Abstract
We consider the Kuramoto–Sivashinsky equation (KSE) on the twodimensional torus in the presence of advection by a given background shear flow. Under the assumption that the shear has a finite number of critical points and there are linearly growing modes only in the direction of the shear, we prove global existence of solutions with data in \(L^2\), using a bootstrap argument. The initial data can be taken arbitrarily large.
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1 Introduction
In this article we consider the Kuramoto–Sivashinsky equation (KSE) in twospace dimension in the presence of advection by a given background shear flow. The KSE is a wellknown model of largescale instabilities, such as those arising in flamefront propagation (see e.g. [26] and references therein).
The KSE comes in a scalar, potential form, and a differentiated, vectorial form. We will confine ourselves to the scalar form, since the addition of a linear transport term is meaningful for the potential:
We solve this equation with periodic boundary conditions on \([0, L_1]\times [0, L_2]\), that is, on a twodimensional torus, which with slight abuse of notation we denote by \({\mathbb {T}}^2\). When \(L_1>2\pi \) or \(L_2>2\pi \), the symbol of the linear operator \(\Delta ^2+\Delta \) is negative on a finite set of low frequencies. Hence, there are (linearly) growing modes in the horizontal or vertical direction, respectively.
Since the Kuramoto–Sivashinsky equation describe the motion of a (plane) frame front [31, 37], one and two space dimensions are the most physically relevant for the problem. For the same reason, taking periodic boundary conditions is physically motivated, even though other geometries, such as that of a channel, are of interest. We confine ourselves to treating periodic boundary conditions.
We consider a modified version of (1), where the potential \(\phi \) is subject to advection by a given steady shear flow, which we write without loss of generality as the horizontal shear \({{\mathbf {v}}}=(u(y),0)\):
where the parameter \(A>0\) represents the amplitude of the flow. As we explain more precisely below, we are interested in considering shear flows which are relaxation enhancing [11], for instance, a sufficient condition [12] is that u has a finite number of critical points. The KSE with general advection term has been utilized in models of turbulent premixedcombustion [15]. By a change of time, the above equation can be rewritten in an equivalent way as
where \(\nu =A^{1}\) and, with slight abuse of notation, we have not relabeled the transformed variables. Because \(\nu \) determines the strength of the dissipation, we will refer to \(\nu \) as a viscosity coefficient. We will refer to the equation above as AKSE.
The main difficulty in dealing with both (1) and (2) is the lack of a priori norm estimates on the solution, which does not allow to bootstrap local existence into global existence via a standard continuation argument. The analysis of the KSE in one space dimension is well developed by now, since in one dimension energy estimates lead to a good control on the \(L^2\) norm of the solution [8, 9, 21,22,23,24, 35]. By contrast, there are only a handful of results concerning the wellposedness of the classical KSE (1) in dimension greater than one. Local wellposedness holds in \(L^p\) spaces [7, 27]. Global existence is known only under fairly restrictive assumptions, such as for thin domains and for the anisotropically reduced KSE [6, 30, 32, 36], without growing modes [1, 17], or with only one growing mode in each direction [2], for small data. The attractor and determining modes were studied in [34], under a uniform bound on higher Sobolev norms that yields global existence of solutions.
In [17], two of the authors proved global existence for AKSE for large data and any number of growing modes, when the advecting velocity field induces a sufficiently small dissipation time, e.g. if the flow is mixing, that leads to a global uniform bound on the \(L^2\) norm of the solution. In this case, the action of the flow is to move energy from large scales to small scales in both directions, where the dissipation can efficiently damp the effect of all the growing modes. We prove in this work that the same result, global existence for large data, holds, if the advecting flow is a shear flow with only isolated critical points and in the absence of growing modes in the direction transverse to the shear, the vertical direction in our set up, which can be achieved by restricting \(0<L_2<2\pi \). The idea of the proof is to exploit the enhanced dissipation arising from the combined action of the hyperdiffusion and the advection to control both the nonlinearity as well as the destabilizing effect of the negative Laplacean at large scale. Intuitively speaking, the shear flow has no influence on purely vertical modes. For instance, the function of y obtained by averaging the solution in the x direction may grow in time. On the other hand, the mixing along streamlines of the flow moves energy from large to small scales. Therefore, the growth generated by growing horizontal modes is damped on a sufficiently large timescale by the dissipation. The nonlinearity then couples all the modes.
