Grenier’s nonlinear iterative scheme

In his paper [Grenier, CPAM 2000], Grenier introduced a nonlinear iterative scheme to prove the instability of Euler and Prandtl equations. Recently, the scheme is also proved to be decisive in the study of water waves: [Ming-Rousset-Tzvetkov, SIAM J. Math. Anal., 2015], and plasma physics: [Han-Kwan & Hauray, CMP 2015] or my recent paper with Han-Kwan (see also my previous blog discussions). I am certain that it can be useful in other contexts as well. In this blog post, I’d like to give a sketch of the scheme to prove instability.

Suppose one wishes to solve the following differential equations:

$\displaystyle \Phi (u): = (\partial_t - L +\epsilon S) u + Q(u,u) = 0\ \ \ \ \ (1)$

in which ${L,S}$ are linear differential operators and ${Q(u,u)}$ denotes some bilinear differential operator. Here, ${\epsilon}$ is some small parameter (e.g., viscosity). In practice, it is more than often to be the case that available good bounds are known only for the (leading) semigroup ${e^{Lt}}$ , but not for ${e^{(L-\epsilon S)t}}$ , and thus controlling nonlinearity would be an issue, not to mention that there are losses of derivatives in the nonlinear term. The main point of the scheme is to overcome this issue. To proceed, assume that the linear problem ${(\partial_t - L) u = g}$ is solvable for arbitrary ${g}$ (equivalently, good bounds on the semigroup ${e^{Lt}}$ ).

The formal scheme is straightforward. The first step is to construct good approximate solutions to the full problem, starting from a linear solution ${u_1}$ , solving ${(\partial_t - L) u_1 =0}$ . One observes that ${\epsilon u_1}$ approximately solves the nonlinear problem: ${\Phi( \epsilon u_1) = \epsilon^2 Su_1 + \epsilon^2 Q(u_1, u_1) }$ , having an error of order ${\epsilon^2}$ . To improve the error, one solves a non-homogenous linear problem that kills off the error term at order ${\epsilon^2}$ : precisely, construct ${u_2}$ solving ${(\partial_t - L) u_2 =Su_1 + Q(u_1, u_1)}$ . The new approximate solution ${\epsilon u_1 + \epsilon^2 u_2}$ then solves the nonlinear problem, leaving an error of order ${\epsilon^3 }$ and smaller. Take ${N}$ arbitrarily large. Inductively, one builds an approximate solution of the form:

$\displaystyle u_\mathrm{app} : = \sum_{n=1}^N \epsilon^n u_n$

in which ${u_n}$ solves a non-homogenous linear problem: ${(\partial_t - L) u_n = R_{n-1}}$ , with ${R_{n-1}}$ being the approximation error caused by the previous step. The approximate solution ${u_\mathrm{app}}$ then solves the nonlinear problem approximately, leaving an error of order:

$\displaystyle \Phi(u_\mathrm{app} ) \approx \epsilon^{N+1} .$

In practice, if ${\epsilon S}$ is not a regular perturbation of ${L}$ and so ${S (\partial_t - L)^{-1}}$ is not well-defined, singular layers may be added for the iterative construction to work; see for instance my previous blog discussions for a toy model as well as for Orr-Sommerfeld equations (stable and unstable cases).

Next, depending on particular problems, exact solutions are obtained from the approximate ones. For instance, with a fixed sufficiently small ${\epsilon}$ , one may take ${N\rightarrow \infty}$ to obtain an exact solution, provided that the series ${u_\mathrm{app}}$ converges in some function space. Below, I give examples of using the scheme to prove instability.

