2}+C\Big{(}\lVert
u_{1}\rVert_{W^{m,p}}^{\frac{4p}{(3+m)p-2}}+\lVert
u_{1}\rVert_{W^{m,p}}\Big{)}\lVert u_{2}\rVert_{L^{2}}^{2}$

by H$\ddot{\mathrm{o}}$lder’s inequality, Sobolev embedding
$W^{m,p}(\mathbb{T}^{2})\hookrightarrow L^{\frac{2p}{2-mp}}(\mathbb{T}^{2})$
if $m\in(0,1)$ so that $p<\frac{2}{m}$ by hypothesis, (3.2a)-(3.2b), and
Young’s inequality. Applying (3.10)-(3.11) to (3.9) implies the desired
result. ∎

######  Proposition 3.4.

Suppose $u_{1}\in L_{T}^{r}W_{x}^{m,p}$ where $m\in[0,1)$ and

$\begin{cases}p\in[1,\frac{2}{m}),r\in(\frac{4}{3},\frac{4}{1+m}]&\text{ if
}m\in(0,1),\\\ p\in(1,\infty],r\in[\frac{4}{3},4)&\text{ if }m=0,\end{cases}$

satisfy

$\displaystyle\frac{2}{p}+\frac{2}{r}=\frac{1}{p}+\frac{3+m}{2}.$

Then $u_{1}\in L_{T}^{\infty}L_{x}^{2}\cap L_{T}^{2}H_{x}^{2}$.

###### Proof.

We take $L^{2}(\mathbb{T}^{2})$-inner products on the first component of
$\eqref{KSE}$ with $u_{1}$ and compute

(3.12) $\displaystyle\frac{1}{2}\frac{d}{dt}\lVert
u_{1}\rVert_{L^{2}}^{2}+\lVert\Delta
u_{1}\rVert_{L^{2}}^{2}=-\int(u\cdot\nabla)u_{1}u_{1}+\lambda\lVert\nabla
u_{1}\rVert_{L^{2}}^{2}$

where $\lambda\lVert\nabla u_{1}\rVert_{L^{2}}^{2}\leq\frac{1}{4}\lVert\Delta
u_{1}\rVert_{L^{2}}^{2}+C\lVert u_{1}\rVert_{L^{2}}^{2}$ similarly to (3.10),
while

$\displaystyle-\int(u\cdot\nabla)u_{1}u_{1}=$ $\displaystyle-\int
u_{2}\partial_{2}u_{1}u_{1}$ $\displaystyle\leq$ $\displaystyle\lVert
u_{2}\rVert_{L^{\infty}}\lVert\nabla u_{1}\rVert_{L^{2}}\lVert
u_{1}\rVert_{L^{2}}$ (3.13) $\displaystyle\lesssim$ $\displaystyle\lVert
u_{2}\rVert_{H^{2}}\lVert u_{1}\rVert_{L^{2}}^{\frac{3}{2}}\lVert\Delta
u_{1}\rVert_{L^{2}}^{\frac{1}{2}}\leq\frac{1}{4}\lVert\Delta
u_{1}\rVert_{L^{2}}^{2}+C\lVert u_{2}\rVert_{H^{2}}^{\frac{4}{3}}\lVert
u_{1}\rVert_{L^{2}}^{2}$

by Sobolev embedding $H^{2}(\mathbb{T}^{2})\hookrightarrow
L^{\infty}(\mathbb{T}^{2})$, (3.10) and Young’s inequality. This implies that
$u_{1}\in L_{T}^{\infty}L_{x}^{2}\cap L_{T}^{2}H_{x}^{2}$ because we know
$\lVert u_{2}(t)\rVert_{H^{2}}\in L_{T}^{2}$ due to Proposition 3.3 ∎

######  Remark 3.5.

Interestingly, the previous proof of Proposition 3.4 would not work for the 2D
Burgers’ equation if we neglect the fact that its solution has the maximum
principle in the following way. Starting from $u_{1}\in L_{T}^{r}L_{x}^{p}$
for some $p$ and $r$, we can still deduce that $u_{2}\in
L_{T}^{\infty}L_{x}^{2}\cap L_{T}^{2}H_{x}^{1}$ analogously to Proposition
3.3. However, this does not seem to be sufficient to lead to $u_{1}\in
L_{T}^{\infty}L_{x}^{2}\cap L_{T}^{2}H_{x}^{1}$ because in the estimate of

$-\int u_{2}\partial_{2}u_{1}u_{1}$

within (3.2), the diffusion for the Burgers’ equation only gives $\lVert\nabla
u_{1}\rVert_{L^{2}}^{2}$ so that we have no choice but to bound this by

$\lVert u_{2}\rVert_{L^{\infty}}\lVert\partial_{2}u_{1}\rVert_{L^{2}}\lVert
u_{1}\rVert_{L^{2}}.$

However, while $\lVert\nabla u_{2}\rVert_{L^{2}}\in L_{T}^{2}$, this estimate
will not go through immediately since
$H^{1}(\mathbb{T}^{2})\not\hookrightarrow L^{\infty}(\mathbb{T}^{2})$. Thus,
Proposition 3.4 really is a special feature of the KSE due to the fourth-order
diffusion.

By Propositions 3.3-3.4 we have $u\in L_{T}^{\infty}L_{x}^{2}$ and thus we
deduce that the 2D KSE is globally well-posed due to Theorem 3.1 as follows:

######  Theorem 3.6.

Suppose $u_{1}\in L_{T}^{r}W_{x}^{m,p}$ where $m\in[0,1)$ and

$\begin{cases}p\in[1,\frac{2}{m}),r\in(\frac{4}{3},\frac{4}{1+m}]&\text{ if
}m\in(0,1),\\\ p\in(1,\infty],r\in[\frac{4}{3},4)&\text{ if }m=0,\end{cases}$

satisfy

$\displaystyle\frac{2}{p}+\frac{2}{r}=\frac{1}{p}+\frac{3+m}{2}.$

Then the 2D KSE is globally well-posed in $H^{1}(\mathbb{T}^{2})$.

######  Remark 3.7.

As mentioned, Theorem 3.6 is very interesting because all previous component
reduction results relied on divergence-free condition (NSE, MHD system, and
SQG equations).

Another remarkable feature of Theorem 3.6 is that we obtained component
reduction at no cost in terms of $\frac{3}{2}+\frac{1}{p}$ (cf. “1” in (2.3a)
and “$\frac{5}{8}$” in (2.12a)). As mentioned, reducing the Serrin criteria
(2.3a) to $u_{3}$ while retaining $\frac{3}{p}+\frac{2}{r}=1$ is a well-known
open problem that has caught much attention in the literature.

We are able to extend our result to the 3D case as follows:

######  Proposition 3.8.

Suppose $u_{1},u_{2}\in L_{T}^{r}W_{x}^{m,p}$ where $m\in[0,1)$ and

$\begin{cases}p\in[\frac{3}{m+2},\frac{3}{m}),r\in(\frac{4}{3},4]&\text{ if
}m\in(0,1),\\\ p\in[\frac{3}{2},\infty],r\in[\frac{4}{3},4]&\text{ if
}m=0,\end{cases}$

satisfy

$\displaystyle\frac{3}{p}+\frac{2}{r}=\frac{3}{2p}+\frac{3+m}{2}.$

Then $u_{3}\in L_{T}^{\infty}L_{x}^{2}\cap L_{T}^{2}H_{x}^{2}$.

