Math 597F, Notes 3: Inviscid limit in the presence of a boundary

ttn12 Math 597F: topics on boundary layers

Most physicists don’t believe there is such an ideal fluid (i.e., no viscosity). It is clear however that the zero viscosity or infinite Reynolds number limit plays a central role in understanding turbulence, as seen in Kolmogorov’s theory, Onsager’s conjecture, and turbulent boundary layers. Hence, understanding the inviscid limit problem is of great practical and analytical importance. As expected in most singular perturbation problems, new phenomena will arise.

Today, I’d like to discuss (part of) what one can expect to prove in the inviscid limit. As we are dealing with the function spaces of infinite dimension (continuous functions, integrable functions, Sobolev spaces, and so on), the convergence of NS to Euler solutions is sensitive to the underlying function spaces, and so we will have to specify it appropriately.

Let {\Omega} be a smooth domain in {\mathbb{R}^n, ~n\ge 2,} that has a nonempty boundary. Consider again the usual NS equations (see equation (1) in the last lecture), accompanied with the classical zero boundary condition:

\displaystyle v_{\vert_{\partial \Omega}} = 0.

This condition means that the fluid on the boundary sticks to the boundary (widely known as no-slip boundary condition). In the zero viscosity limit, formally we get the Euler equation {(\nu =0).} It’s a hyperbolic equation for v^E being transported along the characteristics curves defined by the particle trajectories \dot X(t) = v^E(X(t),t) (for instance, see the first lecture), and so one needs only to prescribe the normal velocity component on the boundary in order to completely determine the particle trajectories and hence the Euler velocity inside the domain. That is, we impose the no-penetration boundary condition for Euler:

\displaystyle v^E_{\vert_{\partial \Omega}} \cdot n = 0,

where {n} denotes the outer normal unit vector on the boundary {\partial \Omega.} Again, there are local smooth solutions to both Euler and NS equations. One might like to prove the following:

A mathematical problem: Given sufficiently smooth, compatible initial data, do NS solutions converge to Euler solutions with the same initial data in the {L^2} energy norm ?

The problem is proved in the case when the initial data are analytic functions and {\Omega} has a flat boundary {\{x_n=0\}:} this is the celebrated work of Sammartino and Caflisch back in 1998 (I am not aware of any such work for curved domains!). This is essentially only positive result, affirmatively proving the convergence. It is noted that analytic functions have exponentially localized spatial frequency, which therefore excludes many interesting fluid instabilities that typically occur in the high frequency regime. Remarkably, other than the result in the analytic setting, the problem is completely open, even within arbitrarily small time; for instance, it remains open for {C^\infty} compactly supported initial data (or even for initial data in Gevrey class {G^s}, for any {s>1}). There are a few other interesting results proving the convergence by my colleague A. Mazzucato at Penn State and her collaborators, for special (NS) flows such as parallel flows. However, the convergence is not known even for an arbitrarily small initial perturbation of these flows. Recently, Maekawa was able to prove the inviscid limit convergence, assuming that initial vorticity (and hence, Euler vorticity) is compactly supported away from the boundary. Under such a (zero) assumption near the boundary, he was able to modify the Cauchy-Kowalewski theorem, previously developed by Sammartino and Caflisch for the analytic setting, and obtain the strong convergence under the zero assumption on initial vorticity near the boundary. The above is essentially the current “state of art” in proving the convergence (unconditionally). Let me now discuss the delicacy of the problem and prove a few conditional theorems.

 

To appreciate the delicacy of the matter, let us for a moment assume that there are approximate solutions {v_\mathrm{app}} that solve the NS equations with the desired boundary conditions, up to an arbitrarily small error {\nu^\infty} as {\nu \rightarrow 0.} Set {w = v - v_\mathrm{app}} to be the difference of the exact and approximate solutions. The difference solves {\nabla \cdot w = 0} and