1.1. Navier-Stokes equations

Let ${L}$ be the linearized Euler operator around a steady state ${u_0}$ , ${S = -\Delta}$ , and ${Q(u,u) = P(u \cdot \nabla u)}$ , where $P$ denotes the Leray projection on the space of divergence-free vector fields. Let ${(\lambda_1, \hat u_1)}$ be the maximal unstable eigenvalue and corresponding eigenvector of ${L}$ , and assume that ${\hat u_1}$ is sufficiently smooth. We build the approximate solution ${u_\mathrm{app}}$ , following the previous iterative scheme starting from ${ u_1 = e^{\lambda_1 t} \hat u_1}$ . Notice that ${u_1 \sim e^{\Re \lambda_1 t}}$ exponentially grows in time, and so ${u_2 = (\partial_t - L)^{-1} (S u_1 + Q(u_1,u_1))}$ must grow at order ${e^{2\Re \lambda_1 t}}$ exponentially in time, due to the quadratic nonlinearity. Inductively,

$\displaystyle \| u_n \|_{H^s} \lesssim e^{n \Re \lambda_1 t}, \qquad n\ge 1\qquad \mbox{and}\qquad \| u_\mathrm{app} \|_{L^2} \sim \epsilon e^{\Re \lambda_1 t}.$

Here, the Sobolev regularity index ${s\ge 0}$ depends on the regularity of the unstable eigenfunction ${\hat u_1}$ . The error of the approximation satisfies

$\displaystyle \| \Phi(u_\mathrm{app} ) \|_{L^2} \lesssim \epsilon^{N+1} e^{(N+1) \Re \lambda_1 t} = \Big( \epsilon e^{\Re \lambda_1 t}\Big)^{N+1}.$

The nonlinear instability now follows straightforwardly. Indeed, write the perturbation ${v := u - u_\mathrm{app}}$ , which solves

$\displaystyle (\partial_t - L - \epsilon \Delta ) v + v \cdot \nabla u_\mathrm{app} + u_\mathrm{app} \cdot \nabla v+ v \cdot \nabla v + \nabla q = \Phi(u_\mathrm{app})$

together with ${\nabla \cdot v =0}$ . The standard energy estimate then gives

$\displaystyle \frac12 \frac{d}{dt} \| v\|_{L^2}^2 \lesssim C_0 \| v\|_{L^2}^2 + \| \Phi(u_\mathrm{app})\|_{L^2} \| v\|_{L^2},$

in which ${C_0: = 1 + \| \nabla u_0 \|_{L^\infty} + \sup_{t\in [0,T]} \| \nabla u_{\mathrm{app}} (t)\|_{L^\infty} }$ . As long as ${C_0}$ is finite, the standard Gronwall inequality (assuming ${v(0)=0}$ ) yields at once

$\displaystyle \| v(t)\|_{L^2} \lesssim \int_0^t e^{C_0(t-s)} \| \Phi(u_\mathrm{app})(s)\|_{L^2} \lesssim \epsilon^N \int_0^t e^{C_0(t-s)} e^{(N+1) \Re \lambda_1 s} \; ds \lesssim \epsilon^{N+1} e^{(N+1) \Re \lambda_1 t} ,$

for sufficiently large ${N}$ so that ${(N+1)\Re\lambda_1 \ge C_0}$ . Here, notice that thanks to the iteration, the approximation error grows at a rate faster than the rough bound ${e^{C_0 t}}$ on the semigroup for ${L- \epsilon S}$ (i.e., no sharp bound is needed). This yields the exact solution of (1), satisfying

$\displaystyle \| u(t)\|_{L^2} \ge \| u_\mathrm{app} (t) \|_{L^2} - \| u - u_\mathrm{app} (t)\|_{L^2} \ge \epsilon e^{\Re \lambda_1 t} - \Big( \epsilon e^{\Re \lambda_1 t}\Big)^{N+1} \ge \frac 12 \epsilon e^{\Re \lambda_1 t}$

as long as ${ \epsilon e^{\Re \lambda_1 t}}$ remains small, but of order one in ${\epsilon}$ . This proves the instability: there are sequences of solutions ${u^\epsilon}$ and times ${t_\epsilon\sim \frac{1}{\Re \lambda_1}\log 1/\epsilon}$ so that

$\displaystyle \| u^\epsilon (0) \|_{H^s} \lesssim \epsilon\qquad \mbox{but}\qquad \| u^\epsilon(t_\epsilon)\|_{L^2} \gtrsim 1.$