###### Proof.

We consider the third component of (1.1), take $L^{2}(\mathbb{T}^{3})$ inner-
products with $u_{3}$ to obtain

(3.14) $\frac{1}{2}\frac{d}{dt}\lVert u_{3}\rVert_{L^{2}}^{2}+\lVert\Delta
u_{3}\rVert_{L^{2}}^{2}=-\int(u\cdot\nabla)u_{3}u_{3}+\lambda\lVert\nabla
u_{3}\rVert_{L^{2}}^{2}$

where $\lambda\lVert\nabla u_{3}\rVert_{L^{2}}^{2}\leq\frac{1}{4}\lVert\Delta
u_{3}\rVert_{L^{2}}^{2}+C\lVert u_{3}\rVert_{L^{2}}^{2}$ similarly to (3.10),
while

$\displaystyle-\int(u\cdot\nabla)u_{3}u_{3}$ $\displaystyle=$
$\displaystyle-\int u_{1}\partial_{1}u_{3}u_{3}+u_{2}\partial_{2}u_{3}u_{3}$
$\displaystyle\lesssim$ $\displaystyle(\lVert
u_{1}\rVert_{L^{\frac{3p}{3-mp}}}+\lVert
u_{2}\rVert_{L^{\frac{3p}{3-mp}}})\lVert
u_{3}\rVert_{L^{\frac{6p}{3p+mp-3}}}\lVert\nabla
u_{3}\rVert_{L^{\frac{6p}{3p+mp-3}}}$ $\displaystyle\lesssim$
$\displaystyle(\lVert u_{1}\rVert_{W^{m,p}}+\lVert
u_{2}\rVert_{W^{m,p}})\lVert u_{3}\rVert_{L^{2}}^{\frac{(3+m)p-3}{2p}}\lVert
u_{3}\rVert_{H^{2}}^{\frac{(1-m)p+3}{2p}}$ (3.15) $\displaystyle\leq$
$\displaystyle\frac{1}{4}\lVert\Delta
u_{3}\rVert_{L^{2}}^{2}+C(\sum_{k=1}^{2}\lVert
u_{k}\rVert_{W^{m,p}}^{\frac{4p}{(3+m)p-3}}+\lVert
u_{k}\rVert_{W^{m,p}})\lVert u_{3}\rVert_{L^{2}}^{2}$

by H$\ddot{\mathrm{o}}$lder’s inequality, Sobolev embedding
$L^{\frac{3p}{3-mp}}(\mathbb{T}^{3})\hookrightarrow W^{m,p}(\mathbb{T}^{3})$
if $m>0$, (3.2a)-(3.2b), and Young’s inequality. Applying these estimates to
(3.14), Grönwall’s inequality completes the proof. ∎

######  Proposition 3.9.

Suppose $u_{1},u_{2}\in L_{T}^{r}W_{x}^{m,p}$ where $m\in[0,1)$ and

$\begin{cases}p\in[\frac{3}{m+2},\frac{3}{m}),r\in(\frac{4}{3},4]&\text{ if
}m\in(0,1),\\\ p\in[\frac{3}{2},\infty],r\in[\frac{4}{3},4]&\text{ if
}m=0,\end{cases}$

satisfy

$\displaystyle\frac{3}{p}+\frac{2}{r}=\frac{3}{2p}+\frac{3+m}{2}.$

Then $u_{1},u_{2}\in L_{T}^{\infty}L_{x}^{2}\cap L_{T}^{2}H_{x}^{2}$.

###### Proof.

We fix $j\in\\{1,2\\}$ arbitrarily and take $L^{2}(\mathbb{T}^{3})$ inner
product on the $j$-th component of (1.1) with $u_{j}$ to obtain

(3.16) $\frac{1}{2}\frac{d}{dt}\lVert u_{j}\rVert_{L^{2}}^{2}+\lVert\Delta
u_{j}\rVert_{L^{2}}^{2}=-\int(u\cdot\nabla)u_{j}u_{j}+\lambda\lVert\nabla
u_{j}\rVert_{L^{2}}^{2}$

where $\lambda\lVert\nabla u_{3}\rVert_{L^{2}}^{2}\leq\frac{1}{4}\lVert\Delta
u_{3}\rVert_{L^{2}}^{2}+C\lVert u_{3}\rVert_{L^{2}}^{2}$ similarly to (3.10),
while

$\displaystyle-\int(u\cdot\nabla)u_{j}u_{j}=-\sum_{k=1}^{3}\int
u_{k}\partial_{k}u_{j}u_{j}$

where $\int u_{j}\partial_{j}u_{j}u_{j}=0$ so that for
$k\in\\{1,2\\}\setminus\\{j\\}$,

$\displaystyle-\int(u\cdot\nabla)u_{j}u_{j}=$ $\displaystyle-\int
u_{k}\partial_{k}u_{j}u_{j}+u_{3}\partial_{3}u_{j}u_{j}$
$\displaystyle\lesssim$ $\displaystyle(\lVert u_{k}\rVert_{L^{3}}+\lVert
u_{3}\rVert_{L^{3}})\lVert\nabla u_{j}\rVert_{L^{6}}\lVert
u_{j}\rVert_{L^{2}}$ (3.17) $\displaystyle\leq$
$\displaystyle\frac{1}{4}\lVert\Delta u_{j}\rVert_{L^{2}}^{2}+C(1+\lVert
u_{k}\rVert_{L^{3}}^{2}+\lVert u_{3}\rVert_{L^{2}}^{\frac{3}{2}}\lVert
u_{3}\rVert_{H^{2}}^{\frac{1}{2}})\lVert u_{j}\rVert_{L^{2}}^{2}$

by H$\ddot{\mathrm{o}}$lder’s inequality, Sobolev embedding
$H^{1}(\mathbb{T}^{3})\hookrightarrow L^{6}(\mathbb{T}^{3})$, Gagliardo-
Nirenberg inequality, and Young’s inequality. We apply these estimates to
(3.16) and because

$\displaystyle\int_{0}^{T}\lVert
u_{k}\rVert_{L^{3}}^{2}\lesssim\int_{0}^{T}\lVert
u_{k}\rVert_{W^{m,\frac{3}{m+1}}}^{2}d\tau<\infty,$
$\displaystyle\int_{0}^{T}\lVert u_{3}\rVert_{L^{2}}^{\frac{3}{2}}\lVert
u_{3}\rVert_{H^{2}}^{\frac{1}{2}}d\tau\leq\lVert
u_{3}\rVert_{L_{T}^{\infty}L_{x}^{2}}^{\frac{3}{2}}\int_{0}^{T}\lVert
u_{3}\rVert_{H^{2}}^{\frac{1}{2}}d\tau<\infty$

due to Sobolev embedding $W^{m,\frac{3}{m+1}}(\mathbb{T}^{3})\hookrightarrow
L^{3}(\mathbb{T}^{3})$ if $m\in(0,1)$, hypothesis, and Proposition 3.8,
Grönwall’s inequality completes the proof. ∎

Due to Propositions 3.8-3.9, we see that $u\in L_{T}^{\infty}L_{x}^{2}$ and
thus Theorem 3.1 again leads to the following theorem.

######  Theorem 3.10.

Suppose $u_{1},u_{2}\in L_{T}^{r}W_{x}^{m,p}$ where $m\in[0,1)$ and

$\begin{cases}p\in[\frac{3}{m+2},\frac{3}{m}),r\in(\frac{4}{3},4]&\text{ if
}m\in(0,1),\\\ p\in[\frac{3}{2},\infty],r\in[\frac{4}{3},4]&\text{ if
}m=0,\end{cases}$

satisfy

$\displaystyle\frac{3}{p}+\frac{2}{r}=\frac{3}{2p}+\frac{3+m}{2}.$

Then the 3D KSE is globally well-posed in $H^{1}(\mathbb{T}^{3})$.

