Dissipation enhancement of cellular flows in general advection diffusion equations

Let $\mathbb{T}^{d}=[0,1]^{d}$ denotes the $d$ dimensional torus, $L_{0}^{2}$ be the mean-zero subspace of $L^{2}(\mathbb{T}^{d})$ with the inner product $\langle\cdot,\cdot\rangle$, and $u$ be a divergence-free vector field. Then we consider the following equation on $\mathbb{T}^{d}$:

(1.1)

$$\partial_{t}\theta+u\cdot\nabla\theta+\gamma\left(-\Delta\right)^{\alpha}\theta=0,$$

with $\alpha,\gamma>0$, and initial data $\theta(0,x)=\theta_{0}\in L_{0}^{2}$. In particular, for $\alpha=1$, it gives the standard advection-diffusion equation

(1.2)

$$\partial_{t}\theta+u\cdot\nabla\theta-\gamma\Delta\theta=0.$$

For $\alpha=2$, (1.1) becomes the advective hyper-diffusion equation

(1.3)

$$\partial_{t}\theta+u\cdot\nabla\theta+\gamma\Delta^{2}\theta=0.$$

The dissipation enhancement effect has been studied extensively in recent decades. Most of these research is carried out based on (1.2), and can be generalized to (1.1) for $\alpha>1$ without any difficulty. However, the results obtained from the maximal principle or stochastic process associated with (1.2) cannot be generalized to (1.1) trivially (for example [2, 12, 8, 9]). In this paper, we show the dissipation enhancement result obtained via a stochastic method in [8] can be generalized to (1.1) and nonlinear diffusion equations, in which the stochastic structure is lost.

To begin with, we define the dissipation time associate to (1.1).

Recent researches indicate that the flows with small dissipation time can be used to suppress the formation of singularities in the nonlinear PDEs (see [8, 3, 7, 5]). In the second-order differential equations, such as the Keller-Segel equations and the ignition type reaction-diffusion equations, one needs to provide flows such that $\tau_{1}^{*}(u,\gamma)$ is small enough [8]. And for the fourth order differential equations, such as Cahn-Hilliard equation [3], Kuramoto-Sivashinsky equation [7] and thin-film equation [5], one need to provide flows such that $\tau_{2}^{*}$ is sufficiently small (see Section 4.1 for more details). Therefore, it is meaningful to construct flows with arbitrarily small dissipation times for either $\alpha=1$ or $2$.

In the seminal paper [1], the main result interpret that given a time-independent velocity field $u$, $\tau^{*}_{1}(Au,\gamma)$ tends to zero as $A$ tends to infinity if and only if the operator $u\cdot\nabla$ has no eigenfunctions in $H^{1}(\mathbb{T}^{d})$ rather than constants. A typical class of such flows is the weakly mixing flows, for which $u\cdot\nabla$ only has continuous spectrums. The proof of [1] relies on the so-called RAGE theorem, as a consequence, contains no quantitative information on the appearance of a faster time scale induced by $u$. After that, the authors in [6, 16] obtain explicit bounds on $\tau_{1}^{*}(u,\gamma)$, by requiring the mixing rate of a mixing flow $u$, associated to the underlying transport equation (see [13]). Here the intuition is based on the fact that the mixing flows create small-scale structures (or high frequencies). And the faster this process happens, the faster the energy of solution to (1.1) is dissipated and, as a result, the smaller $\tau_{1}^{*}(u,\gamma)$ becomes. However, known examples of such flows are either complicated or not regular enough.

Fortunately, many smooth flows are not mixing but have small dissipation times, thus enough to suppress singularities in nonlinear PDEs. For such flows, the dissipation times do not tend to zero as one increases the amplitude of the flow. One typical class of such flows is cellular flows (see [8, 10] for applications). A prototypical example of cellular flow in two dimensions is given by

(1.5)

$$u(x)=\nabla^{\perp}\sin(2\pi x_{1})\sin(2\pi x_{2}).$$

Similar to the example above, all two-dimensional cellular flows have closed trajectories and hence are not mixing. In [8], based on equation (1.2), the authors proved that the dissipation time, $\tau^{*}_{1}$, of such flows could be made arbitrarily small by rescaling the spacial scale and the flow amplitude at the same time. Their proof is based on establishing the relationship between the dissipation time $\tau_{1}(u,1)$ and the effective diffusivity, denote as $D(u)$ (see Section 2 for a more precise definition). For convenience to the reader, we cite their theorem here.

