Ill-posedness of degenerate dispersive equations

Dispersion plays a pivotal role in the analysis of the governing equations for a large number of physical phenomena, ranging from the evolution of surface water waves to the formation of Bose-Einstein condensates. A common theme in such problems is that uniform dispersive effects frequently control nonlinear terms which may otherwise lead to ill-posedness. Consider for instance the following class of quasilinear equations of KdV type studied in [8]:

where $g_{j}=g_{j}(\partial_{x}^{j}u,...,u)$ and $h=h(x,t)$. There it is shown that such equations are well-posed provided $|g_{3}|\geq c>0$ and $g_{2}\geq 0$. The first of these conditions guarantees that dispersive effects from $g_{3}u_{xxx}$ do not vanish while the second prohibits the term $g_{2}u_{xx}$ from acting like a backwards heat operator. Similarly, the results on existence of solutions to the quasilinear dispersive equations found in [15] (about Schrödinger equations), [1] (water waves) and [29] (magma dynamics) share the feature that the dispersive effects are nondegenerate, to name just a few examples.

On the other hand, there are a number of physical problems in which the mechanism which causes the dispersive effects is not only nonlinear but degenerate. Some examples can be found in the study of sedimentation [26], shallow water [5], granular media [19, 11, 20], numerical analysis [14] and elastic rods and sheets [9, 16]. While there are a large number of articles in which degenerate dispersive equations (henceforth referred to as DDE) are derived [19, 20, 11, 5, 16, 9, 26], special solutions computed [24, 25, 11, 20, 19] or numerical computation of solutions performed [28, 27, 24, 10, 17, 6], the existence theory for the initial value problem of these equations is largely undeveloped.¹¹1The notable exception to this is the well-studied Camassa-Holm equation $u_{t}-u_{xxt}=-3uu_{x}+2u_{x}u_{xx}+uu_{xxx}$ which when rewritten as a nonlocal evolution equation could be said to be degenerate and dispersive. This equation is integrable and consequently many results for it do not obviously generalize. The articles [10, 3, 13] prove some a priori estimates for DDE. [3] and [2] prove existence (but not uniqueness) of solutions to a very special class of DDE. [12] contains a semi-rigorous discussion of existence issues for these sorts of equations, including defining “$\delta$-entropy solutions” by borrowing ideas from first order conservation laws. None of these articles fully settles the well-posedness issues for degenerate dispersion, and thus our interest is in studying the Cauchy problem for some simple and common DDE.

In particular we study the following two equations:

and

for $x\in{\bf{R}}$. The first of these is the $K(2,2)$ equation of Rosenau & Hyman [24]. Their goal was to understand the role of degenerate nonlinear dispersion in the formation of patterns and this equation (together with other members of the $K(m,n)$ family $u_{t}=(u^{m})_{x}+(u^{n})_{xxx}$) has come to serve as the paradigm for degenerate dispersive systems. The most celebrated feature of this equation is the existence of compactly supported traveling waves, a.k.a. “compactons.” These are given by

where $\chi_{I}$ is the indicator function for the interval $I=[-2\pi,2\pi]$ and $\lambda\neq 0$. More recently, $K(2,2)$ has been used to model pulses in both elastic and granular media [16, 21]. Equation (3), a “degenerate Airy equation”, appeared in [13] and is amongst the simplest equations which could be said to feature degenerate dispersion. We study it here due to its similarity to the $K(2,2)$ equation and because it is of the form of (1) but does not meet the uniform dispersion hypothesis needed in [8].

Surprisingly, our conclusions are that both equations are ill-posed for data in $H^{2}$. In particular, their solutions do not depend smoothly on their initial conditions. Our results for (2) are computational in nature: we conduct a series of numerical simulations of (2) which demonstrate that data which is very small in $H^{2}$ can be of unit size at a fixed time which is independent of the data’s size. For (3) our results are fully rigorous: we prove the existence of a compactly supported self-similar solution which, when combined with certain scaling invariances, implies ill-posedness. Note that our focus on $H^{2}$ initial data is due to the fact the compacton solutions of $K(2,2)$ have a discontinuity in their second derivative at the edge of their support [18]. Consequently, they reside in $H^{2}$ but not $H^{3}$, and so it is natural to work in this space.