\displaystyle w_t + (v_\mathrm{app} + w)\cdot \nabla w + w\cdot \nabla v_\mathrm{app} + \nabla (p - p_\mathrm{app}) = \nu \Delta w + \mathcal{O}(\nu^{\infty})

whose standard energy estimates yield at once

\displaystyle \frac 12 \frac{d}{dt} \| w\|_{L^2}^2 + \nu \| \nabla w\|_{L^2}^2 = - \int_\Omega (w\cdot \nabla v_\mathrm{app})\cdot w \; dx + \mathcal{O}(\nu^\infty),

in which there is no boundary contribution, thanks to the zero boundary conditions. One immediately encounters a serious difficulty: the convection term is too large as compared to the a priori bound on the left. Indeed, {\nabla v_\mathrm{app}} is generically of order {1/\sqrt \nu} in the sup norm, which by Gronwall’s means the convergence might only hold in the vanishing time of order {\sqrt \nu}. The reason that {\nabla v_\mathrm{app}} is extremely large is due to the mismatch of the boundary conditions of Euler and NS. Indeed, the tangential component of velocity of Euler might be nonzero, whereas that of NS vanishes, on the boundary. As the consequence, locally near the boundary the approximate solution {v_\mathrm{app}} has a rapid change connecting zero boundary value to a nonzero value of the trace of Euler on the boundary. In the other words, extremely large vorticity of order {1/\sqrt \nu} in the sup norm is formed near the boundary. This is not to mention that construction of approximate solutions is another delicate issue, a subject to be discussed in the next lecture.

In search for an affirmative convergence result, Kato in his 1984 paper constructed the “fake” (as opposed to the classical) boundary layer {v^K} so that {v_\mathrm{app} = v^E + v^K} satisfies the zero boundary conditions (recalling that the tangential component of Euler velocity {v^E\cdot \tau} might not be zero). His fake layer is supported near the boundary:

\displaystyle \mathrm{supp}(v^K) \subset \Gamma_\nu : = \{ x\in \Omega ~:~ d(x,\partial\Omega) \le \nu\} .

For instance, in the half-space domain with boundary {\{y=0\}}, one might simply take

\displaystyle v_K = - \nabla^\perp \Big( \chi (\frac{y}{\nu}) y v^E_1\Big),

which is divergence-free and {v^K = - v^E} on the boundary {\{y=0\}.} Here, {\chi(\cdot)} is some smooth cut-off function with support within {[0,1]} and with {\chi(0) = 1} (for general domains, {y} is replaced by the distance function {d(x,\partial \Omega)} to the boundary and {v_1^E} being the tangential component of velocity, locally with respect to the boundary!).

Having defined {v_\mathrm{app} = v^E + v^K}, we now apply the above energy estimates, yielding

\displaystyle \frac 12 \frac{d}{dt} \| w\|_{L^2}^2 + \nu \| \nabla w\|_{L^2}^2 = - \int_\Omega (w\cdot \nabla v_\mathrm{app})\cdot w \; dx + \int_\Omega R_{\mathrm{app}} \cdot w\; dx,\ \ \ \ \ (1)

in which {R_\mathrm{app}} denotes the error of “approximation“, which in our case is simply the whole NS equation as we didn’t take care of the equation for {v^K}:

\displaystyle R_\mathrm{app}: = v^K_t + v^K \cdot \nabla v^K - \nu \Delta v^K.

Since the Kato’s layer is supported in a thin domain having the measure of order {\nu}, there hold trivially the estimates:

\displaystyle \|v^K \|_{L^\infty}\lesssim 1, \qquad \| v^K \|_{L^2} \lesssim \nu^{1/2}, \qquad \|v^K_t\|_{L^2} \lesssim \nu^{1/2}, \qquad \|\nabla v^K \|_{L^2} \lesssim \nu^{-1/2}.

Using this and the fact that the integration of {(w\cdot\nabla v)\cdot v} vanishes, we estimate the right-hand side of (1), yielding at once

\displaystyle \begin{aligned} \int_\Omega (v^K \cdot \nabla v^K) \cdot w \; dx &= \int_\Omega (v^K \cdot \nabla v^K) \cdot (v - v^E - v^K) \; dx = \int_\Omega (v^K \cdot \nabla v^K )\cdot (v-v^E) \; dx \\ &\le \int_{\Gamma_\nu} \nu |v^K \cdot \nabla v^K| \frac{|v-v^E|}{d(x,\partial\Omega)} \; dx \\ &\lesssim \nu \| v^K \cdot \nabla v^K \|_{L^2(\Gamma_\nu)} \| \nabla (v-v^E)\|_{L^2(\Gamma_\nu)} \\ &\lesssim \sqrt \nu \|\nabla (v-v^E)\|_{L^2(\Gamma_\nu)}, \end{aligned}

in which the standard Hardy’s inequality was used. The other two terms in {R_\mathrm{app}} are treated similarly (in fact, trivially). Let us finally give bounds on the convection term

