Small scale formations in the incompressible porous media equation

In this paper, we consider the following three scenarios:

(S1) Let $\Omega=\mathbb{R}^{2}$. Assume $\rho_{0}\in C_{c}^{\infty}(\mathbb{R}^{2})$ is odd in $x_{2}$, and $\rho_{0}\geq 0$ in $\mathbb{R}\times\mathbb{R}^{+}$.

(S2) Let $\Omega=\mathbb{T}^{2}=[-\pi,\pi)^{2}$ be the 2D torus. Assume $\rho_{0}\in C^{\infty}(\mathbb{T}^{2})$ is odd in $x_{2}$.

(S3) Let $\Omega=S:=\mathbb{T}\times[-\pi,\pi]$ be a bounded strip that is periodic in $x_{1}$. Assume $\rho_{0}\in C^{\infty}(S)$.

Our first result shows that in the scenario (S1), $\rho(t)$ must have infinite-in-time growth in $\dot{H}^{s}$ norm for any $s>0$, if it remains regular for all times. Note that $s>0$ is the sharp threshold, since for $s=0$ we know $\|\rho(t)\|_{L^{2}}=\|\rho_{0}\|_{L^{2}}$ does not grow in time.

The next result concerns the torus scenario (S2), where we prove infinite-in-time growth under some additional symmetry and positivity assumptions on $\rho_{0}$. As we will see in the remark afterwards, the same result also holds for the bounded strip scenario (S3).

One can easily check that any horizontal stratified state $\rho_{s}=g(x_{2})$ is a stationary solution of (1.1) in $\mathbb{R}^{2}$, $\mathbb{T}^{2}$ or $S$, since $u=\nabla^{\perp}(-\Delta_{\Omega})^{-1}\partial_{x_{1}}\rho\equiv 0$. (As we will see in Lemma 3.1, all smooth stationary solutions in $S$ are of the form $\rho_{s}=g(x_{2})$. However, in $\mathbb{R}^{2}$ and $\mathbb{T}^{2}$ there are other smooth stationary solutions, e.g. any vertical stratified state is also stationary; see also [2, Section 5] for smooth stationary solutions in $\mathbb{R}^{2}$ supported in an infinite slanted strip.)

Below we briefly summarize the previous stability results for the horizontal stratified state $\rho_{s}=g(x_{2})$. Denoting $\eta(x,t):=\rho(x,t)-\rho_{s}(x)$ and plugging it into (1.1), $\eta$ satisfies

with $u=\nabla^{\perp}(-\Delta_{\Omega})^{-1}\partial_{x_{1}}\eta$. For $\eta$ small, the linearized equation is $\partial_{t}\eta=-g^{\prime}(x_{2})u_{2}$, which can be written as

Since $(-\Delta_{\Omega})^{-1}\partial^{2}_{x_{1}}$ is a negative operator, one would expect the equation to be linearly stable if $g^{\prime}$ is uniformly negative (i.e. lighter density on top, heavier on the bottom).

For the stratified state $\rho_{s}=-x_{2}$, the asymptotic stability of the nonlinear equation (1.6) has been rigorously established by Elgindi [7] in $\mathbb{R}^{2}$ and Castro–Córdoba–Lear [1] in $S$, which also implies the global well-posedness of (1.1) for initial data close to $\rho_{s}=-x_{2}$ in certain Sobolev spaces. More precisely, for $\rho_{s}=-x_{2}$ in $\mathbb{R}^{2}$, if $\|\eta\|_{W^{4,1}(\mathbb{R}^{2})}+\|\eta\|_{H^{s}(\mathbb{R}^{2})}<\epsilon\ll 1$ for $s\geq 20$, [7, Theorem 1.3] proved that $\eta$ remains regular for all time and satisfies $\|\eta(t)\|_{H^{3}}\lesssim\epsilon t^{-1/4}$ for all $t>0$. [7] also obtained asymptotic stability results for periodic perturbation $\eta\in H^{s}(\mathbb{T}^{2})$, where $\rho_{s}$ is still $-x_{2}$ in the whole plane. In [1], the authors proved that for the stratified state $\rho_{s}=-x_{2}$ in $S$ is asymptotically stable in $H^{s}(S)$ for $s\geq 10$, although it may converge to a slightly different stratified state from $\rho_{s}$ as $t\to\infty$.

In this paper, we aim to prove two nonlinear instability results for the horizontal stratified steady state $\rho_{s}=g(x_{2})$ in $\mathbb{T}^{2}$ and $S$ respectively. What sets our approach apart is that we are not following the common path of converting linear instability into a nonlinear one. Rather, we use the monotone quantity - potential energy - to prove infinite-in-time growth of Sobolev norms and then leverage these results to conclude the nonlinear instability. Our first instability result shows that in $\mathbb{T}^{2}$, any horizontal stratified steady state $\rho_{s}$ that is odd in $x_{2}$ is nonlinearly unstable, and the instability can grow “infinitely in time”. Namely, for any arbitrarily large $k>0$, one can construct an initial data $\rho_{0}$ that is arbitrarily close to $\rho_{s}$ in $H^{k}$, such that $\limsup_{t\to\infty}\|\rho(t)-\rho_{s}\|_{\dot{H}^{s}(\mathbb{T}^{2})}=\infty$ for all $s>1$.