######  Remark 3.11.

We leave it as an interesting problem to attain a regularity criterion for the
3D KSE in terms of one component instead of two components as we did in
Theorem 3.10.

###  3.3. Result on $\nabla u$

We aim to obtain a criterion in terms of $\lVert\nabla u\rVert_{L^{p}}$,
motivated the case of the NSE in (2.3b). We can start from (2.4) and rely on
(2.10). For the nonlinear term we can write

(3.18) $\displaystyle\int(u\cdot\nabla)u\cdot\Delta
u=-\sum_{i,j,k}\int\left(\partial_{k}u_{i}\partial_{i}u_{j}\partial_{k}u_{j}-\frac{1}{2}\partial_{i}u_{i}(\partial_{k}u_{j})^{2}\right)\lesssim\int\lvert\nabla
u\rvert^{3}.$

Now let us point out again that if we follow the standard approach of the NSE,
then we would compute for $p\in[2,\infty]$,

$\displaystyle C\int\lvert\nabla u\rvert^{3}\lesssim$
$\displaystyle\lVert\nabla u\rVert_{L^{p}}\lVert\nabla
u\rVert_{L^{2}}\lVert\nabla u\rVert_{L^{\frac{2p}{p-2}}}$
$\displaystyle\lesssim$ $\displaystyle\lVert\nabla u\rVert_{L^{p}}\lVert\nabla
u\rVert_{L^{2}}\lVert\nabla u\rVert_{L^{2}}^{\frac{2p-N}{2p}}\lVert\nabla
u\rVert_{H^{2}}^{\frac{N}{2p}}$ $\displaystyle\leq$
$\displaystyle\frac{1}{4}\lVert\Delta\nabla u\rVert_{L^{2}}^{2}+C(\lVert\nabla
u\rVert_{L^{p}}^{\frac{4p}{4p-N}}+\lVert\nabla u\rVert_{L^{p}})\lVert\nabla
u\rVert_{L^{2}}^{2}$

by H$\ddot{\mathrm{o}}$lder’s, Gagliardo-Nirenberg, and Young’s inequalities.
This only leads to a criterion of $\nabla u\in L_{T}^{r}L_{x}^{p}$ where
$\frac{N}{p}+\frac{2}{r}=2+\frac{N}{2p}$,
$p\in[2,\infty],r\in[1,\frac{8}{8-N}]$. In fact, we approach this estimate
differently from the NSE as follows in both cases $N\in\\{2,3\\}$:

$\displaystyle C\int\lvert\nabla u\rvert^{3}\lesssim$
$\displaystyle\lVert\nabla u\rVert_{L^{p}}\lVert\nabla
u\rVert_{L^{\frac{2p}{p-1}}}^{2}$ $\displaystyle\lesssim$
$\displaystyle\lVert\nabla u\rVert_{L^{p}}\lVert\nabla
u\rVert_{L^{2}}^{\frac{4p-N}{2p}}\lVert\nabla u\rVert_{H^{2}}^{\frac{N}{2p}}$
(3.19) $\displaystyle\leq$ $\displaystyle\tfrac{1}{4}\lVert\Delta\nabla
u\rVert_{L^{2}}^{2}+C(\lVert\nabla
u\rVert_{L^{p}}^{\frac{4p}{4p-N}}+\lVert\nabla u\rVert_{L^{p}})\lVert\nabla
u\rVert_{L^{2}}^{2}$

by H$\ddot{\mathrm{o}}$lder’s inequality, (3.2a) and Young’s inequality. We
also continue from (3.4) and estimate the nonlinear term as

$\displaystyle-\int(u\cdot\nabla)u\cdot u\leq$ $\displaystyle\lVert\nabla
u\rVert_{L^{p}}\lVert u\rVert_{L^{\frac{2p}{p-1}}}^{2}\lesssim\lVert\nabla
u\rVert_{L^{p}}\lVert u\rVert_{L^{2}}^{\frac{4p-N}{2p}}\lVert
u\rVert_{H^{2}}^{\frac{N}{2p}}$ (3.20) $\displaystyle\leq$
$\displaystyle\frac{1}{8}\lVert\Delta u\rVert_{L^{2}}^{2}+C(\lVert\nabla
u\rVert_{L^{p}}^{\frac{4p}{4p-N}}+\lVert\nabla u\rVert_{L^{p}})\lVert
u\rVert_{L^{2}}^{2}$

by H$\ddot{\mathrm{o}}$lder’s inequality (3.2a) and Young’s inequality.
Considering (3.18), (3.19) and (3.20) in (2.4) and (3.4) gives us the
following result.

######  Theorem 3.12.

Suppose $\nabla u\in L_{T}^{r}L_{x}^{p}$ where
$p\in[1,\infty],r\in[1,\frac{4}{4-N}]$ satisfy

$\frac{N}{p}+\frac{2}{r}=2+\frac{N}{2p}.$

Then the $N$D KSE is globally well-posed in $H^{1}(\mathbb{T}^{N})$.

Taking $p=1$ turns $2+\frac{N}{2p}$ to $2+\frac{N}{2}$. We emphasize again
that $2+\frac{N}{2}$ is remarkably larger than 2 in (2.3b).

###  3.4. Results on $\nabla\cdot u$, and $\partial_{2}u_{2}$

Next, we aim to obtain a result on divergence $\nabla\cdot u$, which is an
improvement of $\nabla u$, and in fact, even $\partial_{2}u_{2}$ at the
optimal level.

If we decide to work on the $H^{1}(\mathbb{T}^{N})$-estimate, then we can look
at (2.4) again

$\displaystyle\frac{1}{2}\frac{d}{dt}\lVert\nabla
u\rVert_{L^{2}}^{2}+\lVert\Delta\nabla
u\rVert_{L^{2}}^{2}=\int(u\cdot\nabla)u\cdot\Delta u+\lambda\lVert\Delta
u\rVert_{L^{2}}^{2}.$

Now if we integrate by parts, we can get

$\int(u\cdot\nabla)u\cdot\Delta
u=-\sum_{i,j,k=1}^{N}\int\left((\partial_{i}u_{i})u_{j}\partial_{k}^{2}u_{j}+u_{i}u_{j}\partial_{i}\partial_{k}^{2}u_{j}\right)$

or

$\int(u\cdot\nabla)u\cdot\Delta
u=-\sum_{i,j,k=1}^{N}\int\left(\partial_{k}u_{i}\partial_{i}u_{j}\partial_{k}u_{j}-\frac{1}{2}\partial_{i}u_{i}\lvert\partial_{k}u_{j}\rvert^{2}\right).$

Although we have separated the divergence $\nabla\cdot u$ in
$(\partial_{i}u_{i})u_{j}\partial_{k}^{2}u_{j}$ or
$\frac{1}{2}\partial_{i}u_{i}\lvert\partial_{k}u_{j}\rvert^{2}$, this will
require regularity criteria of an additional term besides the divergence due
to $u_{i}u_{j}\partial_{i}\partial_{k}^{2}u_{j}$ or
$\partial_{k}u_{i}\partial_{i}u_{j}\partial_{k}u_{j}$.

