Abstract It is well-known that the 3D incompressible Euler equations admit some local-in-time solutions for which the vorticity is piecewise smooth and discontinuous across a smooth time-dependent hypersurface which evolves with the flow. In this paper we prove that such a solution can be obtained as zero-viscosity limit of strong solutions to the Navier-Stokes equations whose vorticity has sharp variations near the hypersurface associated with the inviscid limit. Indeed we exhibit some sequences of exact solutions to the Navier-Stokes equations with vanishing viscosity which are given by multi-scale asymptotic expansions involving some characteristic boundary layers given by some linear PDEs. The convergence and the validity of the expansion are guaranteed on the time interval associated with the solution to the Euler equations.

中文翻译：

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具有尖锐涡度梯度的不可压缩 Navier-Stokes 方程的零粘度极限

摘要 众所周知，3D 不可压缩欧拉方程允许一些局部时间解，其中涡度在随流动演化的光滑瞬态超曲面上是分段平滑和不连续的。在本文中，我们证明了这样的解可以作为 Navier-Stokes 方程的强解的零粘度极限获得，该方程的涡度在与无粘性极限相关的超曲面附近具有急剧变化。事实上，我们展示了一些具有粘度消失的 Navier-Stokes 方程的精确解序列，这些方程由涉及一些线性偏微分方程给出的一些特征边界层的多尺度渐近展开给出。在与欧拉方程的解相关的时间间隔上保证了扩展的收敛性和有效性。