For the case at hand of a steady shear flow, the transport operator has a large kernel, namely all the functions on the torus that are constant in the horizontal variable. One needs to project out the kernel to take advantage of the action of the flow. There is no enhanced decay of the energy on the kernel component (at a linear level), but the norm can nevertheless be controlled as they satisfy collectively a modified onedimensional KSE. A key point is to use the fact that the linear operator
is dissipation enhancing [11, 12, 16]. More precisely, for the components of the solution orthogonal to the kernel of the transport operator, it generates an exponentially stable semigroup \(\mathrm{e}^{tH_{\nu }}\) with a rate of decay of the \(L^2\) norm of order \(\lambda _\nu \), where \(\nu /\lambda _\nu \rightarrow 0\) as \(\nu \rightarrow 0\). By contrast, a standard energy estimate shows that the semigroup is contractive with rate \(O(\nu )\). The improved rate in viscosity allows to control both the growing modes as well as the nonlinear terms, provided \(\nu \) is small enough compared to the size of the initial data. Given \(g\in L^2({\mathbb {T}}^2)\), we denote
By Fubini–Tonelli’s Theorem, \(\langle g \rangle \) exists for a.e. y. We observe that \(\langle g\rangle \) corresponds to the projection of g onto the kernel of the advection operator \(u(y) \partial _x\), while \( g_{\ne }\) corresponds to the projection onto the orthogonal complement in \(L^2\).
We define the order of a critical point \(x_0\) of a function u(x) as the smallest \(m\in {\mathbb {N}}\) such that the derivative \(u^{m}(x_0)\ne 0\). A simple critical point has therefore order \(m=2\).
As shown in [4], if u has a finite number of critical points of order at most m, then u is mixing in the sense that for some constant \(C>0\) there holds
for every \(t\ge 0\). Thanks to, [12, Corollary 2.3] this translates into the enhanced dissipation estimate
for some \(\varepsilon _0>0\), independent of \(\nu \), and for every \(t\ge 0\). For AKSE, we use (6) to show that solutions are global, as stated in the next theorem.
Theorem 1.1
Let \(0<L_2<2\pi \) and let \(u:[0,L_2)\rightarrow {\mathbb {R}}\) be a smooth function with a finite number of critical points of order at most m. Then, given \(\phi _0\in L^2({\mathbb {T}}^2)\), there exists \(0<\nu _0<1\) depending on \(L_1,\,L_2, u\), and \(\Vert \phi _0\Vert _{L^2}\) with the following property: for any \(0<\nu <\nu _0\), there exists a unique globalintime weak solution \(\phi \) of (2) with initial data \(\phi _0\) such that \(\phi \in L^\infty ([0,\infty ),L^2)\cap L^2([0,\infty ),H^2)\) \(0<T<\infty \)
We stress that we can allow any number of growing modes in the horizontal direction and the initial data can be arbitrarily large in \(L^2\). The uniqueness of solutions in \(C([0,T],L^2)\) follows similarly to other semilinear parabolic equations (see e.g. [38, Proposition 1.1, page 315]).
The proof is based on a bootstrap argument inspired by [5]. The main steps in this argument are as follows. For any initial data in \(L^2\), there exists a localintime mild solution of (2) on some interval of time \([0,t_0)\), which is also a weak solution in \(L^\infty ([0,t_0),L^2)\cap L^2([0,t_0),H^2)\) and satisfies the energy identity [17]. For \(t_0\) small enough, we can make the \(L^2\) and \(H^2\) norms of the projected component less than a certain multiple of the size of the initial data. By using the stability of the semigroup generated by \(H_\nu \), one then shows that, for \(\nu \) sufficiently small, these norms are in fact half that amount. Hence, the solution can be continued for a longer time than \(t_0\), which allows to bootstrap existence from local to global for the projected component and then conclude using the time evolution of the kernel component of the solution.
As we shall see in Sect. 2, the size of \(\nu _0\) in Theorem 1.1 depends on the rate at which \(\nu /\lambda _\nu \) vanishes as \(\nu \rightarrow 0\). Hence, improving the semigroup estimate (6) automatically implies a better global existence threshold. In Sect. 3, we show that imposing a possibly more restrictive condition on u (see Assumption 3.1), the semigroup bound can be improved. In particular, we consider as a prototypical example the case of
and prove the following result.
Proposition 1.2
Let \(g\in L^2({\mathbb {T}}^2)\), \(0<\nu <1\), and u(y) be given as in (7). There exists \(\varepsilon '_0>0\), independent of \(\nu \), such that
for every \(t\ge 0\).
We observe that \(\ell \) is related to m in (6) in the sense that u in (7) has critical points of order at most \(m=\max \{2,\ell \}\). Hence, a direct comparison between (6) and (8) shows that (8) has a much better decay rate, and in particular \(\nu /\lambda '_\nu \rightarrow 0\) faster as \(\nu \rightarrow 0\).
The derivation of the semigroup estimate (8) is carried out in Sect. 3 via a spectraltheoretic approach. It follows from a general Gearhart–Prüss criterion for maccretive operators devised in [39] based on a quantitative pseudospectral bound. The proof is motivated by that of a similar result for the Laplace operator \(\Delta \) in [20]. For the Laplace operator plus advection, decay rates akin to (8) were obtained in [10, 39] for a shear with infinitely many critical points, using the pseudospectral approach, and for shear flows with finitely many critical points in [4], using hypocoercivity. Such quantitative semigroup estimates are relevant in the investigation of enhanced diffusion for passive scalars [3, 4, 13, 39], in the study of asymptotic stability of particular solutions to the twodimensional Navier–Stokes equations [14, 19, 33, 40], and have also applications to several other nonlinear problems [5, 25, 28, 29].
