Math 597F, Notes 2: Inviscid limit problem: absence of a boundary

ttn12 Math 597F: topics on boundary layers

In this lecture, I will briefly discuss the difficulty of the inviscid limit problem of Navier-Stokes. As it will be clear in the text, the issue is not due to the fact that the million-dollar regularity problem remains unsolved, but rather nature of the singular perturbation problem. Unless otherwise noted, throughout the course the solutions of both Euler and Navier-Stokes are assumed to be sufficiently smooth as one wishes (for instance, one works in the two-dimensional case or with local-in-time solutions, with which smooth solutions are known to exist).

Precisely, consider the incompressible NS equations in {\Omega = \mathbb{R}^n} or {\mathbb{T}^n} (periodic setting), with {n \ge 2},

\displaystyle \left \{ \begin{aligned} v_t + v \cdot \nabla v + \nabla p &= \nu \Delta v \\ \nabla \cdot v &=0 \\ v_{\vert_{t=0}} &= v_0(x), \end{aligned}\right. \ \ \ \ \ (1)

with small viscosity constant {\nu >0}. Unknowns in the equation are velocity field {v} and the pressure {p}. (note here that since {\nabla \cdot v =0}, one gets {\mathrm{div} (v\otimes v) = v \cdot \nabla v =( \sum_j v_j \partial_{x_j} v_k)}, with {v = (v_k)} denoting the vector field {v}). Assume that the initial data {v_0} and the corresponding solutions {v(x,t)} are sufficiently smooth. The most basic quantity associated with (1) is the total energy:
\displaystyle \int_\Omega |v(x,t)|^2 \; dx .

Let us calculate the rate of change of the total energy with respect to time:
\displaystyle \begin{aligned} \frac 12 \frac{d}{dt} \int_{\Omega} |v(x,t)|^2 \; dx &= \int_{\Omega} v_t \cdot v \; dx \\ &= \int_{\Omega} \Big( \nu \Delta v - \nabla p - v\cdot \nabla v \Big) \cdot v\; dx. \end{aligned}

We now apply the integration by parts (that is, using the divergence theorem and noting that there is no boundary contribution in our case as {\partial \Omega = \emptyset}). The middle term vanishes, since {\nabla \cdot v =0}. The last term is computed as follows:
\displaystyle \int_\Omega (v \cdot \nabla v) \cdot v\; dx = \int_\Omega (v\cdot \nabla ) \frac{|v|^2}{2} \; dx = - \int_{\Omega} \nabla \cdot v \frac{|v|^2}{2}\; dx = 0.

This calculation (we shall refer to it as the standard energy estimate!) yields the energy balance for NS solutions:
\displaystyle \begin{aligned} \frac 12 \frac{d}{dt} \int_{\Omega} |v(x,t)|^2 \; dx = - \nu \int_\Omega |\nabla v (x,t) |^2 \; dx. \end{aligned}

That is, smooth solutions of NS equations dissipate energy, and in particular, smooth solutions of Euler ({\nu =0}) conserve energy (i.e., constant in time).
A side remark: this latter fact turns out to be false for low-regularity Euler solutions. Precisely, Onsager (1949) conjectured that the {C^{\alpha}} Euler weak solutions conserve energy when {\alpha >1/3}, and in fact dissipate energy when {\alpha<1/3}. The first part of the conjecture was proved in 1994 by Eyink, and then by Constantin-E-Titi. The second part is essentially proved by the recent breakthrough of De Lellis and Székelyhidi Jr., and also Isett and Buckmaster (see, for instance this paper, or Isett’s PhD thesis, and the references therein), using convex integration techniques (introduced by Nash in his famous isometric embedding theorem, and later developed by Gromov in his study of h-principle). This {\frac 13} power is closely related to the theory of turbulence of Kolmogorov in 1941. I plan to expand this side remark on another blog post in the near future!