Up to a minor modification in the iterative scheme, the initial data can be sufficiently small in ${H^s}$ , of order ${\epsilon^M}$ , for arbitrary ${s, M\ge 0}$ . Unlike the classical instability analysis of [Friedlander-Strauss-Vishik, 1997] or of [Bardos-Guo-Strauss, 2002], the above yields ${L^2}$ , in fact ${L^p}$ , instability (instead of ${W^{1,p}}$ ); see also [Z. Lin, 2004] for an interesting different proof of ${L^p}$ instability of unstable steady states of Euler. In application to boundary layers, the unstable eigenvalue is typically of order ${\lambda_1 \sim \frac{1}{\sqrt \epsilon}}$ (in case of an inflection point in the steady state solutions), and thus the instability occurs within an arbitrarily short time in the inviscid limit ${\epsilon \rightarrow 0}$ , since the instability occurs within ${t_\epsilon \sim \sqrt \epsilon \log 1/\epsilon}$ . This was done in his original paper [Grenier, CPAM 2000] with ${L^\infty}$ instability up to order ${\epsilon^{1/4}}$ . The advantage of the iterative scheme is again that only instability of Euler is needed.

1.2 Grenier’s boundary layer solutions

In case of boundary layers, we consider the Navier-Stokes problem, with small viscosity $\nu$ , on the half-space with no-slip boundary condition. We analyze the stability of time-dependent shear flows (special boundary layers; see the boundary layer theory) of the form: ${u_s = [U(t,\frac {y}{\sqrt{\nu}}),0]^t}$ . In the scaled variables: ${(t,x,y) = (t,x,y)/\sqrt \nu}$ , one easily writes the perturbation in the form

$\displaystyle \Phi(u): = (\partial_t - L) u + \nu^{\frac 18} S u - \sqrt \nu \Delta u + Q(u,u) =0,\ \ \ \ \ (2)$

in which ${L}$ denotes the linearized Euler operator around the stationary boundary layer ${U(0,y)}$ , ${Q(u,u)}$ denotes the quadratic term ${\mathbb{P}(u \cdot \nabla u)}$ , and ${S}$ denotes the perturbed operator defined by

$\displaystyle Su: = \nu^{-1/8}[u_s (\sqrt \nu t, y) - u_s (0, y)] \cdot \nabla u + \nu^{-1/8} u \cdot \nabla [u_s (\sqrt \nu t, y) - u_s (0, y)] .$

Within the time ${T_\nu \sim \log 1/\nu}$ (the instability time as foreseen from the above nonlinear iteration analysis), we observe that in the function space of boundary layers of size ${\nu^{1/4}}$ , ${Su}$ is bounded, since ${\nu^{-1/8}\nu^{1/4} t \lesssim 1}$ . This shows that we can consider ${S}$ as a perturbation in the construction of approximate solutions.

The construction of the approximate solutions follows as discussed formally above: namely, it starts from ${u_{\mathrm{e},0} = e^{\lambda_0 t} \hat u_{\mathrm{e},0}}$ , where ${(\lambda_0, \hat u_{\mathrm{e},0})}$ is the maximal growing mode of the linearized Euler problem ${(\partial_t - L) u_{\mathrm{e},0} =0}$ . Clearly, $u_{\mathrm{e},0}$ does not satisfy the zero no-slip boundary conditions (only the normal component of the velocity vanishes).

To correct the zero boundary conditions, we add to the Euler solution ${u_{\mathrm{e},0}}$ a boundary layer ${u_{\mathrm{bl},0}}$ solving the Stokes problem

$\displaystyle \partial_t u_{\mathrm{bl},0} - \sqrt \nu \Delta u_{\mathrm{bl},0} + \nabla p_{\mathrm{bl},0} =0,\qquad \nabla \cdot u_{\mathrm{bl},0} = 0$

with the boundary condition ${u_{\mathrm{bl},0} + u_{\mathrm{e},0} = 0}$ on ${y=0}$ . The boundary layer solution ${u_{\mathrm{bl},0}}$ is of the form ${[ v_{\mathrm{bl},0}, \nu^{1/4} w_{\mathrm{bl},0}] ^t (t,x,\frac{y}{\nu^{1/4}}) }$ . Then, one observes that ${\nu^p u_0: = \nu^p (u_{\mathrm{e},0}+u_{\mathrm{bl},0} )}$ approximately solves the nonlinear problem (2).

