Existence and Stability of Noncharacteristic Boundary Layers for the Compressible Navier–Stokes and Viscous MHD Equations
Existence and Stability of Noncharacteristic Boundary Layers for the Compressible Navier–Stokes and Viscous MHD Equations
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DOI：
10.1007/s00205-009-0277-y
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发表时间：
2008-05
影响因子：
2.5
通讯作者：
O. Guès;G. Métivier;Mark E. Williams;K. Zumbrun
中科院分区：
文献类型：
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作者：
O. Guès;G. Métivier;Mark E. Williams;K. Zumbrun
对于一类包含可压缩Navier-Stokes方程和可压缩MHD方程的双曲抛物方程组，我们证明了在包括经典Navier-Stokes边界条件在内的各种边界条件下非特征粘性边界层的存在性和稳定性.我们的第一个主要结果，使用的抽象框架建立的作者在同伴的工作（Guidalal。在J Differ Equ，244，309-387（2008）中）是为了显示任意振幅精确边界层解的存在性和稳定性，该边界层解遵循关于层轮廓的一致谱稳定性条件，该条件可以用Evans函数表示（一致Evans稳定性）。只要这个条件成立，我们给出了一个严格的描述的小粘度极限的解决方案的双曲型问题的“剩余”的边界条件。我们的第二个是表明，均匀埃文斯稳定性的小振幅层是等价的埃文斯稳定性的限制常数层，这反过来又可以检查的线性代数计算。最后，对于一类包含上述物理例子的非线性耗散系统，我们进行了能量估计，结果表明常数（因此小振幅）层总是满足一致Evans稳定性。这就产生了可压缩Navier-Stokes方程和MHD方程的小振幅多维边界层的存在性。对于这两个方程，这些似乎是第一个这样的结果，在可压缩的情况下。
For a general class of hyperbolic–parabolic systems including the compressible Navier–Stokes and compressible MHD equations, we prove existence and stability of noncharacteristic viscous boundary layers for a variety of boundary conditions including classical Navier–Stokes boundary conditions. Our first main result, using the abstract framework established by the authors in the companion work (Gueset al. in J Differ Equ,244, 309–387 (2008)), is to show that existence and stability of arbitrary amplitude exact boundary layer solutions follow from a uniform spectral stability condition on layer profiles that is expressible in terms of an Evans function (uniform Evans stability). Whenever this condition holds we give a rigorous description of the small viscosity limit as the solution of a hyperbolic problem with “residual” boundary conditions. Our second is to show that uniform Evans stability for small-amplitude layers is equivalent to Evans stability of the limiting constant layer, which in turn can be checked by a linear-algebraic computation. Finally, for a class of symmetric-dissipative systems including the physical examples mentioned above, we carry out energy estimates showing that constant (and thus small-amplitude) layers always satisfy uniform Evans stability. This yields existence of small-amplitude multi-dimensional boundary layers for the compressible Navier–Stokes and MHD equations. For both equations these appear to be the first such results in the compressible case.