 

 

We are now back to the inviscid limit problem (for smooth solutions). Naturally, one expects to recover the Euler solutions of Euler equations: \displaystyle \left \{ \begin{aligned} v^E_t + v^E \cdot \nabla v^E + \nabla p^E &= 0 \\ \nabla \cdot v^E &=0 \\ v^E_{\vert_{t=0}} &= v^E_0(x). \end{aligned}\right. \ \ \ \ \ (2) Again, we assume the initial data {v^E_0} is sufficiently smooth, and hence there are (at least, local-in-time) smooth solutions to Euler. One quickly proves the following theorem:

 

Theorem 1 Let {v^E,v} be smooth solutions to Euler, NS equations in {[0,T]}. Then, in the energy norm, solutions to Navier-Stokes converge to those of Euler as {\nu \rightarrow 0}, if the convergence holds at initial time. Precisely, there holds \displaystyle \sup_{t\in [0,T]} \|v(t) - v^E(t)\|_{L^2} \le e^{C_0 t}\|v_0 - v_0^E\|_{L^2}+ C_0 \nu, as {\nu \rightarrow 0}, for some positive constant {C_0} that depends on certain norm of Euler solution and on time {T}. Here, the energy norm is defined by {\|\cdot \|_{L^2} = (\int_{\Omega} |\cdot|^2 \; dx)^{1/2}}.

 

Proof: The difference {w = v - v^E} satisfies {\nabla \cdot w =0} and solves \displaystyle w_t + (v^E + w)\cdot \nabla w + w\cdot \nabla v^E + \nabla (p - p^E) = \nu \Delta w + \nu \Delta v^E.

The standard energy estimate (as done above) yields

\displaystyle \frac{1}{2}\frac{d}{dt} \int_\Omega |w|^2\; dx + \nu \int_\Omega |\nabla w|^2 \; dx = - \int_\Omega (w\cdot \nabla v^E) \cdot w \; dx + \nu \int_\Omega \Delta v^E \cdot w\; dx.

Now, the first term on the right is bounded by {C_0 \| w\|_{L^2}^2}, assuming that the Euler solutions have bounded derivatives. One can use the standard Holdër’s and then Young’s inequality to treat the last term as

\displaystyle \nu \int_\Omega \Delta v^E \cdot w\; dx \le \| w\|_{L^2}^2 + \nu^2 \|\Delta v^E\|_{L^2}^2.

Putting this into the above energy equality, one gets

\displaystyle \frac{1}{2}\frac{d}{dt} \int_\Omega |w|^2\; dx + \nu \int_\Omega |\nabla w|^2 \; dx \le C_0 \|w\|_{L^2}^2 + C_0 \nu^2

with which the standard Gronwall’s inequality finishes the theorem. \Box

 

Remark: One could in fact obtain a strong convergence of Navier-Stokes to Euler in {H^s} space with {s>\frac n2+1} and lower the regularity of the initial data in the same {H^s} space. The original proof was due to Swann ’71 and Kato ’72, also by Constantin ’86, and later improved by Masmoudi ’06. Many authors study the inviscid limit problem and obtain the convergence for even less regular data; see, for instance, the study of vortex patches (vorticity is a characteristic function of a smooth domain) by P. Constantin and J. Wu ’95.
Out of the scope of this course for a moment (meaning I don’t want to go into too much details of weak solutions just yet, but the proof is in fact straightforward!): there holds the following theorem for a very weak solutions of Euler and Navier-Stokes:

 

Theorem 2 (Bardos – Titi, 2013) For given initial data {v_0} in {L^2(\Omega)}, then any {L^2} weak limit {v^E}, as {\nu \rightarrow 0}, of the weak Leray-Hopf solutions {v(x,t)} of NS equations with initial data {v_0}, is a dissipative solution of Euler.
The notion of dissipative solutions was introduced by DiPerna and Lions in the 80s. It is indeed very weak notion of weak solutions (it’s defined in term of a stability inequality, instead of equality), nevertheless it has a nice stability property. For instance, it yields the uniqueness of dissipative solutions, within the class of strong solutions in the neighborhood of a priori known smooth solution. In particular, if there exists a smooth solution, then all dissipative solutions to Euler with the same initial data must coincide with the smooth solution. Using the notion of relative entropy solutions introduced by Dafermos ’79 and also the notion of suitable weak solutions by Feireisl, Jin, and Novotn\’y, we also extend the above theorem to the compressible case and prove several criteria for the inviscid limit to hold in the case of a boundary; see this paper of Bardos and myself.

 

Open problem: prove or disprove the convergence of Navier-Stokes to Euler solutions in the case when {\Omega} has a boundary. We will discuss this issue in the next lecture (in fact, in the remaining lectures!).

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4 thoughts on “Math 597F, Notes 2: Inviscid limit problem: absence of a boundary”

  1. I am interested in your side remark. In fact, I am trying to read DeLellis, et al’s paper of non-uniqueness of admissible solutions to 2d Riemann problem during Christmas, but never complete. I am looking forward to your new post.

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