In Sect. 3, we prove a more general version of Proposition 1.2, namely Proposition 1.1, for shear flows satisfying a certain condition, Assumption 3.1, again inspired by [20]. This condition can be readily verified for u in (7). This is a main reason while we chose it as prototypical example. In fact, by refining the method of proof, we expect an analog of Proposition 1.2 to hold for any shear flow with critical points of order m.
In what follows, C denotes a generic constant that may depend on the domain, i.e., on \(L_1\) and \(L_2\). We utilize standard notation to denote function spaces, e.g. \(H^k({\mathbb {T}}^2)\) is the usual \(L^2\)based Sobolev space. We denote the \(L^2\)norm on the whole \({\mathbb {T}}^2\) as \(\Vert \cdot \Vert _{L^2}\) while \(\Vert \cdot \Vert _{L^2_y}\) is the \(L^2\)norm on the torus in the vertical direction.
Finally, the paper is organized as follows. In Sect. 2, we obtain the bootstrap estimates and prove Theorem 1.1. Then, in Sect. 3, we establish the exponential stability of the semigroup generated by \(H_\nu \) with the improved decay rate, using spectral estimates.
2 Global existence for the KSE with shear
In this section, we establish global existence of solutions of the KSE in the presence of advection by a shear flow with a finite number of critical points. The semigroup estimate (6) allows to control these growing modes through a suitable decomposition of the solution and a bootstrap argument.
2.1 Decomposition of the solution and proof of the main result
In this section, we derive the system of coupled equations that describe the time evolution of the component \(\langle \phi \rangle \) of the solution in the kernel of the transport operator and the time evolution of the component \(\phi _{\ne }\) in the orthogonal complement.
We will refer informally to \(\langle \phi \rangle \) and \(\phi _{\ne }\) as the kernel and projected components, respectively. Then \(\langle \phi \rangle \) satisfies
while \(\phi _{\ne }\) satisfies
We remark that in the equation above the kernel component interacts with the projected ones through the term \({\partial }_y \langle \phi \rangle \). Denoting \(\psi =\partial _y\langle \phi \rangle \) for notational ease, we have
It was proved in [17] that the unique local mild solution to (2) is also a weak solution satisfying the energy identity on the time of existence of the mild solution. In particular, \(\phi _{\ne }\in L^\infty ((0,t_0);L^2({\mathbb {T}}^2))\cap L^2((0,t_0);H^2({\mathbb {T}}^2))\), at least for a sufficiently small time \(t_0>0\). Furthermore, it was shown in [17] that the mild and weak solution persists as long as its \(L^2\) norm is finite, that is, if \(T^*\) is the maximal time of existence of the solution, then
Our goal is to obtain a global bound on the \(L^2\) norm of the solution via a bootstrap argument, from which global existence follows. We will employ both energy estimates as well as semigroup estimates to exploit enhanced dissipation arising from the addition of the advection term on \(\phi _{\ne }\).
Let \({\mathcal {S}}_t\) be the solution operator from 0 to time \(t\ge 0\) for the transporthyperdiffusion equation:
that is, \({\mathcal {S}}_t=\mathrm{e}^{t H_\nu }\). Then \(\phi _{\ne }\) satisfies for \(0\le {\bar{t}}\le t\),
by Duhamel’s principle. We note that the “forcing” term under the integral sign on the righthand side of this equation is well controlled as long as \(\phi _{\ne }\in L^\infty ((0,t_0);L^2({\mathbb {T}}^2))\cap L^2((0,t_0);H^2({\mathbb {T}}^2))\), provided \(\psi \) is also controlled.
Using the decay of \({\mathcal {S}}_t\) on the projected component given by (6), it follows from (12) that, for \(0\le s\le t\),
where in the above estimate we used the fact \(\left\Vert \psi \right\Vert _{L^2_y}\le C\left\Vert \partial ^2_y\psi \right\Vert _{L^2_y}\) by applying Poincaré’s inequality twice (we exploit here that \(\psi \) and, hence, all its derivatives have zero average by definition), and the following GagliardoNirenberg interpolation inequalities:
We next derive some energy estimates that will be needed for the bootstrap argument. Multiplying (10) by \(\phi _{\ne }\) and integrating by part, using the periodic boundary conditions, yields:
We recall the GagliardoNirenberg interpolation inequalities in (14) and
These estimate imply:
where we have integrated by parts in the last term in (15). Applying Young’s inequality, we further get
We also recall that the enhanced diffusion estimate (6) for \(S_t\):
for any \(g\in L^2({{\mathbb {T}}}^2)\) with \(\int _{{{\mathbb {T}}}^1}g(x,y)\,\mathrm{d}x=0\). Above \(\lambda _\nu \) satisfies
In view of (12), the regularity of the mild and weak solution and the continuation principle, for all sufficiently small times \(t\ge s\ge 0\) we can assume that

(H1)
\(\left\Vert \phi _{\ne }(t)\right\Vert _{L^2}\le 8 \mathrm{e}^{\lambda _\nu (ts)/4}\left\Vert \phi _{\ne }(s)\right\Vert _{L^2}\),

(H2)
\(\nu \int _s^t \left\Vert \Delta \phi _{\ne }(\tau )\right\Vert _{L^2}^2\,\mathrm{d}\tau \le 4\left\Vert \phi _{\ne }(s)\right\Vert _{L^2}^2\).
