Convergence of solutions to Navier-Stokes to Euler's equation for viscosity $\to$ zero

Question

Let $$ \partial_t u + \nabla_u u = - \nabla p $$ be Euler's equation (Wikipedia) for an ideal incompressible fluid. Let

$$ \partial_t u + \nabla_u u - \nu \Delta u = - \nabla p $$ be the Navier-Stokes equations for a Newtonian incompressible fluid with a real parameter $\nu$, the viscosity. Obviously this equation reduces to Euler's equation if we set $\nu = 0$ (and domains and initial conditions are equal and suitable boundary conditions are prescribed).

What is known about the convergence of solutions $u_{\nu}$ of the Navier-Stokes equation as stated above to a solution of Euler's equation in the limit $\nu \to 0?$ That is, are there topological vector spaces T and sequences or nets of solutions $u_{\nu_k}$ of the Navier-Stokes equation with viscosity $\nu_k$ in T such that the limit $\lim_{\nu_k \to 0} u_{\nu_k}$ exists and is a solution to Euler's equation?

Answer 1

To complete Michael's answer, the only situation that is under control is that of the Cauchy problem: the spatial domain is ${\mathbb R}^d$ or ${\mathbb T}^d$ (case of periodic solutions). This means that there is no boundary condition.

If $d=2$, both systems are globally well-posed for $t>0$, with uniformly bounded (in $L^2$) solutions, and $u_\nu$ converges strongly to the solution of the Euler equation. Notice that it is not a trivial fact: the reasons why both Navier-Stokes and Euler Cauchy problems are globally well-posed have nothing in common; for Navier-Stokes, it comes from the Ladyzhenskaia inequality (say, $\|w\|_{L^4}^2\le c\|w\|_{L^2}\|\nabla w\|_{L^2}$), while for Euler, it is the transport of the vorticity.

If $d=3$, both Cauchy problems are locally-in-time well-posed for smooth enough initial data. One has a convergence as $\nu\rightarrow0+$ on some time interval $(0,\tau)$, but $\tau$ might be strictly smaller than both the time of existence of Euler and the $\lim\inf$ of the times of existence for Navier-Stokes.

To my knowledge, the initial-boundary value problem is a nightmare. The only result of convergence is in the case of analytic data (Caflisch & Sammartino, 1998). From time to time, a paper or a preprint appears with a "proof" of convergence, but so far, such papers have all be wrong.

By the way, your question is incorrectly stated, when you say boundary conditions are equal. The boundary condition for NS is $u=0$, whereas that for Euler is $u\cdot\vec n=0$, where $\vec n$ is the normal to the boundary. This discrepancy is the cause of the boundary layer. One may say that the difficulty lies in the fact that this boundary layer is characteristic. Non-characteristic singular limits are easier to handle.

Another remark is that some other boundary condition for NS are better understood. For instance, there is a convergenece result (Bardos) when $u=0$ is replaced by $$u\cdot\vec n=0,\qquad {\rm curl}u\cdot\vec n=0.$$

Answer 2

There is a rather extensive literature on this, starting with the following paper of Kato: T. Kato, Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary, In: Seminar on nonlinear partial differential equations (Berkeley, Calif., 1983), volume 2 of Math. Sci. Res. Inst. Publ., pages 85–98. Springer, New York, 1984.

One of the obstacles to obtaining better results is that the Prandtl boundary layer equations are difficult to analyze and not necessarily well posed. This is shown in recent work by Gerard-Varet and Dormy: D. Gerard-Varet and E. Dormy, On the ill-posedness of the Prandtl equation, J. Amer. Math. Soc., 23 (2010), 591–609.