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On wellposedness of Prandtl: a contradictory claim?

Yesterday, Nov 17, Xu and Zhang posted a preprint on the ArXiv, entitled “Well-posedness of the Prandtl equation in Sobolev space without monotonicity” (arXiv:1511.04850), claiming to prove what the title says. This immediately causes some concern or possible contrary to what has been known previously! Here, monotonicity is of the horizontal velocity component in the normal direction to the boundary. It’s well-known that monotonicity implies well-posedness of Prandtl (e.g., Oleinik in the 60s; see this previous post for Prandtl equations). It is then first proved by Gerard-Varet and Dormy that without monotonicity, the Prandtl equation is linearly illposed (and some followed-up works on the nonlinear case that I wrote with Gerard-Varet, and then with Guo). Is there a contradictory to what it’s known and this new preprint of Xu and Zhang? The purpose of this blog post is to clarify this.

For sake of clarity, it suffices to focus discussions on the linearized Prandtl equations around a stationary shear flow ${u_s(y)}$, for ${y\in\mathbb{R}_+}$. Assume that the shear profile has a non-degenerate critical point: precisely, there is a point ${a>0}$ so that

$\displaystyle \partial_y u_s(a) = 0 , \quad \partial_y^2 u_s(a) \not =0, \qquad u_s(0) = \partial_y^2 u_s(0) =0. \ \ \ \ \ (1)$

Under the assumption on the asymptotic behavior of the shear flow

$\displaystyle \lim_{y\rightarrow \infty} u_s(y) = 1, \qquad |\partial_y^ku_s| \lesssim e^{-\eta y}, \qquad k\ge 1, \quad y \gg 1,$

Gerard-Varet and Dormy (JAMS 2010) proved that the linearized Prandtl is ill-posed in Sobolev spaces, making use of the non-monotonicity assumption. The illposedness is extremely strong in the sense that they construct a growing mode of the form ${e^{in x}e^{t\sqrt n }}$, for sufficiently large spatial frequency number ${n}$. This strong growth prevents a general local-in-time well-posedness theory for the Prandtl equation in Sobolev spaces, with arbitrary short time. For instance, Gerard-Varet and I showed that for arbitrary short time, we can construct initial data so that the linearized problem has no distributional solution! In fact, linearly, this strong growth even prevents a general well-posedness theory in Gevrey class of any order greater than ${2}$ (the closest result in this direction is by Masmoudi and Gerard-Varet, Ann. Sci. Ecole Norm. Sup, to appear, with which they were only able to push up to Gevrey of order ${7/4}$ for the nonlinear problem).

Xu-Zhang studies the same Prandtl problem in the perturbative setting near a non-monotonic shear flow ${u_s}$, with a polynomial decaying rate at infinity; namely, they assume that

$\displaystyle \partial_y^{p+1} u_s \approx \langle y \rangle^{-k - p}, \qquad p\ge 0, \quad k>1.$

Both lower and upper bounds with the same decaying rate turns out to be very crucial in their analysis. They claim to establish a well-posedness theory of the (nonlinear) Prandtl equation around ${u_s}$. The initial perturbations that they take are precisely in a weighted Sobolev spaces, which in particular obeys the following asymptotic behavior

$\displaystyle |u_0 - u_s| \lesssim \langle y \rangle^{-2k -1} \ \ \ \ \ (2)$

in which ${\langle y\rangle ^{-k}}$ is the decay rate of ${\partial_yu_s}$. Since there are no apparent nonlinear cancellations used in their paper, the paper also yields a corresponding wellposedness theory for the linearization (for the obvious reason!).

An obvious conclusion is that the shear flows behave differently at infinity, so apparently no contradiction between the two papers. But, there are more delicate / technical reasons that I shall now discuss !

Illposedness of shear flows with polynomial decay rates? With a close examination, it appears that the above shear flow with a fast enough polynomial decay remains linearly unstable in the sense of Gerard-Varet and Dormy! This, however, won’t contradict with the wellposedness of Xu and Zhang, if correct. Indeed, one observes that Gerard-Varet and Dormy’s growing modes are of the form ${[u,v] = e^{in(x-ct)} [\partial_y \phi, -i n \phi] }$, where ${\phi(y)}$ denotes the corresponding stream function, which solves

$\displaystyle (u_s - c) \phi'' - u_s'' \phi = i\epsilon \phi''''$

with $\epsilon = 1/n$, (which is the classical Orr-Sommerfeld equations, for the zero spatial frequency). Their construction of the growing modes starts with ${\phi = u_s - c}$, and hence the asymptotic behavior at infinity is as follows:

$\displaystyle u \approx \partial_y \phi \approx u_s' .$

Hence, in order to observe the instability, one takes the initial perturbation to be exactly that of the growing modes: ${u = e^{inx} \partial_y \phi}$, which decays as ${\partial_y u_s \approx \langle y \rangle^{-k} }$, which is much slower than that allowed by Xu and Zhang’s perturbations: see (2). In the other words, Xu and Zhang work with a smaller set of initial perturbations, which excludes the possible growing eigenfunctions constructed by Gerard-Varet and Dormy. This yields no contradictory.

A further remark on shear flows with exponential decay rates: would Xu-Zhang’s approach work for Gerard-Varet and Dormy’s illposed shear flows with exponential decay rates, that is, establishing well-posedness in a smaller set of initial perturbations (still in Sobolev spaces)? This does NOT appear possible, for a rather technical reason, that in order to recover the velocity ${u}$ from the Masmoudi-Wong’s good unknown ${g}$ (heavily used in the paper), one writes

$\displaystyle u = - u_s' \int_y^\infty \frac{g}{u_s'}$

which requires a different weight function, due to the $y$-integration, to close the analysis. It works precisely with the polynomial weight, with exact lower and upper bounds for each derivatives of ${u_s}$. For polynomial decay shear profiles ${u_s'}$, one forces to use the same weight for both ${u}$ and ${g}$, which is not possible due to the ${y}$-integration.

UPDATED Feb 22, 2016: the authors Xu – Zhang have withdrawn their paper from the arxiv (and replaced by a different paper / results). To conclude, there is no known well-posedness theory in Sobolev spaces for Prandtl with non-monotonic setting (other than illposedness results mentioned above).