Let \(t_0>0\) be the maximal time such that the estimates above hold on \([0,t_0]\). Following [7], we refer to (H1)–(H2) with \(t\in [0,t_0]\) as the bootstrap assumptions. The next lemma ensures suitable bounds on \(\psi \) once the bootstrap assumptions (H1) and (H2) hold.
Lemma 2.1
Let \(0<L_2<2\pi \). Assume the bootstrap assumptions (H1) and (H2). There exists a \(\nu \)independent constant \(C_1=C_1(\left\Vert \phi _{\ne }(0)\right\Vert _{L^2},\left\Vert \psi (0)\right\Vert _{L^2_y})\), which can be explicitly computed, such that
for all \(t\in [0,t_0]\).
Proof
First from the energy estimate and Poincaré’s inequality, we have
It follows from the GagliardoNirenberg inequality that
Appealing to (14) and the bound above, estimate (23) becomes
We now exploit in a crucial way the hypothesis \(L_2<2\pi \). By Young’s inequality we deduce that
We define an integrating factor \(\mu =\exp \big (C\nu \int _0^t\left\Vert \phi _{\ne }\right\Vert _{L^2}^{2/3}\left\Vert \Delta \phi _{\ne }\right\Vert _{L^2}^2\,\mathrm{d}s\big )\). Then solving (26) gives
where the last inequality follows by the bootstrap assumptions (H1) and (H2). By using (27) in (25), we get (22). \(\square \)
We show below in Sect. 2.2 that, in fact, there exists \(\nu _0>0\) small enough such that, if \(\nu <\nu _0\) and (H1)–(H2) hold on \([0,t_0]\), then for any \(0\le s\le t\le t_0\)

(B1)
\(\left\Vert \phi _{\ne }(t)\right\Vert _{L^2}\le 4 \mathrm{e}^{\lambda _\nu (ts)/4}\left\Vert \phi _{\ne }(s)\right\Vert _{L^2}\),

(B2)
\(\nu \int _s^t \left\Vert \Delta \phi _{\ne }(\tau )\right\Vert _{L^2}^2\,\mathrm{d}\tau \le 2\left\Vert \phi _{\ne }(s)\right\Vert _{L^2}^2\).
We refer to (B1)–(B2) as the bootstrap estimates. Thus, under the assumptions (H1)–(H2), the stronger conclusions (B1)–(B2) hold. As a consequence, (B1)–(B2) hold for all positive times. Assuming temporarily this fact, we proceed with the proof of Theorem 1.1.
Proof of Theorem 1.1
From Lemmas 2.2 and 2.5 and the definition of \(t_0\), we must have \(t_0=\infty \). In particular, \(\phi _{\ne }\in L^\infty ([0,\infty );L^2({\mathbb {T}}^2))\cap L^2([0,\infty );H^2({{\mathbb {T}}}^2))\). By Lemma 2.1, we have \(\psi \in L^\infty ([0,\infty );L^2({{\mathbb {T}}}^1))\cap L^2([0,\infty );H^2({{\mathbb {T}}}^1))\). By the triangle and Poincaré’s inequalities, it then follows \(\Delta \phi \in L^2([0,\infty );L^2({{\mathbb {T}}}^2))\). If we further denote
then from (9) we have
By integrating the above equation and applying estimate (22) and (B2), we obtain \({\bar{\phi }}\in L^\infty ([0,\infty ))\). By Lemma 2.1, it follows that \(\psi \in L^\infty ([0,\infty );L^2({{\mathbb {T}}}^1))\). Hence, (28) and the Poincaré inequality imply that \(\langle \phi \rangle \in L^\infty ([0,\infty );L^2({{\mathbb {T}}}^1))\). As a consequence, \(\phi \in L^\infty ([0,\infty );L^2({{\mathbb {T}}}^2))\). Finally, we note that \(\nabla ^2 \phi =\nabla ^2 \phi _{\ne }+\nabla \psi \) so that \(\phi \in L^2([0,\infty );H^2({{\mathbb {T}}}^2))\). This concludes the proof. \(\square \)
2.2 Bootstrap estimates
It remains to establish the bootstrap estimates (B1)–(B2), which we accomplish through a series of lemmas. We address (B2) first.
Lemma 2.2
Let \(0<L_2<2\pi \). Assume the bootstrap assumptions (H1) and (H2). There exists \(\nu _0=\nu _0(\left\Vert \phi _{\ne }(0)\right\Vert _{L^2})\), explicitly computable, with the following property: for any \(0\le s\le t \le t_0\) and for any \(\nu \le \nu _0\), it holds that
In particular, (B2) holds.