In fact, thanks to Theorem 3.1 we can work in $L^{2}(\mathbb{T}^{N})$-norm and
avoid this issue. Indeed, we can restart from (3.4) where we can estimate
$\lambda\lVert\nabla u\rVert_{L^{2}}^{2}\leq\frac{1}{4}\lVert\Delta
u\rVert_{L^{2}}^{2}+C\lVert u\rVert_{L^{2}}^{2}$ identically to (3.10), while

$-\int(u\cdot\nabla)u\cdot u=-\int(u\cdot\nabla)\frac{1}{2}\lvert
u\rvert^{2}=\frac{1}{2}\int(\nabla\cdot u)\lvert u\rvert^{2}$

(note that this term vanishes for the NSE but not for the $N$D KSE when
$N>1$). Now if we estimate

$\int(\nabla\cdot u)\lvert u\rvert^{2}\leq\lVert\nabla\cdot
u\rVert_{L^{\infty}}\lVert u\rVert_{L^{2}}^{2},$

we can get a criterion in terms of $\int_{0}^{T}\lVert\nabla\cdot
u\rVert_{L^{\infty}}d\tau$ analogously to the 3D Euler equations ([3]). But to
make full use of the fourth-order diffusion, we compute similarly to (3.19)

$\displaystyle\frac{1}{2}\int(\nabla\cdot u)\lvert u\rvert^{2}\leq$
$\displaystyle\frac{1}{2}\lVert\nabla\cdot u\rVert_{L^{p}}\lVert
u\rVert_{L^{\frac{2p}{p-1}}}^{2}\lesssim\lVert\nabla\cdot
u\rVert_{L^{p}}\lVert u\rVert_{L^{2}}^{\frac{4p-N}{2p}}\lVert
u\rVert_{H^{2}}^{\frac{N}{2p}}$ (3.21) $\displaystyle\leq$
$\displaystyle\frac{1}{4}\lVert\Delta u\rVert_{L^{2}}^{2}+C(\lVert\nabla\cdot
u\rVert_{L^{p}}^{\frac{4p}{4p-N}}+\lVert\nabla\cdot u\rVert_{L^{p}})\lVert
u\rVert_{L^{2}}^{2}$

by H$\ddot{\mathrm{o}}$lder’s inequality, (3.2a), and Young’s inequality.
Because Theorem 3.1 implies that $L_{T}^{\infty}L_{x}^{2}$-bound leads
immediately to the global well-posedness of the $N$D KSE, we obtain the
following result.

######  Theorem 3.13.

Suppose $\nabla\cdot u\in L_{T}^{r}L_{x}^{p}$ where
$p\in[1,\infty],r\in[1,\frac{4}{4-N}]$ satisfy

$\frac{N}{p}+\frac{2}{r}=2+\frac{N}{2p}.$

Then the $N$D KSE is globally well-posed in $H^{1}(\mathbb{T}^{N})$.

By a similar trick to the proof of Theorem 3.6, we can also obtain the
following result in the 2D case:

######  Proposition 3.14.

Suppose $\partial_{2}u_{2}\in L_{T}^{r}L_{x}^{p}$ where
$p\in[1,\infty],r\in[1,2]$ satisfy

$\frac{2}{p}+\frac{2}{r}=2+\frac{1}{p}.$

Then $u_{1}\in L_{T}^{\infty}L_{x}^{2}\cap L_{T}^{2}H_{x}^{2}$.

###### Proof.

This is almost identical to the previous computation for the criteria of
$\nabla\cdot u$. We restart from (3.12) where $\lambda\lVert\nabla
u_{1}\rVert_{L^{2}}^{2}$ may be handled again identically to (3.10) while
similarly to (3.21)

$\displaystyle-\int(u\cdot\nabla)u_{1}u_{1}=$
$\displaystyle\frac{1}{2}\int\partial_{2}u_{2}(u_{1})^{2}\leq\frac{1}{2}\lVert\partial_{2}u_{2}\rVert_{L^{p}}\lVert
u_{1}\rVert_{L^{\frac{2p}{p-1}}}^{2}$ $\displaystyle\lesssim$
$\displaystyle\lVert\partial_{2}u_{2}\rVert_{L^{p}}\lVert
u_{1}\rVert_{L^{2}}^{\frac{2p-1}{p}}\lVert u_{1}\rVert_{H^{2}}^{\frac{1}{p}}$
(3.22) $\displaystyle\leq$ $\displaystyle\frac{1}{4}\lVert\Delta
u_{1}\rVert_{L^{2}}^{2}+C(\lVert\partial_{2}u_{2}\rVert_{L^{p}}^{\frac{2p}{2p-1}}+\lVert\partial_{2}u_{2}\rVert_{L^{p}})\lVert
u_{1}\rVert_{L^{2}}^{2}.$

Now applying (3.22) to (3.12) completes the proof of Proposition 3.14. ∎

We can raise this regularity immediately as follows:

######  Proposition 3.15.

Suppose $\partial_{2}u_{2}\in L_{T}^{r}L_{x}^{p}$ where
$p\in[1,\infty],r\in[1,2]$ satisfy

$\frac{2}{p}+\frac{2}{r}=2+\frac{1}{p}.$

Then $u_{2}\in L_{T}^{\infty}L_{x}^{2}\cap L_{T}^{2}H_{x}^{2}$.

###### Proof.

We restart from (3.9) again where $\lambda\lVert\nabla
u_{2}\rVert_{L^{2}}^{2}$ may be handled identically to (3.10) while

$\displaystyle-\int(u\cdot\nabla)u_{2}u_{2}=$ $\displaystyle-\int
u_{1}\partial_{1}u_{2}u_{2}\leq\lVert
u_{1}\rVert_{L^{\infty}}\lVert\partial_{1}u_{2}\rVert_{L^{2}}\lVert
u_{2}\rVert_{L^{2}}$ $\displaystyle\lesssim$ $\displaystyle(\lVert\nabla
u_{2}\rVert_{L^{2}}^{2}+\lVert u_{1}\rVert_{L^{\infty}}^{2}\lVert
u_{2}\rVert_{L^{2}}^{2})$ $\displaystyle\lesssim$ $\displaystyle(\lVert
u_{2}\rVert_{L^{2}}\lVert\Delta u_{2}\rVert_{L^{2}}+\lVert
u_{1}\rVert_{H^{2}}^{2}\lVert u_{2}\rVert_{L^{2}}^{2})$ (3.23)
$\displaystyle\leq$ $\displaystyle\frac{1}{4}\lVert\Delta
u_{2}\rVert_{L^{2}}^{2}+C(1+\lVert u_{1}\rVert_{H^{2}}^{2})\lVert
u_{2}\rVert_{L^{2}}^{2}$

by the Gagliardo-Nirenberg inequality as in (3.10), the Sobolev embedding
$H^{2}(\mathbb{T}^{2})\hookrightarrow L^{\infty}(\mathbb{T}^{2})$, and Young’s
inequality. Therefore, applying (3.23) to (3.9) leads to

$\frac{1}{2}\frac{d}{dt}\lVert u_{2}\rVert_{L^{2}}^{2}+\lVert\Delta
u_{2}\rVert_{L^{2}}^{2}\leq\frac{1}{2}\lVert\Delta
u_{2}\rVert_{L^{2}}^{2}+C(1+\lVert u_{1}\rVert_{H^{2}}^{2})\lVert
u_{2}\rVert_{L^{2}}^{2}.$

Now applying Proposition 3.14 completes the proof of Proposition 3.15. ∎

By Propositions 3.14-3.15, we deduce that $u\in L_{T}^{\infty}L_{x}^{2}$ and
thus by Theorem 3.1 we see that the 2D KSE is globally well-posed in
$H^{1}(\mathbb{T}^{2})$ as follows:

######  Theorem 3.16.

Suppose $\partial_{2}u_{2}\in L_{T}^{r}L_{x}^{p}$ where
$p\in[1,\infty],r\in[1,2]$ satisfy

$\frac{2}{p}+\frac{2}{r}=2+\frac{1}{p}.$

Then 2D KSE (1.1) is globally well-posed in $H^{1}(\mathbb{T}^{2})$.

