Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation. (English) Zbl 0961.35101

The author reviews resent progress on the analysis of Prandtl’s equation and the related issue of the zero-viscosity limit for the solution of the Navier-Stokes equations.
Let \(v(x,y,t)=(v_1,v_2)\) be the vector of velocity of a fluid and \(p(x,y,t)\) be the pressure. Prandtl’s equations \[ \frac{\partial v_1}{\partial t}+v_1 \frac{\partial v_1}{\partial x_1}+v_2 \frac{\partial v_1}{\partial x_2}-\frac{\partial^2 v_1}{\partial x^2_2}+ \frac{\partial p}{\partial x_1}=0,\quad \frac{\partial v_1}{\partial x_1}+\frac{\partial v_2}{\partial x_2}=0, \] with the initial condition \(v(x,0)=v_0(x)\) are considered in \( \Omega=\{x\in \mathbb{R}^2: x_2\geqslant 0\}\). The corresponding steady problem is considered too. The author presents solvability results for these problems including an unpublished one. In addition, the limit of the solution to the Navier-Stokes equations \[ \frac{\partial v^\varepsilon}{\partial t} +(v^\varepsilon\cdot\nabla)v^\varepsilon -\varepsilon\Delta v^\varepsilon+p^\varepsilon=0, \quad \nabla\cdot v^\varepsilon=0 \quad\text{in} \;\Omega \] for \(\varepsilon\to 0\) is discussed.
Finally, the author indicates some directions where progress is expected in the near future. The proofs are not presented.


35Q30 Navier-Stokes equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
76D10 Boundary-layer theory, separation and reattachment, higher-order effects
76D05 Navier-Stokes equations for incompressible viscous fluids
Full Text: DOI


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