Proof
The bootstrap assumptions, Lemma 2.1 and the energy estimate (20) give
We observe that, since \(\nu /\lambda _\nu \rightarrow 0\) as \(\nu \rightarrow 0\), there exists \(\nu _0\) such that
Hence, by combining the choice \(\nu _0\) with (31) we conclude the proof of the Lemma. \(\square \)
It remains to prove (B1), which we accomplish in different steps. The next lemma states that within a fixed length of time, the quantity \(\left\Vert \phi _{\ne }\right\Vert _{L^2}\) will never grow too fast.
Lemma 2.3
Let \(0<L_2<2\pi \). Assume the bootstrap assumptions (H1) and (H2), and fix \(\tau ^*=4/\lambda _\nu \). For any \(0\le t_1<t_0\), there exists \(\nu _0=\nu _0(\left\Vert \phi _{\ne }(0)\right\Vert _{L^2})\) such that for any \(\nu \le \nu _0\) there holds
for all \(t\in [t_1, t_1+\tau ^*]\cap [0,t_0]\).
Proof
We will assume that \(\nu _0\) is small enough so that Lemma 2.2 applies. Again by Lemma 2.1, the energy estimate (20), and the bootstrap assumption (H1), for some positive \(C_2=C_2(\left\Vert \phi _{\ne }(0)\right\Vert _{L^2},\left\Vert \psi (0)\right\Vert _{L^2_y})\) we have that
Now, we define T(B) as
It is easy to see that \(T(\cdot )\) is a decreasing function and, since by the bootstrap assumption (H1) we have that \(\left\Vert \phi _{\ne }(t_1)\right\Vert _{L^2}\le 8 \left\Vert \phi _{\ne }(0)\right\Vert _{L^2}\), it follows that \(T(\left\Vert \phi _{\ne }(t_1)\right\Vert _{L^2})\ge T(8\left\Vert \phi _{\ne }(0)\right\Vert _{L^2})\). In light of (33), by the definition of T(B) we have that for any \(t\in [t_1, t_1+T(8\left\Vert \phi _{\ne }(0)\right\Vert _{L^2})]\cap [0,t_0]\), the estimate \(\left\Vert \phi _{\ne }(t)\right\Vert _{L^2}\le \sqrt{2}\left\Vert \phi _{\ne }(t_1)\right\Vert _{L^2}\) holds. The lemma is now proved if we choose \(\tau ^*\le T(8\left\Vert \phi _{\ne }(0)\right\Vert _{L^2})\), which is equivalent to asking for
Finally, \(\nu /\lambda _\nu \rightarrow 0\) as \(\nu \rightarrow 0\), so it is enough to satisfy (35) for \(\nu _0\). This concludes the proof. \(\square \)
The next lemma shows that by, selecting \(\nu \) sufficiently small, a constant fraction of \(\left\Vert \phi _{\ne }\right\Vert _{L^2}\) decays after a fixed length of time.
Lemma 2.4
Let \(0<L_2<2\pi \). Assume the bootstrap assumptions (H1) and (H2), and fix again \(\tau ^*=4/\lambda _\nu \). If \(t_0\ge \tau ^*\) then there exists \(\nu _0=\nu _0(\left\Vert \phi _{\ne }(0)\right\Vert _{L^2})\), explicitly computable, with the following property: for any \(s\in [0,t_0\tau ^*]\) and for any \(\nu \le \nu _0\),
Proof
In the course of this proof, we assume that \(\nu _0\) is small enough so that Lemma 2.2 can be applied. By the definition of \(\tau ^*\), we have
Using this inequality in (13) yields
Using the bootstrap assumptions (H1)–(H2) and Lemma 2.1, it then follows that
where we used the fact that \(\nu \tau ^*\ll 1\) when \(\nu _0 \) is small enough. By further restricting \(\nu _0\) so that
the desired result follows from (39). \(\square \)
Now we are ready to show that the bootstrap assumption (H1) can be refined.
Lemma 2.5
Let \(0<L_2<2\pi \). Assume the bootstrap assumptions (H1) and (H2). There exists \(\nu _0=\nu _0(\left\Vert \phi _{\ne }(0)\right\Vert _{L^2})\), explicitly computable, with the following property: for any \(0\le s\le t\le t_0\) and for any \(\nu \le \nu _0\), it holds that
In particular, (B1) holds.
Proof
We fix \(\nu _0 =\nu _0(\left\Vert \phi _{\ne }(0)\right\Vert _{L^2})\) so that all the restrictions in Lemmata 2.2–2.4 are fulfilled. If \(t_0<\tau ^*\) then (B1) directly follows by Lemma 2.3 since \(\sqrt{2}\mathrm{e}<4\). When \(t_0\ge \tau ^*\), by Lemma 2.4, we have
For any \(0\le s\le t\le t_0\), there exists n such that \(t\in [n\tau ^*+s,~(n+1)\tau ^*+s)\). From Lemma 2.3 with \(t_1=n\tau ^*+s\), it follows that
This concludes the proof of the lemma. \(\square \)
3 Semigroup estimates
In this section, we prove Proposition 1.2, namely an improved decay estimate for the semigroup generated by \(H_\nu \) in \(L^2\), under a general condition on the shear velocity profile u.