\displaystyle \begin{aligned} \int_\Omega (w\cdot \nabla v_\mathrm{app})\cdot w \; dx &\le \| \nabla v^E\|_{L^2} \| w\|_{L^2}^2 + \int_\Omega (w\cdot \nabla v^K)\cdot (v - v^E) \; dx \\ &\le \| \nabla v^E\|_{L^2} \| w\|_{L^2}^2 + \nu^2 \|\nabla v^K \|_{L^\infty} \int_{\Gamma_\nu} \Big| \frac{v-v^E}{d(x,\partial \Omega)}\Big|^2\; dx + \int_\Omega (v^K\cdot \nabla v^K)\cdot (v - v^E) \; dx \\ &\le \| \nabla v^E\|_{L^2} \| w\|_{L^2}^2 +\nu \|\nabla (v-v^E)\|_{L^2(\Gamma_\nu)}^2 + \sqrt \nu \|\nabla (v-v^E)\|_{L^2(\Gamma_\nu)},\end{aligned}

in which again the Hardy’s inequality and the previous estimate were used. Putting these together, one gets at once the energy inequality

\displaystyle \frac 12 \frac{d}{dt} \| w\|_{L^2}^2 + \nu \| \nabla w\|_{L^2}^2 \lesssim \| w\|_{L^2}^2 +\sqrt \nu \|\nabla v\|_{L^2(\Gamma_\nu)} + \delta(\nu), \ \ \ \ \ (2)

for some constant {\delta(\nu)\rightarrow 0} as {\nu \rightarrow 0}.

Kato then assumes that the anomalous dissipation within a very thin layer near the boundary tends to zero

\displaystyle \nu \int_0^T \int_{\Gamma_\nu} |\nabla v|^2\; dx dt \rightarrow 0,\ \ \ \ \ (3)

widely known as Kato’s criterium for the inviscid limit to hold. Indeed, if (3) holds, the standard Gronwall’s inequality applied to (2) yields that the {L^2} energy norm of {w = v - v^E - v^K} tends to zero, uniformly in time. As the {L^2} energy norm of Kato layer is of order {\nu^{1/2}}, this proves the {L^2} convergence of NS to Euler solutions, under the assumption (3). Now, the necessity of this condition follows trivially from the energy equality of NS solutions; see the beginning of this lecture. Let me summarize Kato’s criteria as follows:

Theorem 1 (Kato’s criterium, 1984) The followings are equivalent in the inviscid limit {\nu \rightarrow 0}:

(i) {v \rightarrow v^E} strongly in {L^2(\Omega)}, uniformly in time;

(ii) {v \rightarrow v^E} weakly in {L^2(\Omega)}, a.e. in time;

(iii) {\nu \int_0^T \int_{\Omega} |\nabla v|^2\; dx dt \rightarrow 0};

(iv) {\nu \int_0^T \int_{\{ d(x,\partial\Omega)\le \nu\}} |\nabla v|^2\; dx dt \rightarrow 0}.

Since Kato’s work, there are many variants of his criteria for the inviscid limit to hold. For instance, this very recent paper by J. Kelliher has many interesting observations, centering around Kato-type conditions. Two other equivalent kinds of conditions that I thought to mention:

(1) Bardos-Titi, 2013: the inviscid limit holds in a very weak sense if and only if {\nu (\omega \times n)\cdot \tau } converges to zero weakly in {L^2(\partial \Omega)} on the boundary, where {\omega = \nabla \times v} denotes the vorticity.