Our suspicion that (2) is ill-posed has its origin in the various numerical simulations of collisions between compactons in $K(m,n)$ equations. A wide variety of numerical methods has been used to compute solutions: pseudo-spectral [24], particle [6], Padé [28, 27], discontinuous Galerkin [17], and finite difference [10]. Each of these simulations shows that the computed collisions between compactons are nearly elastic: the compactons emerge from the collision nearly identical to their original state but phase shifted. As is common in collisions between solitary waves in “nearly” integrable PDE, the simulations show that in addition to the outgoing waves there is a small amplitude disturbance behind the waves. In nondegenerate problems, similar disturbances are sometimes called “dispersive ripples” and are viewed as being linear effects—the tail (approximately) solves the linearized equation [30, 31]. However, since the linearization of (2) about zero is trivial this heuristic cannot apply. The disturbance is described variously as: a compacton/anti-compacton pair in [24]; a numerical artifact in [28, 10]; and as a shock in [27]. No such disturbance is present at all in [6]. Lastly, the authors of [17] describe a “high-frequency oscillation” but go on to state “we do not know what is the source of these oscillations…” In each case, the qualitative features of the disturbance are quite different. Our feeling is that these discrepancies are not related to differences in the numerical methods or the implementation thereof but rather are due to an underlying issue with the equation itself.

Moreover, inspection of the right hand side of (2) provides evidence that the equation may be ill-posed. Observe that when $u_{x}<0$, the term $6u_{x}u_{xx}$ is formally a backwards heat operator. The key question, initially raised in [24], is this: can dispersive effects due to $2uu_{xxx}$ counteract the instability caused by the backwards heat operator $6u_{x}u_{xx}$? Thus we should seek to understand the dispersive effects due to $2uu_{xxx}$—hence our interest in (3). Our results show that the answer to the key question is “no”. As we stated above, our results show that $2uu_{xxx}$ is, in its own right, just as problematic as the backwards heat term. Differentiation of (3) with respect to $x$ yields for $w:=u_{x}$

Note that the second term on the RHS above is also a backwards heat operator when $u_{x}<0$. The backwards heat term evident in (2) is, in this way, hidden in (3). Thus we expect blow up of $u_{x}$ when $u_{x}$ is negative. Moreover, this blow up should be particularly catastrophic when the function crosses the $x$-axis with negative slope.

This motivates our approach: the self-similar solution constructed has such a transverse crossing and the resulting ill-posedness is due to blow up in the first and second derivatives of the solution. While our proof that (3) is ill-posed does not naturally carry over to (2), the numerics are highly suggestive. Most tellingly, since the numerics are performed using strictly positive initial data, they suggest the ill-posedness can manifest even in the absence of such a transverse crossing.

The remainder of this paper is organized as follows. In Section 2 we show that the existence of a self-similar solution to (3) gives, as a consequence, the ill-posedness of the equation. In Section 3 we construct the self-similar solution using techniques from spatial dynamics. In Section 4 we discuss our numerical strategy for demonstrating ill-posedness of (2) and present the results of our simulations.

We will prove there is a self-similar solution for (3) of the form

where $A$ gives the profile. Though other choices for the scaling are possible, we choose this particular set so that our proposed solution exhibits blow up (as $t\to 1^{-}$) in the first and second derivative while the $L^{\infty}$ norm remains bounded. Inserting this into (3) yields:

Here $\tau=x(1-t)^{-1/3}$ is the self-similar coordinate upon which $A$ depends and we have $(\square)^{\prime}=d/d\tau(\square)$. We specify $A(0)=0$ and $A^{\prime}(0)=-1$ so that the self-similar solution crosses the $x$-axis transversely with negative slope, the “bad” situation identified in the Introduction.

We prove the following proposition concerning solutions of (4):

The proof of this proposition is technical and difficult and we postpone it until the next section. First we utilize it to prove our main result for (3):

This section is devoted to proving Proposition 1. We rewrite (4) as the initial value problem (IVP):

Our strategy will be as follows: first we will show that the IVP (7) has a unique solution, which depends on $\mu$ continuously. Next we will show that for $\mu$ sufficiently large the solution of (7) goes to zero in finite time. Then we will show that for $\mu=0$ there is a smallest time $0<\tau_{2}<\infty$ such that the solution of (7) has its first local maximum at $\tau_{2}$ and remains strictly negative for all $\tau\in(0,\tau_{2}]$. See Figure 2.

Then an open/closed argument will give the special solution $A_{*}$ for a choice $\mu=\mu_{*}>0$.

The differential equation (4) is nonlinear, nonautonomous and singular at $A=0$. As such, carrying out the details of our strategy is quite challenging. The key observation for proving Proposition 1 is that (4) is, in a loose sense, close to being hamiltonian. Specifically, one can rewrite the right hand side of (4) as $\displaystyle\frac{\tau}{6}\frac{d}{d\tau}\ln\left|A(\tau)\right|$. If we then consider the related equation

obtained from (4) by replacing the nonautonomous $\tau/6$ with the constant $1$, one sees that this equation is in fact hamiltonian. Specifically one can integrate both sides to find:

The conserved energy is

and most of the dynamics of (8) can be determined by analyzing this quantity. In the following proof of Proposition 1, we will frequently make use of quantities similar to this energy which, though not conserved for solutions of (4), do provide quite a bit of dynamical information about solutions.