Finally, we prove an instability result in the bounded strip $S=\mathbb{T}\times[-\pi,\pi]$ for any stratified steady state $\rho_{s}\in C^{\infty}(S)$, including those monotone stratified states that are linearly stable such as $\rho_{s}=-x_{2}$. Namely, we can construct a smooth perturbation small in $H^{2-\gamma}$ norm for any $\gamma>0$, such that $\limsup_{t\to\infty}\|\rho(t)-\rho_{s}\|_{\dot{H}^{s}(S)}=\infty$ for any $s>1$.

In Section 2 we discuss some preliminaries and the local well-posedness results in the scenarios (S1)–(S3). In Section 3 we show the monotonicity of the potential energy in the three scenarios, and use it to prove the infinite-in-time growth results in Theorems 1.1–1.2. We take a brief detour in Section 4 to derive some infinite-in-time growth results for less restrictive initial data, which we call the “bubble” solution and the “layered” solution. This will enable us to obtain nonlinear instability results in Section 5 for initial data close to stratified steady states, where we prove Theorems 1.4–1.5.

For any $f\in L^{2}(\mathbb{R}^{2})$, its Fourier transform is defined as usual as

and the Plancherel theorem yields $\|\hat{f}\|_{L^{2}(\mathbb{R}^{2})}^{2}=\|f\|_{L^{2}(\mathbb{R}^{2})}^{2}.$ As usual, we define

and

For (1.1) in $\mathbb{R}^{2}$, Córdoba–Gancedo–Orive [5, Theorem 3.2] proved local-wellposedness for initial data $\rho_{0}\in H^{s}(\mathbb{R}^{2})$ with $s>2$. They also established a regularity criteria, showing that $\rho(t)$ remains regular as long as $\int_{0}^{t}\|\nabla\rho(t)\|_{BMO\,}ds<\infty$. They also obtained another regularity criteria with a geometric flavor, and we refer the reader to [5, Theorem 3.4] for details.

For any $f\in L^{1}(\mathbb{T}^{2})$, let us denote its Fourier series as

where the Fourier coefficient $\hat{f}(k)$ for $k=(k_{1},k_{2})\in\mathbb{Z}^{2}$ is given by

By Parseval’s theorem, for any $f,g\in L^{2}(\mathbb{T}^{2})$ we have $\int_{\mathbb{T}^{2}}f(x)\overline{g(x)}dx=(2\pi)^{2}\sum_{k\in\mathbb{Z}^{2}}\hat{f}(k)\overline{\hat{g}(k)},$ which in particular implies

For $s\neq 0$, throughout this paper, $\|f\|_{\dot{H}^{s}(\mathbb{T}^{2})}$ is defined by

Finally, for mean-zero $f$ (in particular this is the case for $\partial_{x_{1}}\rho$), its inverse Laplacian is given by $(-\Delta)^{-1}f=\sum_{k\in\mathbb{Z}^{2}\setminus\{(0,0)\}}|k|^{-2}\hat{f}(k)e^{ik\cdot x}$.

Below we sketch a-priori estimates that can be used to establish local regularity as well as conditional criteria for global regularity. With these estimates, a fully rigorous argument can be given in a standard way, using either smooth mollifier approximations like in [14] or Galerkin approximations.

Suppose that $\rho(x,t)$ is a smooth solution of $\partial_{t}\rho+(u\cdot\nabla)\rho=0,$ $u=\nabla^{\perp}(-\Delta)^{-1}\partial_{x_{1}}\rho,$ where $\nabla^{\perp}=(-\partial_{x_{2}},\partial_{x_{1}}).$ Observe that all the $L^{p}$ norms of $\rho$ are conserved by evolution. Multiplying the equation by $(-\Delta)^{s}\rho$ and integrating we obtain

Here $s\geq 1$ is an integer. In the integral above, we can expand the power of the Laplacian and integrate by parts, then use the periodicity to transfer exactly half of the derivatives on $u\cdot\nabla\rho$. What we get is a sum of terms of the form

where $D^{s}$ stands for some partial derivative of order $s.$ Next we apply Leibniz rule to open up the derivative $D^{s}$ falling on $(u\cdot\nabla)\rho.$ Note that when all derivatives fall on $\rho,$ we get

due to incompressibility of $u.$ Therefore we obtain a sum of the terms of the form

where $1\leq j\leq s.$ Let us apply Hölder inequality to control such integral by

where $p_{j},$ $q_{j}$ satisfy $p_{j}^{-1}+q_{j}^{-1}=\frac{1}{2}.$ Let us recall a particular case of Gagliardo-Nirenberg inequality [9, 15]

for any $f\in C_{0}^{\infty}(\mathbb{R}^{2}),$ where in the 2D case $\theta=\frac{j-\frac{2}{q}}{s-1}.$ The inequality is valid for $s>j,$ $0<\theta<1.$ While (2.5) is usually stated in $\mathbb{R}^{2},$ an extension to $\mathbb{T}^{2}$ is straightforward. Taking now $p_{j}=\frac{2(s+1)}{j}$ and $q_{j}=\frac{2(s+1)}{s+1-j}$ and applying (2.5), we get

where in the first step we used $L^{p}-L^{p}$ bound on singular integral operators for $1<p<\infty.$ Similarly,

