## 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 The author presents solvability results for these problems including an unpublished one.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.

### MSC:

 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

### Keywords:

boundary layer; zero-viscosity limit; Prandtl’s equations
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### References:

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