Both the proofs in [8] and [9] rely on the stochastic processes associated to equation (1.2), and therefore cannot be generalized to (1.1) directly. On the other hand, in [3] the authors derived an upper bound of $\tau_{2}^{*}(u,\gamma)$ in terms of $\tau_{1}^{*}(u,\gamma)$ for any smooth divergence-free vector field. We cite their result here.

Once velocity fields with small $\tau_{1}^{*}(u,\gamma)$ are known, one can use Lemma 1.4 to produce velocity fields for which $\tau_{2}^{*}(u,\gamma)$ is small. For instance, if $u$ is mixing at a sufficiently fast rate, then the results of Wei[15], Coti Zelati et al.[16], Feng and Iyer[6] along with Lemma 1.4 can be used to produce velocity fields for which $\tau_{2}^{*}(u,\gamma)$ is arbitrarily small. However, Lemma 1.4 is not tight enough to produce arbitrary small $\tau_{2}^{*}(u,\gamma)$ for cellular flows. Indeed, with the $\tau_{1}^{*}$ bound of cellular flows in [8] and [9], the right-hand side of (1.7) blows up as $\lVert u\rVert_{C^{2}}$ increases.

There are two main contributions of this paper. Firstly, we show that cellular flows indeed can be used to produce $\tau_{\alpha}^{*}(u,\gamma)$ arbitrarily small for any $\alpha\geqslant 1$, that is Theorem 1.2 remains true if we replace $\tau_{1}^{*}$ by $\tau_{\alpha}^{*}$ with $\alpha>1$, the formal conclusion is established in Theorem 3.1. As a byproduct, we also show that cellular flows can be used to enhance dissipation in nonlinear diffusion equations, such as porous medium equations and p-Laplacian equations (see Theorem 4.7 and Theorem 4.7). In the second part of this paper, we prove that there exist smooth incompressible flows that are relaxation enhancing with respect to the advective hyper-diffusion equation but not to the standard advection-diffusion equation (Proposition 5.2).

We devote this section to introduce the notations and establish some elementary estimates.

For a function $f(x)$ in defined in $\mathbb{T}^{d}$, we define the homogeneous $L^{2}$-Sobolev norm $\dot{H}^{s}$ ($s\neq 0$) to be

$$\|f\|_{\dot{H}^{s}}=\left(\sum\limits_{|n|\neq 0}|\hat{f}(n)|^{2}|n|^{2s}\right)^{\frac{1}{2}},$$

where $\hat{f}(n)$ is the $n^{\text{th}}$ Fourier coefficient of $f$.

We say that a flow $u$ is symmetric in $x_{n}$ if we have

(2.1)

$$u(R_{n}x)=R_{n}u(x)\qquad\text{for all }x\in\mathbb{T}^{d}.$$

Here $R_{n}v\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}v-2v_{n}e_{n}$ for $v=\sum\limits_{n=1}^{d}v_{n}e_{n}\in\mathbb{R}^{d}$ is the reflection in the $n^{\text{th}}$ coordinate. A periodic flow that is symmetric in all $d$ coordinates is said to have a cellular structure and we call it a cellular flow.

Consider (1.2) on $\mathbb{R}^{d}$, with initial data $\theta_{0}$, and let $u$ be time-independent, mean-zero, divergence-free, and Lipschitz continuous vector fields. For simplicity, we assume $\gamma=1$ from now on. Define the stochastic process $X_{t}^{x}(\omega)$:

(2.2)

$$dX_{t}^{x}=\sqrt{2}dB_{t}-u(X_{t}^{x})dt,\quad X_{0}^{x}=x,$$

where $B_{t}$ is a normalized Brownian motion on $\mathbb{R}^{d}$ with $B_{0}=0$, defined on some probability space $(\Omega,\mathcal{B}_{\infty},\mathbb{P}_{\Omega})$. Let $G_{t}(x,y)$ denotes the fundamental solution to (1.2) and $\mathbb{E}_{\Omega}$ the expectation with respect to $\omega\in\Omega$, then the corresponding solution to (1.2) can be written as $\theta(t,x)=\int_{\mathbb{R}^{d}}G_{t}(x,y)\theta_{0}(y)dy=\mathbb{E}_{\Omega}(\theta_{0}(X_{t}^{x}))$. Then we define the effective diffusivity of $u$ in the direction $e$ by

(2.3)

$$D_{e}(u)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\lim_{t\rightarrow\infty}\mathbb{E}_{\Omega}\left(\frac{\lvert\left(X_{t}^{x}-x\right)\cdot e\rvert^{2}}{2t}\right)\qquad(\geqslant 1),$$

with the limit being independent of $x\in\mathbb{R}^{d}$. In addition, let

(2.4)

$$D(u)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\min\left\{D_{e_{1}}(u),\dots,D_{e_{d}}(u)\right\}\geqslant 1$$

denotes the minimum of effective diffusivities in all the coordinate directions (see [17] for more details). Intuitively, $D_{e}(u)$ is the average distance a point can go along with the process (2.2).

Let $\mathcal{L}_{m}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}iu(mx)\cdot\nabla$ with $u$ divergence free, note that $\mathcal{L}_{m}$ is a self-adjoint operator with respect to $\langle\cdot,\cdot\rangle$. Let $P_{c}$ and $P_{p}$ be the spectral projection on its continuous and discrete spectral subspace correspondingly. On the other hand, we denote by $0<\lambda_{1}\leqslant\lambda_{2}\leqslant\dots$ the eigenvalues of the operator $-\Delta$ on the torus, and $P_{N}$ be the orthogonal projection on the subspace spanned by its first $N$ eigen-modes, and by $S=\left\{\theta\in L^{2}:\lVert\theta\rVert_{L^{2}}=1\right\}$ the unit sphere in $L^{2}$. For the convection field with an amplitude $A>0$, equation (1.1) can be represented in the following way:

(2.5)

$$\frac{d}{dt}\theta_{m}^{A}(t)=iA\mathcal{L}_{m}\theta_{m}^{A}(t)-(-\Delta)^{\alpha}\theta_{m}^{A}(t),$$

with initial date $\theta_{m}^{A}(0)=\theta_{0}$. Here we assume $\alpha\geqslant 1$. For convenient, we rescale (2.5) in time by the factor $\varepsilon^{-1}=A$, thus we obtain the following equivalent reformulation:

(2.6)

$$\frac{d}{dt}\theta_{m}^{\varepsilon}(t)=i\mathcal{L}_{m}\theta_{m}^{\varepsilon}(t)-\varepsilon(-\Delta)^{\alpha}\theta_{m}^{\varepsilon}(t),\qquad\theta_{m}^{\varepsilon}(0)=\theta_{0}.$$

Multiplying $\theta^{\varepsilon}_{m}(t)$ and integration by parts in $x$, one can easily get

(2.7)

$$\frac{d}{dt}\lVert\theta_{m}^{\varepsilon}\rVert_{L^{2}}^{2}=-2\varepsilon\lVert\theta_{m}^{\varepsilon}\rVert_{\dot{H}^{\alpha}}^{2},$$

and we have the following lemma on the decay of $\|\theta_{m}^{\varepsilon}\|_{L^{2}}$.

We also need an estimate on the growth of the difference between the viscous and inviscid problem in terms of $L^{2}$ norm. First notice that for any smooth incompressible flow $u(mx)$, and define $\mathcal{L}_{m}$ as before, then for any $\phi\in\dot{H}^{\alpha}$, $\alpha\geqslant 1$ and $t>0$ the following estimates hold:

(2.9)

$$\lVert\mathcal{L}_{m}\phi\rVert_{L^{2}}\leqslant C\|u(m\cdot)\|_{L^{\infty}}\lVert\phi\rVert_{\dot{H}^{\alpha}}\quad\text{and}\quad\lVert e^{i\mathcal{L}_{m}t}\phi\rVert_{\dot{H}^{\alpha}}\leqslant F_{m}(t)\lVert\phi\rVert_{\dot{H}^{\alpha}}$$

with both the constant $C$ and the function $F_{m}(t)<\infty$ independent of $\phi$ and $F_{m}(t)\in L_{\text{loc}}^{2}(0,\infty)$. Here $F_{m}(t)$ depends on $\|u(m\cdot)\|_{H^{\alpha}}$.