######  Remark 3.17.

We emphasize again that Theorem 3.16 is surprising due to the KSE lacking
divergence-free property and that $2+\frac{1}{p}$ in Theorem 3.16 is same as
the condition in Theorem 3.12; in comparison, “$\frac{3(p+2)}{4p}$ for $p>2$”
in (2.12b) is much smaller than “2” in (2.3b).

Lastly, we aim for $\partial_{2}u_{1}$. It is well-known that in general, such
a criterion is expected to be more difficult than $\partial_{2}u_{2}$ where
the components of $\nabla$ and $u$ match; indeed,
$\frac{p+3}{2p}<\frac{3(p+2)}{4p}$ in (2.12b)-(2.12c). Now if we return again
to (3.4) where we can rely on the same estimates in (3.10) to handle the term
$\lambda\lVert\nabla u\rVert_{L^{2}}^{2}$, then we are only faced with a
nonlinear term of

$\displaystyle\int(u\cdot\nabla)u\cdot
u=-\int\left((u\cdot\nabla)u_{1}u_{1}+(u\cdot\nabla)u_{2}u_{2}\right).$

The first term allows us to separate $\partial_{2}u_{1}$ as follows:

(3.24)
$\displaystyle-\int(u\cdot\nabla)u_{1}u_{1}=-\int\left(u_{1}\partial_{1}u_{1}u_{1}+u_{2}\partial_{2}u_{1}u_{1}\right)=-\int
u_{2}\partial_{2}u_{1}u_{1};$

however, for the second term we are faced with

$\displaystyle\int(u\cdot\nabla)u_{2}u_{2}=-\int\left(u_{1}\partial_{1}u_{2}u_{2}+u_{2}\partial_{2}u_{2}u_{2}\right)=-\int
u_{1}\partial_{1}u_{2}u_{2}.$

We seem to face a significant difficulty here. We can integrate by parts to
deduce $\frac{1}{2}\int\partial_{1}u_{1}u_{2}u_{2}$, but $\partial_{2}u_{1}$
is nowhere to be found. One idea would be to rely on some analogue of
anisotropic Sobolev inequality that has proven to be useful in the theory of
fluid PDEs such as the NSE and the MHD system (e.g., [10, Lemma 1.1]).
However, those results are typically all on the whole space such as
$\mathbb{R}^{2}$ because they rely on the decay at infinity; i.e., while

$f(x)=\int_{-\infty}^{x}\partial_{z}f(z)dz,$

an analogous attempt in $\mathbb{T}^{2}$ gives

$f(x)=\int_{-\pi}^{x}\partial_{z}f(z)dz+f(-\pi).$

Remarkably, there is a trick that seems to be special to the KSE here. Let us
make our argument precise by working on the equation of $\phi$ rather than $u$
and attain a criterion in terms of $\partial_{12}\phi$, informally
$\partial_{1}u_{2}$ as desired (see Theorem 3.20).

###  3.5. Results on $\phi$ and $\partial_{12}\phi$

We emphasize that the following two results focus on (1.2), independently of
(1.1). Let us start with a criterion in terms of $\phi$.

######  Theorem 3.18.

Suppose $\phi\in L_{T}^{r}W_{x}^{m,p}$ where $m\in[0,1)$, and

$\displaystyle\begin{cases}p\in[1,\frac{N}{m}),r\in(2,\frac{4}{m}]&\text{ if
}N=2,m\in(0,1),\\\ p\in(\frac{N}{m+2},\frac{N}{m}),r\in(2,\infty)&\text{ if
}N=3,m\in(0,1),\\\ p\in(\frac{N}{2},\infty],r\in[2,\infty)&\text{ if
}N\in\\{2,3\\},m=0.\end{cases}$

satisfy

$\displaystyle\frac{N}{p}+\frac{2}{r}=\frac{2+m}{2}+\frac{N}{2p}.$

Then the $N$D KSE (1.2) is globally well-posed in $L^{2}(\mathbb{T}^{N})$.

######  Remark 3.19.

Very recently, Feng and Mazzucato [26] showed a blowup criterion of $\phi\in
L_{T}^{\infty}L_{x}^{2}$ for 2D KSE. If $N=2$, we only require
$\frac{2}{p}+\frac{2}{r}\leq 1+\frac{1}{p}$, allowing cases such as
$L_{T}^{\infty}L_{x}^{p}$ for any $p>1$ and $L_{T}^{4}L_{x}^{2}$.

###### Proof.

We take $L^{2}(\mathbb{T}^{N})$-inner products on (1.2) with $\phi$ to deduce

(3.25)
$\frac{1}{2}\frac{d}{dt}\lVert\phi\rVert_{L^{2}}^{2}+\lVert\Delta\phi\rVert_{L^{2}}^{2}=-\lambda\int\Delta\phi\phi-\frac{1}{2}\int\lvert\nabla\phi\rvert^{2}\phi.$

Now

(3.26)
$-\int\Delta\phi\phi\leq\lVert\Delta\phi\rVert_{L^{2}}\lVert\phi\rVert_{L^{2}}\leq\frac{1}{4}\lVert\Delta\phi\rVert_{L^{2}}^{2}+C\lVert\phi\rVert_{L^{2}}^{2}$

by Young’s inequality, while

$\displaystyle-\frac{1}{2}\int\lvert\nabla\phi\rvert^{2}\phi\leq$
$\displaystyle\frac{1}{2}\lVert\phi\rVert_{L^{\frac{Np}{N-mp}}}\lVert\nabla\phi\rVert_{L^{\frac{2Np}{Np+mp-N}}}^{2}$
$\displaystyle\lesssim$
$\displaystyle\lVert\phi\rVert_{W^{m,p}}\lVert\phi\rVert_{L^{2}}^{\frac{(m+2)p-N}{2p}}\lVert\phi\rVert_{H^{2}}^{\frac{(2-m)p+N}{2p}}$
(3.27) $\displaystyle\leq$
$\displaystyle\frac{1}{4}\lVert\Delta\phi\rVert_{L^{2}}^{2}+C\Big{(}\lVert\phi\rVert_{W^{m,p}}^{\frac{4p}{(2+m)p-N}}+\lVert\phi\rVert_{W^{m,p}}\Big{)}\lVert\phi\rVert_{L^{2}}^{2}$

by H$\ddot{\mathrm{o}}$lder’s inequality, Sobolev embedding
$W^{m,p}(\mathbb{T}^{N})\hookrightarrow L^{\frac{Np}{N-mp}}(\mathbb{T}^{N})$
in case $m\in(0,1)$ so that $p<\frac{N}{m}$ by hypothesis, (3.2b), and Young’s
inequality. By applying (3.26)-(3.27) to (3.25), we deduce that $\phi\in
L_{T}^{\infty}L_{x}^{2}\cap L_{T}^{2}H_{x}^{2}$ and thus the desired result. ∎

In the following theorem, we assume $N=2$ and aim for a result analogous to a
criterion of (1.1) in terms of $\partial_{1}u_{2}$ as promised.

######  Theorem 3.20.

Suppose $\partial_{12}\phi\in L_{T}^{r}L_{x}^{p}$ where
$p\in[1,\infty],r\in[1,2]$ satisfy

$\frac{2}{p}+\frac{2}{r}=2+\frac{1}{p}.$

Then 2D KSE (1.2) is globally well-posed in $L^{2}(\mathbb{T}^{2})$.