Precisely,

$\displaystyle \Phi (\nu^pu_0 ) = \nu^{p+\frac 18} S u_0 + \nu^{2p}Q(u_0 ,u_0 ) - \nu^{p+\frac 12} \Delta u_{\mathrm{e},0} + \nu^{p+\frac 14} \frac{U \cdot \nabla u_{\mathrm{bl},0} + u_{\mathrm{bl},0} \cdot \nabla U}{\nu^{1/4}}$

which is of order ${\nu^{p+\frac 18}}$ or smaller in the limit ${\nu\rightarrow 0}$ . Indeed, as discussed above, ${Su_0}$ is of order one. Note that ${U(0) =0}$ and so ${U\cdot \nabla u_{\mathrm{bl},0} = U \partial_x u_{\mathrm{bl},0} \lesssim \nu^{1/4} \frac{y}{\nu^{1/4} } \partial_x u_{\mathrm{bl},0} \lesssim \nu^{1/4}}$ . Similarly, since ${ u_{\mathrm{bl},0} = [ v_{\mathrm{bl},0}, \nu^{1/4} w_{\mathrm{bl},0} ]^t}$ , we have ${ u_{\mathrm{bl},0} \cdot \nabla U \lesssim \nu^{1/4}}$ . This shows that the last term is also of order ${\nu^{p+1/4}}$ or smaller.

Finally, we observe that ${ Q(u_0 ,u_0 ) \lesssim 1}$ , upon noting that the normal component of ${u_{\mathrm{e},0}}$ vanishes on the boundary ${y=0}$ and that of boundary layer ${u_{\mathrm{bl,0}}}$ is of order ${\nu^{1/4}}$ , which balance with ${\nabla u_{\mathrm{bl},0} \sim \nu^{-1/4}}$ .

To improve the error, we construct ${u_1 = u_{\mathrm{e},1} + u_{\mathrm{bl},1}}$ , with zero initial data, so that

$\displaystyle (\partial_t - L) u_{\mathrm{e},1} = S u_{\mathrm{e},0} , \qquad \nabla \cdot u_{\mathrm{e,1}} =0$

and

$\displaystyle \partial_t u_{\mathrm{bl},1} - \sqrt \nu \Delta u_{\mathrm{bl},1} + \nabla p_{\mathrm{bl},1} = S u_{\mathrm{bl},0},\qquad \nabla \cdot u_{\mathrm{bl},1} =0,$

together with the boundary condition ${u_1 =0}$ on ${y=0}$ . Formally, one can check easily that ${\Phi(\nu^p u_0 + \nu^{p+\frac 18} u_1)}$ is of order ${\nu^{p+\frac 14}}$ or smaller. Inductively, we construct

$\displaystyle u_\mathrm{app} = \sum_{n=0}^N \nu^{p + \frac{n}{8}} u_n, \qquad u_n: = u_{\mathrm{e},n} (t,x,y) + u_{\mathrm{bl},n} \Big (t,x,\frac{y}{\nu^{1/4}}\Big) \ \ \ \ \ (3)$

in which the inviscid solution ${u_{\mathrm{e},n}}$ solves the linearized Euler problem:

$\displaystyle (\partial_t - L) u_{\mathrm{e},n} = R_{\mathrm{e},n}, \qquad \nabla \cdot u_{\mathrm{e,n}} =0\ \ \ \ \ (4)$

and the boundary layer solution ${u_{\mathrm{bl},n}}$ solves the Stokes problem:

$\displaystyle \partial_t u_{\mathrm{bl},n} - \sqrt \nu \Delta u_{\mathrm{bl},n} + \nabla p_{\mathrm{bl},n} = R_{\mathrm{bl},n},\qquad \nabla \cdot u_{\mathrm{bl},n} =0,\ \ \ \ \ (5)$

together with the boundary condition ${u_n =0}$ on ${y=0}$ , and both ${u_{\mathrm{e},n}}$ and ${u_{\mathrm{bl},n}}$ have zero initial data. Here, ${R_{\mathrm{e},n}}$ and ${R_{\mathrm{bl},n}}$ denote the error of order ${\nu^{p+\frac n8}}$ introduced from the approximation: precisely, for all ${n\ge 0}$ , we set ${u_{-n} \equiv 0}$ and denote

$\displaystyle R_n = S u_{n-1} - \Delta u_{\mathrm{e},n-4} + \frac{U \cdot \nabla u_{\mathrm{bl},n-2} + u_{\mathrm{bl},n-2} \cdot \nabla U}{\nu^{1/4}} + \sum_{k = 0}^{n-1} Q(u_k, u_{n-k-8p}).$

We then denote by ${R_{\mathrm{bl},n}}$ all terms in ${R_n}$ defined above involving the boundary layer solutions ${u_{\mathrm{bl},k}}$ . The source for the inviscid problem is defined by ${R_{\mathrm{e,n}}: = R_n - R_{\mathrm{bl},n}}$ . The error of the approximation can be computed by

$\displaystyle \begin{aligned} \Phi (u_\mathrm{app}) &= \nu^{p+\frac{N+1}{8}} S u_{N} - \sum_{k=0}^3 \nu^{p+\frac{N+k+1}{8}} \Delta u_{\mathrm{e},N+k-3} \\&\quad + \sum_{k=0}^1 \nu^{p+\frac{N+k+1}{8}} \frac{U \cdot \nabla u_{\mathrm{bl},N+k-1} + u_{\mathrm{bl},N+k-1} \cdot \nabla U}{\nu^{1/4}} \\&\quad + \sum_{k+j > N+1 -8 p; 1\le k,j\le N} \nu^{2p+ \frac{j+k}{8}}Q(u_k, u_j). \end{aligned}\ \ \ \ \ (6)$

It is easy to show that the error ${\Phi(u_\mathrm{app})}$ is indeed of order ${\nu^{p+\frac{N+1}{8}}}$ or smaller. In addition, we can rigorously prove that in appropriate norms

$\displaystyle u_n \lesssim e^{\left(1+ \frac{n}{8p}\right)\Re \lambda_0 t}, \qquad n\ge 0$

and so the error is bounded by

$\displaystyle \Phi (u_\mathrm{app}) \lesssim \sum_{k=0}^N \nu^{p+\frac{N+k+1}{8}} e^{\left(1+ \frac{N+k+1}{8p}\right)\Re \lambda_0 t} = \sum_{k=0}^N \Big( \nu^{p} e^{\Re \lambda_0 t}\Big)^{\left(1+ \frac{N+k+1}{8p}\right)}.$

In the nonlinear instability analysis, we take ${N}$ sufficiently large so that ${(1+ \frac{N+1}{8p})\Re \lambda_0}$ is larger than the rough bound on the spectral radius of the semigroup of the linearized Navier-Stokes problem. The instability time is of order ${\log 1/\nu}$ so that ${ \nu^{p} e^{\Re \lambda_0 t}}$ remains small, and so the error ${\Phi(u_\mathrm{app})}$ is indeed controlled by the linear growth.

1.3. Vlasov-Maxwell systems

The iterative scheme is also proved to be useful in the context of Vlasov-Maxwell systems, giving the nonlinear instability in the classical limit (the speed of light tends to infinity) and in the quasineutral limit (the Debye length tends to zero), in which only instability of Vlasov (leading system) is needed; see my previous blog discussions, OR precisely Section 3, my paper with Han-Kwan for the construction of approximate solutions and the instability analysis.

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