(2) The one-sided condition by Constantin-Kukavica-Vicol, 2014: roughly speaking, the inviscid limit holds for no backflow Euler solution {v^E \cdot \tau \ge 0} if and only if {\nu (\omega \times n)\cdot \tau \ge M(t)} near the boundary, with {\lim_{\nu \rightarrow 0} \int_0^T M(t) \; dt =0}.
The Kato’s criteria and its variants appear equally hard to verify. They do however indicate that there would be violent behaviors occurring in the thin layer of size \nu near the boundary, should the inviscid limit fail to hold. The size \nu is much smaller than the size \sqrt \nu of the classical boundary layers, predicted by Prandtl’s theory (will be discussed in the next lecture). For instance, Prandtl predicted that vorticity is created of order \frac{1}{\sqrt \nu} near the boundary, whereas the mentioned criteria indicate that the vorticity would be much larger of order \frac{1}{\nu} near the boundary, should the inviscid limit fail.

 

Additional notes (optional): Let me go a bit further in details explaining the above two recent criteria for the inviscid limit to hold. Unlike Kato’s, let us perform the energy estimate for {w = v - v^E}, without adding the fake layer to take care of the nonzero boundary conditions of {v^E}. It follows that {\nabla \cdot w = 0} and

\displaystyle w_t + (v^E+ w)\cdot \nabla w + w\cdot \nabla v^E + \nabla (p - p^E) = \nu \Delta v .

Note that {w \cdot n = 0} on the boundary, and hence integration by part can still be performed for the second and forth terms on the left, leaving no contribution from the boundary. Certainly, the Laplacian term now causes trouble, since {w\cdot \tau = - v^E \cdot \tau} might not be zero. This immediately yields

\displaystyle \frac 12 \frac{d}{dt} \| w\|_{L^2}^2 = - \int_\Omega (w\cdot \nabla v^E) \cdot w \; dx - \nu \int_\Omega \nabla v : \nabla w \; dx - \nu \int_{\partial\Omega} (n\cdot \nabla v) \cdot v^E \; d\sigma,

in which {\nabla v : \nabla w = \sum_{jk} \partial_{x_j} v_k \partial_{x_j}w_k}. The first two terms on the right are treated exactly as done in the case without a boundary (see last lecture). The Bardos-Titi’s criterium is really the obvious condition that {(n\cdot \nabla v) \cdot \tau \rightarrow 0} weakly in {L^2(\Omega)} in the inviscid limit, so that the above (only) boundary term vanishes in the limit (recalling that {v^E\cdot n = 0} on the boundary). One could state it in term of vorticity {\omega = \nabla \times v} by observing the trivial calculations:

\displaystyle (\nabla v - (\nabla v)^t) n \cdot \tau = (\omega \times n)\cdot \tau

and

\displaystyle (\nabla v)^t n \cdot \tau = \sum_{k,j} \tau_k\partial_k v_j n_j = \tau \cdot \nabla (v\cdot n) - (\tau \cdot \nabla) n \cdot v = - (\tau \cdot \nabla) n \cdot v,

 

in which the last identity was due to the assumption that {v \cdot n=0} on the boundary and the fact that {\tau} is tangent to the boundary. The term {\sqrt \nu \| v\|_{L^2(\partial \Omega)}} can be absorbed to the left by the energy dissipation {\sqrt \nu \|\nabla v\|_{L^2}}.

 

The main point in Bardos-Titi’s criterium is that they allow the solution to Euler to be very weak (even weaker than usual distributional weak solutions!): dissipative solutions. Due to the nature of dissipative solutions, which are defined via a stability inequality in term of the energy difference of the solution and test functions, the proof again follows directly from the basic energy estimate as done above and a use of the Gronwall’s inequality.

 

As for the Constantin-Kukavica-Vicol’s criterium, their motivation is to use the sign of the tangential component of Euler flows on the boundary, which is assumed to be nonnegative. They then reexamine the most crucial term in the energy estimate; namely,

\displaystyle \begin{aligned} - \nu \int_{\Gamma_\nu} \nabla v : \nabla v^K & =- \nu \int_{\Gamma_\nu} \nabla v : \Big( \tau^t n (n\cdot \nabla v^K) + n^t \tau (\tau\cdot \nabla v^K)\Big) \\&\le - \nu \int_{\Gamma_\nu} (\nabla v)n \cdot \tau (n\cdot \nabla v^K) + C \nu^{3/2} \| \nabla v\|_{L^2(\Gamma_\nu)}. \end{aligned}