We denote by \(\mathring{L}^2({\mathbb {T}}^2)\) the closed subspace of \(L^2({\mathbb {T}}^2)\) of elements for which \(\langle g \rangle =0\). By Fubini–Tonelli’s Theorem, such elements are also meanzero on the torus. We will be concerned with the restriction of the operator \(H_\nu \) to \(\mathring{L}^2({\mathbb {T}}^2)\) viewed as an unbounded operator. By slight abuse of notation, we denote the restriction also by \(H_\nu \). It is straightforward to check that the projection onto \(\mathring{L}^2({\mathbb {T}}^2)\) commutes with the semigroup \(\mathrm{e}^{tH_\nu }\) generated by \(H_\nu \).
Let \((X,\Vert \cdot \Vert )\) be a complex Hilbert space and let H be a closed, densely defined operator on X. As shown in [39], if H is an maccretive operator on X, then the decay properties of the semigroup \(\mathrm{e}^{t H}\) can be understood in terms of the following quantity:
is related to the pseudospectral properties of the operator [18]. Following [39], for \(\frac{L_1k}{2\pi }\in {\mathbb {Z}}{\setminus }\{0\}\) and \(\nu \in (0,1]\), we consider the operator \(H_\nu \) localized to the kth Fourier mode in the direction of the shear, namely, the operator
Following the arguments in [39] for the Laplace operator, it can be shown that \(H_{\nu ,k}\) is an maccretive operator on \(L^2({\mathbb {T}}^1)\) with domain \(H^4({\mathbb {T}}^1)\). Here, \(L^2\) is a space of complexvalued functions. Then, as a consequence of [39]*Theorem 1.3,
where \(\Vert \cdot \Vert _\text {op}\) denotes the operator norm. To establish lower bounds on \(\Psi (H_{\nu ,k})\), we assume the following condition on the shear flow.
Assumption 3.1
There exist \(m, N\in {\mathbb {N}}\), \(c_1>0\) and \(\delta _0\in (0,L_2)\) with the property that, for any \(\lambda \in {\mathbb {R}}\) and any \(\delta \in (0,\delta _0)\), there exist \(n\le N\) and points \(y_1,\ldots y_n\in [0,L_2)\) such that
Remark 3.1
Assumption 3.1 is heavily inspired by a similar property of the velocity field associated to the Oseen’s vortex [33]. In [20], Gallay previously observed that the method of proof in [33] can be extended to more general shear flows assuming a condition similar to (47). We observe that a strictly monotone shear flow satisfies Assumption 3.1 with \(m =1\). However, no such flows exists on the torus \({\mathbb {T}}^2\), where any shear flow must have at least a simple critical point, so that we always have \(m\ge 2\). See for instance the example in Proposition 1.2.
The following is the main result of this section.
Proposition 1.1
Let u satisfy Assumption 3.1. Assume \(k\ne 0\) and \(\nu k^{1}\le 1\). There exists a constant \(\varepsilon '_0>0\), independent of \(\nu \) and k, such that
We state next a direct consequence of the theorem.
Corollary 3.1
In the hypotheses of Proposition 1.1, let \(P_k\) denote the \(L^2\) projection onto the kth Fourier mode in the horizontal direction. Then, for every \(t\ge 0\),
In particular, \(H_\nu \) generates an exponentially stable semigroup in \(\mathring{L}^2({\mathbb {T}}^2)\) with rate:
where \(\lambda '_\nu =\varepsilon '_0 \nu ^\frac{m}{m+4}\) for some \(\varepsilon _0'>0\).
Before proving Proposition 1.1, we show that the Assumption 3.1 is satisfied with \(m\ge 2\) for u as in (7).
Example 3.1
We consider the case of \(u(y)=(\sin (y))^\ell \), \(\ell \in {\mathbb {N}}\), defined on \([0,2\pi )\). For a general period \(L_2\), the result follows by a standard rescaling argument. Without loss of generality, we may assume that \(\delta _0>0\) is small enough so that \(\cos (\delta _0)\ge 1/2\). In particular, \(\delta _0\) and hence \(\delta <1\) and, for every \(C\ge 1\),
Given \(\lambda \in {\mathbb {R}}\), we choose the set of points \(y_1,\dots ,y_n\) to be the union of the set of the critical points of u with the set \(u^{1}(\{\lambda \})\) (possibly empty). More precisely, we consider
There are at most \(N=8\) points in Y. Observe that the critical points at \(y=\pi /2, 3\pi /2\) are such that \(u''(\pi /2),u''(3\pi /2)\ne 0\) for any \(\ell \ge 1\). On the other hand, for \(\ell \ge 2\) the critical points at \(y=0,\pi \) are such that \(u^{(j)}(0)=u^{(j)}(\pi )=0\) for every \(j=1,\ldots ,\ell 1\) and \(u^{(\ell )}(0),u^{(\ell )}(\pi )\ne 0\). We denote \(Y=\{y_{i}\}_{i=1}^n\) and we order the points in such a way that \(0\le y_i\le y_{i+1}\le 2\pi \) for \(i=1,\ldots ,N\). We fix \(\delta \in (0,\delta _0)\) and consider different cases.