###### Proof.

Let us consider the equation (1.2), take $L^{2}(\mathbb{T}^{2})$-inner
products with $-\Delta\phi$ so that

$\displaystyle\frac{1}{2}\frac{d}{dt}\lVert\nabla\phi\rVert_{L^{2}}^{2}+\lVert\Delta\nabla\phi\rVert_{L^{2}}^{2}$
$\displaystyle=$
$\displaystyle\lambda\lVert\Delta\phi\rVert_{L^{2}}^{2}-\frac{1}{2}\int\partial_{1}((\partial_{1}\phi)^{2}+(\partial_{2}\phi)^{2})\partial_{1}\phi-\frac{1}{2}\int\partial_{2}((\partial_{1}\phi)^{2}+(\partial_{2}\phi)^{2})\partial_{2}\phi$
(3.28)
$\displaystyle\triangleq\lambda\lVert\Delta\phi\rVert_{L^{2}}^{2}+I+II.$

It is straight-forward to estimate

(3.29)
$\lambda\lVert\Delta\phi\rVert_{L^{2}}^{2}\leq\frac{1}{8}\lVert\Delta\nabla\phi\rVert_{L^{2}}^{2}+C\lVert\nabla\phi\rVert_{L^{2}}^{2}$

by Young’s inequality. Next, we estimate

(3.30) $\displaystyle I=$
$\displaystyle-\int\frac{1}{3}\partial_{1}(\partial_{1}\phi)^{3}+\partial_{2}\phi\partial_{12}\phi\partial_{1}\phi$
$\displaystyle\leq$
$\displaystyle\lVert\partial_{2}\phi\rVert_{L^{\frac{2p}{p-1}}}\lVert\partial_{12}\phi\rVert_{L^{p}}\lVert\partial_{1}\phi\rVert_{L^{\frac{2p}{p-1}}}$
$\displaystyle\lesssim$
$\displaystyle\lVert\partial_{2}\phi\rVert_{L^{2}}^{\frac{2p-1}{2p}}\lVert\partial_{2}\phi\rVert_{H^{2}}^{\frac{1}{2p}}\lVert\partial_{12}\phi\rVert_{L^{p}}\lVert\partial_{1}\phi\rVert_{L^{2}}^{\frac{2p-1}{2p}}\lVert\partial_{1}\phi\rVert_{H^{2}}^{\frac{1}{2p}}$
$\displaystyle\leq$
$\displaystyle\frac{1}{8}(\lVert\partial_{1}\Delta\phi\rVert_{L^{2}}^{2}+\lVert\partial_{2}\Delta\phi\rVert_{L^{2}}^{2})$
$\displaystyle+C(\lVert\partial_{12}\phi\rVert_{L^{p}}+\lVert\partial_{12}\phi\rVert_{L^{p}}^{\frac{4p}{4p-1}}+\lVert\partial_{12}\phi\rVert_{L^{p}}^{\frac{2p}{2p-1}})\lVert\nabla\phi\rVert_{L^{2}}^{2}$

by H$\ddot{\mathrm{o}}$lder’s inequality, (3.2a) and Young’s inequality. Next,
we get a break here as it turns out that we can apply identical estimates in
$I$ to $II$ because

$\displaystyle II=$
$\displaystyle-\frac{1}{2}\int\partial_{2}((\partial_{1}\phi)^{2}+(\partial_{2}\phi)^{2})\partial_{2}\phi$
(3.31) $\displaystyle=$
$\displaystyle-\int[(\partial_{1}\phi)(\partial_{12}\phi)+(\partial_{2}\phi)(\partial_{22}\phi)]\partial_{2}\phi=-\int\partial_{1}\phi\partial_{12}\phi\partial_{2}\phi.$

By applying (3.29)-(3.31) to (3.28), we obtain $\phi\in
L_{T}^{\infty}\dot{H}_{x}^{1}\cap L_{T}^{2}\dot{H}_{x}^{3}$. It is not
difficult to ensure the lower regularity at $L_{T}^{\infty}L_{x}^{2}$ from
such high regularity by going back to (3.25). Indeed, we can compute from
(3.25)

$\displaystyle\frac{1}{2}\frac{d}{dt}\lVert\phi\rVert_{L^{2}}^{2}$
$\displaystyle+\lVert\Delta\phi\rVert_{L^{2}}^{2}\leq\lambda\lVert\nabla\phi\rVert_{L^{2}}^{2}+\frac{1}{2}\lVert\phi\rVert_{L^{2}}\lVert\nabla\phi\rVert_{L^{2}}\lVert\nabla\phi\rVert_{L^{\infty}}$
(3.32) $\displaystyle\lesssim$
$\displaystyle\lVert\phi\rVert_{L^{2}}\lVert\Delta\phi\rVert_{L^{2}}+\lVert\phi\rVert_{L^{2}}\lVert\nabla\phi\rVert_{L^{2}}(\lVert\nabla\phi\rVert_{L^{2}}+\lVert\Delta\nabla\phi\rVert_{L^{2}})$

where we used the same estimate in (3.10) and the Sobolev embedding
$H^{2}(\mathbb{T}^{2})\hookrightarrow L^{\infty}(\mathbb{T}^{2})$. This
estimate (3.32) immediately implies

(3.33)
$\displaystyle\frac{d}{dt}\lVert\phi\rVert_{L^{2}}\lesssim\lVert\nabla\phi\rVert_{L^{2}}+\lVert\Delta\nabla\phi\rVert_{L^{2}}+\lVert\nabla\phi\rVert_{L^{2}}(\lVert\nabla\phi\rVert_{L^{2}}+\lVert\Delta\nabla\phi\rVert_{L^{2}})$

due to Young’s inequality. Integrating (3.33) over $[0,t]$ now and relying on
the fact that $\phi\in L_{T}^{\infty}\dot{H}_{x}^{1}\cap
L_{T}^{2}\dot{H}_{x}^{3}$ complete the proof of Theorem 3.20. ∎

##  4\. Computational Results

In this section, we present some computational results. In particular, we
simulate solutions to the 2D KSE. All simulations are done for the scalar form
of the system (1.2), (except for the simulation used to generate Figure 1.1,
which directly simulated (1.1)), with the slight modification of subtracting
the spatial average at each time step, as is common in simulations of the KSE
(see, e.g., [39]). Note that this modification formally preserves the
differential form of the KSE, system (1.1). In this section, we assume that
$u=\nabla\phi$; that is, here, $u$ is essentially a notation for $\nabla\phi$.

The existing literature on computational studies of the 2D KSE is fairly
limited, but we refer to [39] where an exhaustive computational study of the
KSE was carried out in various rectangular domains. Various modifications to
the 2D KSE have been investigated computationally in several works; see, e.g.,
[72, 50].

Our simulations were computed via MATLAB (R2020a) on the periodic square
domain $[-\pi,\pi)^{2}$, using a uniform rectangular grid $256\times 256$.
Derivatives were computed using spectral methods (also called pseudo-spectral
methods), specifically using Matlab’s fftn and ifftn implementations of the
$N$D discrete Fourier transform, making sure to respect the 2/3’s dealiasing
law for quadratic nonlinearities, as well as the vanishing of the Nyquist
frequency for only odd-order derivatives. Time stepping was carried out using
a 4th-order ETD-RK4 method (see, e.g., [40, 41]) to handle the linear terms
implicitly, while the nonlinear term was computed explicitly and in physical
space. In particular, a complex contour with 64 nodes (technically 32 nodes
plus a symmetry condition) was used to handle the removable singularities in
the ETD-RK4 coefficients, as proposed in [40]. We used a uniform time-step of
$\Delta t=2.1227\times 10^{-5}$, well below the advective CFL given by $\Delta
t<\Delta
x/(\tfrac{1}{2}\|\nabla\phi\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{T}^{2}))})\approx
4.4\times 10^{-5}$. All simulations presented here were well-resolved at each
time step, in the sense that the magnitude of the energy spectrum of $\varphi$
was below machine precision ($\epsilon=2.2204\times 10^{-16}$ in MATLAB) for
all wave numbers above the $2/3$’s dealising cutoff (i.e., for all
$\vec{k}\in\mathbb{Z}^{2}$ with $|\vec{k}|\geq 256/3$). This can be seen in
the plots of the spectrum at each time step in Figure 4.1. We used roughly the
largest $\lambda$ possible at this resolution ($256^{2}$) while maintaining
well-resolvedness; namely $\lambda=29.1$ (found experimentally), meaning that
96 modes were unstable (not counting the zero mode, which was set to zero);
that is,
$\\#\\{\vec{k}\in\mathbb{Z}^{2}\setminus\\{\vec{0}\\}:|\vec{k}|^{2}\triangleq
k_{1}^{2}+k_{2}^{2}<29.1\\}=96$. However, we observed in tests that the
behavior of the solution presented here was qualitatively similar for a wide
range of $\lambda$ values.