Here, {n,\tau} are local normal and tangential curvilinear coordinates in the thin region {\Gamma_\nu} near the boundary. As the Kato’s layer only changes rapidly in the normal direction, {\tau \cdot \nabla v^K} is uniformly bounded. Thus, the last term in the above can be absorbed into the energy dissipation term. As for the first term, one can write

\displaystyle (\nabla v)n \cdot \tau = (\omega \times n)\cdot \tau + \tau \cdot \nabla (v\cdot n) - (\tau \cdot \nabla) n \cdot v

and estimate

\displaystyle \begin{aligned} \nu \int_{\Gamma_\nu} \tau \cdot \nabla (v\cdot n) (n\cdot \nabla v^K) &= - \nu \int_{\Gamma_\nu} (v\cdot n) (\tau \cdot \nabla )(n\cdot \nabla v^K) \le C \int_{\Gamma_\nu}|v | \\ \nu \int_{\Gamma_\nu} (\tau \cdot \nabla) n \cdot v (n\cdot \nabla v^K)\Big| &\le C \int_{\Gamma_\nu}|v | \le C \nu \int_{\Gamma_\nu} |\nabla v| \le C \nu^{3/2} \| \nabla v\|_{L^2(\Gamma_\nu)}, \end{aligned}

both of which are neglected in the energy estimate. It remains to estimate the term involving the vorticity. By a view of Kato’s construction, the largest term in {n\cdot \nabla v^K} is when the derivative hits the cut-off function, yielding

\displaystyle - \frac{1}{\nu} \chi' (\frac{d(x,\partial\Omega)}{\nu}) v^E \cdot \tau + \mathcal{O}(1).

Now, observing that the cut-off function can be constructed so that it is decreasing: {\chi'\le 0}, using the sign condition on the Euler flow: {v^E \cdot \tau \ge 0}, and assuming a sign condition on vorticity: {\nu (\omega \times n)\cdot \tau \ge M(t)}, one can estimate

\displaystyle \begin{aligned} - \nu \int_{\Gamma_\nu} (\omega \times n)\cdot \tau (n\cdot \nabla w^\tau) &= -\nu \int_{\Gamma_\nu} (\omega \times n)\cdot \tau \Big( - \frac 1\nu \chi' (\frac{d(x,\partial\Omega)}{\nu}) v^E \cdot \tau + \mathcal{O}(1)\Big) \\ &\le \frac{M(t)}{\nu} \int_{\Gamma_\nu} \chi' (\frac{d(x,\partial\Omega)}{\nu}) v^E \cdot \tau + C\nu \int_{\Gamma_\nu} |\nabla v| \\ &\le C M(t) + C \nu^{3/2} \| \nabla v\|_{L^2(\Gamma_\nu)}. \end{aligned}

Hence, under the assumption {\int_0^T M(t)\; dt \rightarrow 0} in the inviscid limit, this most crucial term in the energy estimate is taken care of. Other terms are treated similarly. Of course, interested readers should consult their paper for precise and careful treatments of all the terms in the energy estimate.

Print

Share this post:

Share on LinkedIn

2 thoughts on “Math 597F, Notes 3: Inviscid limit in the presence of a boundary”

  1. Consider an unbounded domain with a simple curve as boundary. if we assume the boundary is analytic, can we just adapt Sammartino and Caflisch’s proof to this curved boundary case?

    • The point is that the smoothness is needed for the initial data (not just the smoothness of a boundary). In fact, even in the flat boundary, Sammartino and Caflisch proved less than what’s mentioned in the blog: they were able to prove the asymptotic expansion of Navier-Stokes = Euler, plus a Prandtl layer, plus small perturbation, where all data for Euler, Prandtl, and perturbation are in the analytic function space. In particular, the result implies the inviscid limit. This type of results is sometimes referred to as “inviscid limit with well-prepared initial data”). Since there is a loss of derivative in the estimates for Prandtl layers, their use of analytic function space was very crucial to control this loss of derivatives. I don’t think such a result is extended to a curved boundary, for various reasons, which I could only think of: (1) Prandtl equation now depends on the curvature of the boundary, essentially the equation is different near each region on the boundary, and (2) one might need to work with micro-local analysis, as now if you’d straight the boundary, the Laplace operator becomes a second order operator with variable coefficients…., and there might be others more serious that I didn’t know!

Leave a Reply