Case \(\lambda \in [0,1]\). Denote by \(y_\lambda \) the smallest element of \(u^{1}(\{\lambda \})\). Due to the symmetries of u, we know that \(y_\lambda \in [0,\pi /2]\). We consider three situations.

\(0\le y_\lambda <\delta /4\): In this case, thanks to (51), we have
$$\begin{aligned} u(y)\lambda &=(\sin (y))^\ell (\sin (y_\lambda ))^\ell  \ge \sin (\delta )^\ell \sin (y_\lambda )^\ell \nonumber \\&\ge \frac{\delta ^\ell }{2^\ell }  \sin (\delta /4)^\ell \ge \frac{\delta ^\ell }{2^\ell } \frac{\delta ^\ell }{4^\ell }\ge \frac{\delta ^\ell }{2^{\ell +1}}, \end{aligned}$$(53)where we used that \(0\le \sin (y)\le y\).

\(y_\lambda \pi /2<\delta /4\): In this case we have
$$\begin{aligned} u(y)\lambda &=(\sin (y))^\ell (\sin (y_\lambda ))^\ell \ge (\sin (\pi /2\delta ))^\ell (\sin (y_\lambda ))^m \nonumber \\&\ge (\sin (\pi /2\delta )))^\ell (\sin (\pi /2)))^\ell  (\sin (\pi /2)))^\ell (\sin (\pi /2\delta /4)))^\ell . \end{aligned}$$(54)Since \({\partial }_y (\sin (y))^\ell = \ell (\sin (y))^{\ell 1}\cos (y)\) and \({\partial }^2_y (\sin (y))^\ell = \ell (\ell 1)(\sin (y))^{\ell 2}\cos ^2 (y)\ell (\sin (y))^\ell \),
$$\begin{aligned} (\sin (\pi /2\delta )))^\ell (\sin (\pi /2)))^\ell =\left \left( 1\frac{\delta ^2}{2}+O(\delta ^4)\right) ^ ell1\right =\frac{\ell }{2}\delta ^2+O(\delta ^4). \end{aligned}$$(55)so that, by possibly restricting the size of \(\delta _0\) further (depending on \(\ell \)), we also have
$$\begin{aligned} (\sin (\pi /2\delta )))^\ell (\sin (\pi /2)))^\ell \ge \frac{\ell }{4}\delta ^2. \end{aligned}$$(56)On the other hand,
$$\begin{aligned} (\sin (\pi /2)))^\ell (\sin (\pi /2\delta /4)))^\ell \le \frac{\ell }{8}\delta ^2. \end{aligned}$$(57)Consequently,
$$\begin{aligned} u(y)\lambda \ge \frac{\ell }{8}\delta ^2. \end{aligned}$$(58) 
\(y_\lambda \ge \delta /4\) and \(y_\lambda \pi /2\ge \delta /4\): In this case, we have
$$\begin{aligned} u(y)\lambda &=(\sin (y))^\ell (\sin (y_\lambda ))^\ell  \nonumber \\&\ge \min \left\{ (\sin (y_\lambda +\delta /8))^\ell (\sin (y_\lambda ))^\ell ,(\sin (y_\lambda \delta /8))^\ell (\sin (y_\lambda ))^\ell \right\} . \end{aligned}$$(59)We observe that \(y_\lambda +\delta /8<\pi /2\delta /8\) and \(y_\lambda \delta /8>\delta /8\). Then if \(y_\lambda \ge \pi /4\), it follows by the Mean Value Theorem that for some \(\xi \in (y_\lambda ,y_\lambda +\delta /8)\),
$$\begin{aligned} (\sin (y_\lambda +\delta /8))^\ell (\sin (y_\lambda ))^\ell &=\ell (\sin (\xi ))^{\ell 1}\cos (\xi )\frac{\delta }{8} \nonumber \\&\ge \ell (\sin (\pi /4))^{\ell 1}\cos (\pi /2\delta /8)\frac{\delta }{8} \nonumber \\&\ge \frac{\ell }{2^{\frac{\ell 1}{2}}}\sin (\delta /8)\frac{\delta }{8}\ge \frac{\ell }{2^{\frac{\ell 1}{2}}}\frac{\delta ^2}{128}, \end{aligned}$$(60)also using (51). Otherwise, if \(y_\lambda \le \pi /4\), we have
$$\begin{aligned} (\sin (y_\lambda +\delta /8))^\ell (\sin (y_\lambda ))^\ell &=\ell (\sin (\xi ))^{\ell 1}\cos (\xi )\frac{\delta }{8} \nonumber \\&\ge \ell (\sin (\delta /4))^{\ell 1}\cos (\pi /4+\delta /8)\frac{\delta }{8} \nonumber \\&\ge \ell (\sin (\delta /4))^{\ell 1}\cos (\pi /3)\frac{\delta }{8}\ge \frac{\ell }{8^{\ell +1}}\delta ^{\ell }. \end{aligned}$$(61)A lower bound for \((\sin (y_\lambda \delta /8)))^\ell (\sin (y_\lambda )))^\ell \) can be proved in a similar way.