Therefore, for all $1\leq j\leq s$ we have

and hence

Such inequality can be used to show local well-posedness in $H^{s}$ provided that $s>2.$

To obtain a criteria for blow up, we can run a similar calculation but using $\|\nabla\rho\|_{L^{\infty}}$ and $\|\nabla u\|_{L^{\infty}}$ instead of $\|\rho\|_{L^{\infty}}$ and with $p_{j}=2s/j,$ $q_{j}=2s/(s-j).$ This way instead of (2.6) we obtain the differential inequality

We can conclude control of $\|\rho\|_{\dot{H}^{s}}$ up to any time $T$ provided that $\int_{0}^{T}\left(\|\nabla\rho\|_{L^{\infty}}+\|\nabla u\|_{L^{\infty}}\right)\,dt$ remains finite. Let us state a proposition summarizing the observations of this section.

Remarks. 1. Uniqueness of the solution can be shown in a standard way; blow up is understood in the sense of leaving the class $C([0,T],H^{s}(\mathbb{T}^{2})).$
2. The Proposition can certainly be improved in terms of the condition on $s$ and the regularity criterion, but we do not pursue it in this paper.

When the domain is a bounded strip $S:=\mathbb{T}\times[-\pi,\pi]$, due to the presence of the top and bottom boundaries, the functional spaces and the local-wellposedness results are more involved than the periodic case. Below we briefly describe the results by Castro–Córdoba–Lear [1], and we refer the readers to the paper for more details.

Biot-Savart law and functional space. In the strip case, the velocity field $u$ is given by $u=\nabla^{\perp}\psi$, where the stream function $\psi$ solves the Poisson’s equation with zero boundary condition (see [1, Section 2.2] for a derivation):

so that $u=\nabla^{\perp}(-\Delta_{S})^{-1}\partial_{x_{1}}\rho$. One can check that the operator $-\Delta_{S}$ (with zero boundary condition) is a positive self-adjoint operator, and it has a family of eigenfunctions $\{\omega_{p,q}\}_{p\in\mathbb{Z},q\in\mathbb{N}}$ that form an orthonormal basis for $L^{2}(S)$, given by

where

and

The eigenfunction expansion allows us to define $(-\Delta_{S})^{s}f$ for any $s\in\mathbb{R}$ and $f\in L^{2}(S)$. We can then define the $\dot{H}^{s}$ homogenous Sobolev norm as

and one can check that the spaces $\dot{H}^{s}(S)$ and $\dot{H}^{-s}(S)$ are dual with respect to the $L^{2}$ norm (see e.g. [17] for the general construction of a scale of Sobolev spaces associated with a positive self-adjoint operator).

For $s\geq 0$, let us define $\|f\|_{H^{s}(S)}^{2}:=\|f\|_{\dot{H}^{s}(S)}^{2}+\|f\|_{L^{2}(S)}^{2}$ as usual. For $s\in\mathbb{N}$, the above definition of $\|f\|_{{H}^{s}(S)}^{2}$ is comparable to $\sum_{0\leq m\leq s}\|D^{m}f\|_{L^{2}(S)}^{2}$ if $\partial_{x_{2}}^{n}f|_{\partial S}=0$ for all even $n$ with $n<s$, where $D^{m}$ is any partial derivative of order $m\leq s$.

Local/global well-posedness results in the strip. Since the goal of [1] was to establish stability results near the steady state $\rho_{s}(x)=-x_{2}$, (1.1) was written into an equivalent equation (1.6) (with $g(x_{2})=-x_{2}$) describing the evolution of $\eta=\rho-\rho_{s}$. Here $u$ can be expressed in terms of $\eta$ similarly to (2.7), except that the right hand side $\partial_{x_{1}}\rho$ is replaced by $\partial_{x_{1}}\eta$ (using that $\rho_{s}$ has zero contribution to $u$ since it is a steady state).

When the initial data $\eta(0)$ belongs to the functional space $X^{k}(S)$, given by

(which in fact coincides with $H^{k}(S)$ defined above), the authors proved local-wellposedness of (1.6) for $\eta(0)\in X^{k}(S)$ for any $k\geq 3$ [1, Theorem 4.1], and gave a regularity criteria showing that $\eta(t)$ remains in $X^{k}(S)$ as long as $\int_{0}^{t}(\|\nabla\eta(s)\|_{L^{\infty}(S)}+\|\nabla u(s)\|_{L^{\infty}(S)})ds<\infty$. As we discussed in the introduction, for $k\geq 10$, they proved the asymptotic stability of $\eta(t)$ (which implies global regularity) for $\eta(0)\in X^{k}(S)$ with $\|\eta(0)\|_{X^{k}(S)}\ll 1$.