For concreteness, we use the initial data $\phi^{in}$ studied in [39]; namely

(4.1) $\displaystyle\phi^{in}(x,y)=\sin(x+y)+\sin(x)+\sin(y).$

However, we also tested several other choices of initial data, including data
based on randomly generated Fourier amplitudes, and found qualitatively
similar results to those presented here. Thus, for simplicity of presentation,
we only include the results from simulations with initial data (4.1).

Figure 4.1.  The spectrum at time steps $t=0.0,t=0.001,t=0.002,\ldots,10.0.$
(log-log scale), starting in grey and getting darker with increasing time.
Highlighted are the initial spectrum (cyan  $+$), the final ($t=10.0$)
spectrum (red $*$), and the average spectrum (green  $\circ$), (averaged over
$9\leq t\leq 10$).

We computed the $L^{p}$ norms for $1\leq p\leq\infty$ in physical space (using
Riemann sums for $p<\infty$), with the modification of dividing by the area of
the domain; that is, in this section only, unless otherwise noted, we denote

(4.2)
$\displaystyle\|f\|_{L^{p}}^{p}\triangleq\frac{1}{4\pi^{2}}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}|f(x,y)|^{p}\,dx\,dy.$

for $p<\infty$, so that
$\|f\|_{L^{p}}\leq\|f\|_{L^{\infty}}\triangleq\operatorname*{ess\,sup}_{(x,y)\in\mathbb{T}^{2}}|f(x,y)|$.
This is done only for aesthetic purposes so that it is easier to see the plots
on the same figure, see Figures 4.2 and 4.3. From these graphics, it appears
that all the criteria proven in the present work hold. Indeed, it appears that
$\phi$, $\phi_{x}$, $\phi_{y}$, $u_{1}\triangleq\phi_{x}$,
$u_{2}\triangleq\phi_{y}$, $\phi_{xx}$, $\phi_{xy}\equiv\phi_{yx}$,
$\phi_{yy}$, and $\text{div}(u)=\Delta\phi$ are all in
$L^{\infty}([0,T],L^{\infty})$ for all $T>0$, at least for the initial data
and $\lambda$-values we tested. Thus, in these tests, we do not see evidence
for ill-posedness.

Figure 4.2.  $L^{p}$ norms of $\phi$ (left), $\phi_{x}$ (middle) and
$\phi_{y}$ (right). Note that $u_{1}=\phi_{x}$, and $u_{2}=\phi_{y}$.
($p=1,\tfrac{3}{2},2,3,4,5,6,\infty.$)

Figure 4.3.  $L^{p}$ norms of $\phi_{xx}$ (top left), $\phi_{xy}$ (top right)
and $\phi_{yy}$ (bottom left), and $\Delta\phi$ (bottom right). Note that
$\text{div}(u)=\Delta\phi$. ($p=1,\tfrac{3}{2},2,3,4,5,6,\infty.$)

In the context of the 2D NSE, it is common to look for regions in, e.g., the
energy-enstrophy plane or enstrophy-palenstrophy plane, etc., where the
attractor lies (see, e.g., [11, 20, 21, 22, 23, 24]). In the same spirit, we
consider analogous planes combining $\|\varphi\|_{L}^{2}$ (“Energy”),
$\|\nabla\varphi\|_{L}^{2}$ (“Enstrophy”), and $\|\Delta\varphi\|_{L}^{2}$
(“Palenstrophy”), where we have borrowed the names for these quantities from
the NSE setting even though they have different physical meanings in the
context of the KSE. In Figure 4.4, these plots are displayed. Here, we see
that the large-time dynamics are constrained to a relatively small region of
these planes with a roughly elliptical shape.

Figure 4.4.  Plots of combinations of “Energy” $\|\varphi\|_{L}^{2}$,
“Enstrophy” $\|\nabla\varphi\|_{L}^{2}$ and “Palenstrophy”
$\|\Delta\varphi\|_{L}^{2}$ for times $0\leq t\leq 3$. At $t=0$, the solutions
are close to the origin, and move roughly northeast until reaching a region in
the northeast where they tend to remain, moving about chaotically.

We also display some quantities that occur in standard energy estimates. For
example, in taking the inner product with $\phi$ in (1.2) as in (3.25), the
nonlinear term $|\nabla\phi|^{2}/2$ (bottom left of Figure 4.6) yields the
term $\frac{1}{2}\int_{\Omega}\phi|\nabla\phi|^{2}\,dx$ (Figure 4.5). If this
term happened to be positive, or even if it had a positive integral over
aribitrary time intervals of the form $[0,T]$, $T>0$ this would be sufficient
to obtain a bound on the $L^{2}$ norm on finite time intervals, and hence a
proof of global well-posedness of the KSE. Our numerical tests in Figure 4.5
(left) show that this term is typically positive (at least, for the
simulations we performed), but can intermittantly become negative for small
windows of time. In Figure 4.6 (bottom middle), the integrand of this term,
$\phi|\nabla\phi|^{2}$ can also be seen at time $t=10.0$. Here, we see that
the integrand is mostly positive in space, but has regions where it is
strongly negative. A deeper understanding of these negative regions may be
important to understanding the well-posedness of the KSE. Note also that,
strictly speaking, the positivity of
$(\phi,|\nabla\phi|^{2})+\|\Delta\phi\|_{L^{2}}^{2}$ would be sufficient to
bound the $L^{2}$ norm, as from (3.25), one has

$\displaystyle\frac{1}{2}\frac{d}{dt}\lVert\phi\rVert_{L^{2}}^{2}+\tfrac{1}{2}(\lvert\nabla\phi\rvert^{2},\phi)+\lVert\Delta\phi\rVert_{L^{2}}^{2}=-\lambda(\Delta\phi,\phi)\leq\tfrac{1}{2}\|\Delta\phi\|_{L^{2}}^{2}+\tfrac{\lambda^{2}}{2}\|\phi\|_{L^{2}}^{2},$

so that

$\displaystyle\frac{d}{dt}\lVert\phi\rVert_{L^{2}}^{2}+(\lvert\nabla\phi\rvert^{2},\phi)+\lVert\Delta\phi\rVert_{L^{2}}^{2}\leq\lambda^{2}\|\phi\|_{L^{2}}^{2}.$

A nearly identical calculation yields

$\displaystyle\frac{d}{dt}\lVert u\rVert_{L^{2}}^{2}+2(u\cdot\nabla
u,u)+\lVert\Delta u\rVert_{L^{2}}^{2}\leq\lambda^{2}\|u\|_{L^{2}}^{2}.$

Thus, in Figure 4.5 we also display
$(\phi,|\nabla\phi|^{2})+\|\Delta\phi\|_{L^{2}}^{2}$ and $2(u\cdot\nabla
u,u)+\lVert\Delta u\rVert_{L^{2}}^{2}$. In the figure, we see that the terms
coming quantities coming from the non-linear terms are mostly positive, but
occasionally become negative. However, after adding the norm of the Laplacian,
the quantity remains positive for all time. Of course, if one could prove that
such positivity holds for general initial data, one could prove global well-
posedness.