Case \(\lambda >1\). In this case \(u^{1}(\{\lambda \})=\emptyset \). Hence, since \(\mathrm{dist}(y,Y)\ge \delta \), we simply have
where we used (51) in the last inequality.
Case \(\lambda <0\). This case is only relevant for \(\ell \) even, hence \(\ell \ge 2\). In fact, for \(\ell \) odd we proceed as for the case \(\lambda \in [0,1]\) by symmetry. Here, again \(u^{1}(\{\lambda \})=\emptyset \). Hence, since \(\mathrm{dist}(y,Y)\ge \delta \), we have
by (51).
Since \(\delta > \delta ^2\), combining all these estimates together gives that there exists a constant \(c_\ell >0\) such that
as we wanted.
We now turn our attention to the proof of Proposition 1.1.
Proof of Proposition 1.1
The theorem follows by establishing a lower bound on \(\Psi (H_{\nu ,k})\). In the following, \(\Vert \cdot \Vert \) denotes the \(L^2\) norm and \(\langle \cdot ,\cdot \rangle \) denotes the Hermitian inner product in \(L^2\).
We fix \(\lambda \in {\mathbb {R}}\) and pick \(g\in D(H_{\nu ,k})\) with \(\Vert g\Vert =1\). For notational convenience, we set
Let \(\chi :[0,L_2)\rightarrow [1,1]\) be a smooth approximation of \(\mathrm{sign}(u(y)\lambda )\) such that \(\Vert \chi '\Vert _{L^\infty }\le c_2\delta ^{1}\), \(\Vert \chi ''\Vert _{L^\infty }\le c_2\delta ^{2}\), \(\chi (u\lambda )\ge 0\) and
where \(y_j\) are the points in Assumption 3.1. The function \(\chi \) can be constructed via a standard mollification argument. A double integration by parts in y yields the identity
which implies that
On the other hand, we have
so that
In particular, from the properties of the function \(\chi \) and the interpolation inequality \( \Vert {\partial }_yg\Vert ^2\le \Vert \Delta _k g \Vert \Vert g\Vert \), it follows that
We denote
where \(y_j\) are the points in Assumption 3.1. By (47) we have
Utilizing (67) and (70), we find that there exists a positive constant \({\tilde{c}}_2\) such that
On the other hand, since \(E^c\) is of size at most \(N\delta \), we have
where we made use of (67) in the last inequality. Without loss of generality, we can assume that \(\delta _0\) in Assumption 3.1 is small enough so that \(\delta \le L_2/(4N)\) for all \(\delta \in (0,\delta _0)\). Thus we can add (73) and (74) to conclude that
We now take \(\delta \) satisfying
for some sufficiently small constant \(c_3>0\), which is independent of \(\nu ,k\) since \(\nu k^{1}\le 1\). We conclude that
for a large enough constant \(c_4>0\) independent of \(\nu \) and k. By definition then
for some constant \(\varepsilon '_0>0\) independent of \(\nu \) and k, whence proving the proposition. \(\square \)
Remark 3.2
The proof of Proposition 1.1 carries over to the slightly more general case of the semigroup generated by the hypoelliptic operator
for which the same semigroup estimate (50) holds.
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Acknowledgements
The authors thank Tarek Elgindi and Thierry Gallay for insightful discussions.
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On behalf of all authors, the corresponding author states that there is no conflict of interest. Data sharing not applicable to this article as no datasets were generated or analyzed during the current study. A.M. was partially supported by the US National Science Foundation Grant DMS1909103. Part of this work was conducted while A.M. participated remotely in a program hosted by the Mathematical Sciences Research Institute (MSRI) in Berkeley, California, during the Spring 2021 semester. MSRI receives major support from the US National Science Foundation under Grant DMS1928930. M.C.Z. and M.D. acknowledges funding from the Royal Society through a University Research Fellowship (URF/R1/191492)
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Coti Zelati, M., Dolce, M., Feng, Y. et al. Global existence for the twodimensional Kuramoto–Sivashinsky equation with a shear flow. J. Evol. Equ. 21, 5079–5099 (2021). https://doi.org/10.1007/s00028021007529
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DOI: https://doi.org/10.1007/s00028021007529
Keywords
 Two dimension
 Kuramoto–Sivashinsky
 Mixing
 Global existence
 Mild solutions
 Enhanced diffusion
 Diffusion time