The proofs of Lemma 2.1 and Lemma 2.2 are standard, the reader can find them on [1].

To prove the main theorem, we first establish and prove some lemmas.

First, let us consider the eigenfunctions of $\mathcal{L}_{m}$. We will show that the eigenfunctions of $\mathcal{L}_{m}$ will have big $\dot{H}^{\alpha}$ norm. In fact, for the cellular flow in dimension two, the eigenfunction always exists and is of class $H^{\alpha}$. However, for the sake of simplicity we ignore the existence discussion here.

Let $\phi_{m}^{0}$ be any normalized ($\lVert\phi_{m}^{0}\rVert_{L^{2}}=1$) eigenfunction of $\mathcal{L}_{m}$ that belongs to $\dot{H}^{\alpha}$ (with $\alpha\geqslant 1$) associated with eigenvalue $E_{m}$. Then the following lemma shows that for fixed sufficiently large $m$, the normalized point spectrums of $\mathcal{L}_{m}$ have an uniform lower bound in terms of the $\dot{H}^{\alpha}$ norm, otherwise it contradicts with the result for $\alpha=\gamma=1$ proved in [8].

With the help of Lemma 3.5, we can further control from below the growth of $\dot{H}^{\alpha}$ norm of solutions, to the underlying inviscid problem, corresponding to discrete eigenfunctions.

On the other hand, for the continuous spectrum of $\mathcal{L}_{m}$ we can control it by the following RAGE type theorem.

The proof of Lemma 3.7 can be found in [1].

Now, we turn to the main theorem. We may decomposite the proof into two parts:

(1). For the iniviscid problem, by a decomposition of the initial value $\theta_{0}$ into continuous spectrum and discrete spectrum of $\mathcal{L}_{m}$, one can use Lemma 3.6 and 3.7 lower bound of high frequencies in the sense of time average.

(2). By Lemma 3.5 and rescaling of time, one can control the difference of the solutions to the inviscid problem and the viscid problem, and based on Lemma 2.1 to get the desired decay estimate.

This argument is similar to the proof of the Theorem 1.4 in [1]. For completion, we provide the whole proof in Appendix.

In this chapter, we will construct a relaxation enhancing flow for hyper-diffusion but not for standard diffusion.

To begin with, we recall that a number $\alpha\in\mathbb{R}$ is called $\beta$-Diophatine if there exists a constant $C$ such that for each $k\in\mathbb{Z}\setminus\{0\}$ we have

$$\inf_{p\in\mathbb{Z}}|\alpha\cdot k+p|\geqslant\frac{C}{|k|^{1+\beta}}.$$

And the number $\alpha$ is Liouvillean if it is not Diophantine for any $\beta>0$. More specifically, for any $n(>1)$ and constant $C>0$, one can find $q_{n}\in\mathbb{Z}\setminus\{0\}$ and $p_{n}\in\mathbb{Z}$, such that

(5.1)

$$|\alpha\cdot q_{n}+p_{n}|\leqslant\frac{C}{|q_{n}|^{n}}.$$

One can understand the Liouvillean numbers are the ones which can be very well approximated by rationals.

On the other hand, we denote by $\Phi^{u}_{t}$ the flow on the torus generated by $u$, and by $U^{t}$ the evolution operator on $L^{2}(\mathbb{T}^{2})$ generated by $\Phi_{t}^{u}$:$(U^{t}f)(x)=f(\Phi_{-t}^{u}(x))$. By now, we are ready to state our result.

To prove Proposition 5.2, we first prove following auxiliary lemma. Let $\mathbb{S}^{1}=[-\frac{1}{2},\frac{1}{2}]$ be the one-dimensional circle.