Figure 4.5.  Quantities of interest in energy estimates:
$\tfrac{1}{2}(\phi,\lvert\nabla\phi\rvert^{2})$ (top left),
$(\phi,|\nabla\phi|^{2})+\|\Delta\phi\|_{L^{2}}^{2}$ (top right),
$(u\cdot\nabla u,u)=-\frac{1}{2}(\Delta\phi,|\nabla\phi|^{2})$ (bottom left),
and $2(u\cdot\nabla u,u)+\|\Delta u\|_{L^{2}}^{2}$ (bottom right). While the
quantities in the left column have intermittent negativity, in our
simulations, the quantities in the right column were positive for all $t\geq
0$. (Only $0\leq t\leq 1$ displayed (for visibility), but similar behavior was
observed out to $t=10$. Norms were not normalized as in (4.2).)

Similar observations can be made at the level of the $u$ equation (1.1). In
particular, regarding the cubic term in (3.4), we notice

(4.3) $\displaystyle(u\cdot\nabla
u,u)=\frac{1}{2}\int_{\Omega}u\cdot\nabla|u|^{2}\,dx=-\frac{1}{2}\int_{\Omega}(\nabla\cdot
u)|u|^{2}\,dx=-\frac{1}{2}\int_{\Omega}\Delta\phi|\nabla\phi|^{2}\,dx.$

Note that, according to (3.4), positivity of this term would immediately imply
$u\in L^{\infty}(0,T;L^{2})$, and hence global existence for the KSE. Thus, we
also display $\Delta\phi|\nabla\phi|^{2}/2$ in Figure 4.6 (bottom right),
where we make the perhaps unsurprising observation that integrand in (4.3) is
not of a single sign; however, it is surprising that integrand evidently has
positive values that are far greater than the magnitude of the negative
values. Moreover, the regions where the integrand is negative seem to be
concentrated in small, relatively thin pockets, while the positive regions
dominate the domain. Indeed, in simulations negative values of $(u\cdot\nabla
u,u)$ were observed only for short intervals of time (no larger than roughly
$[t,t+0.003]$), but after a short time (roughly $t>0.08$), $(u\cdot\nabla
u,u)$ remained positive. This phenomenon was observed repeatedly in
simulations, indicating that one possible route to proving global well-
posedness of (1.1) might be to show that the negative part of
$\Delta\phi|\nabla\phi|^{2}/2$ remain sufficiently small in comparison to its
positive part.

Figure 4.6.  Solution $\phi$ to (1.2) at time $t=10.0$, and various quantities
derived from $\phi$. Here, $\Delta=\partial_{xx}+\partial_{yy}$. Note that in
terms of $u\triangleq\nabla\phi$, one has $\text{div}(u)=\Delta\phi$.

##  5\. Appendix

Here, we leave details concerning the local well-posedness of the KSE (1.1) in
$H^{1}(\mathbb{T}^{N})$ for $N\in\\{2,3\\}$, formally stated in Theorem 2.2.
The proof follows the argument in [50], which in turn follows [51]; we only
sketch the main steps. We consider a Galerkin approximation with $P_{n}$ being
the projection onto the Fourier modes of order up to
$n\in\mathbb{N}\cup\\{0\\}$:

$P_{n}u(x)\triangleq\sum_{\lvert k\rvert\leq n}\hat{u}(k)e^{ix\cdot k}.$

We consider the following Galerkin approximation system

(5.1a)
$\displaystyle\partial_{t}u^{n}+P_{n}((u^{n}\cdot\nabla)u^{n})=-\lambda\Delta
u^{n}-\Delta^{2}u^{n},$ (5.1b) $\displaystyle u^{n}(\cdot,0)\triangleq
P_{n}u^{in}(\cdot).$

Relying on [51, Theorem 3.1] we can deduce the following statement:

######  Proposition 5.1.

Given initial data $u^{in}\in H^{1}(\mathbb{T}^{N})$, there exists $T=T(\lVert
u^{in}\rVert_{H^{1}})>0$ such that (5.1a)-(5.1b) has a solution

(5.2) $u^{n}\in L^{\infty}([0,T];H^{1}(\mathbb{T}^{N}))\cap
L^{2}([0,T];H^{3}(\mathbb{T}^{N}));$

additionally, $\partial_{t}u^{n}\in L^{2}([0,T];H^{-1}(\mathbb{T}^{N}))$.
Moreover, these bounds are independent of $n$. Finally, if $T^{\ast}$ is the
maximal existence time and $T^{\ast}<\infty$, then $\limsup_{t\to
T^{\ast}}\lVert u^{n}(t)\rVert_{H^{1}}=+\infty$.

###### Proof.

By [51, Theorem 3.1], it can be immediately shown that given $u^{in}\in
H^{1}(\mathbb{T}^{N})$, there exists a unique solution $u^{n}\in
C^{1}([0,T_{n}),H^{1}(\mathbb{T}^{N})\cap O^{M})$ for some $T_{n}>0$ and
$O^{M}\triangleq\\{f\in H^{1}(\mathbb{T}^{N}):\lVert f\rVert_{H^{1}}\leq
M\\}$. Now we can take $L^{2}(\mathbb{T}^{N})$-inner products on (5.1a) with
$u$ and then with $-\Delta u^{n}$, sum the resulting equations and prove that
there exists a constant $c\geq 0$ such that

$\frac{d}{dt}\lVert u^{n}\rVert_{H^{1}}\leq c(1+\lVert
u^{n}\rVert_{H^{1}})^{3}$

We can fix such a constant $c>0$ and deduce that $H^{1}(\mathbb{T}^{N})$-norm
does not blow up for all $t<T^{\ast}\triangleq\frac{1}{2c(1+\lVert
u^{in}\rVert_{H^{1}})^{2}}$. Hence, $T_{n}>T\triangleq\frac{T^{\ast}}{2}$ for
all $n\in\mathbb{N}$ and $u^{n}$ has the regularity of (5.2). This regularity
leads to to
$\int_{0}^{T}\lVert\partial_{t}u^{n}\rVert_{H^{-1}}^{2}d\tau\lesssim 1$ by
(1.1) and a standard argument (see [51, Cor. 3.2]) completes the proof of
Proposition 5.1. ∎

From the regularity of the solution to the Galerkin approximation due to
Proposition 5.1, we can deduce the following convergence results by standard
compactness lemma (e.g., [63, Theorem 5], [64, Lem. 4]): weak∗ in
$L^{\infty}(0,T;H^{1}(\mathbb{T}^{N}))$; weak in
$L^{2}(0,T;H^{3}(\mathbb{T}^{N}))$; strong in
$L^{2}(0,T;H^{s}(\mathbb{T}^{N}))$ for $s\in[2,3)$ and
$C([0,T];H^{s}(\mathbb{T}^{N}))$ for $s\in[0,1)$. Thereafter, verifying that
the limiting solution indeed solves the KSE and proving its uniqueness is
standard. We refer to, e.g., [50] for details.

## Acknowledgments

Author A.L. was partially supported by NSF grant CMMI-1953346. We wish to
thank the anonymous reviewers for valuable comments that have improved our
manuscript significantly.

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