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首页> 外文期刊>Annales scientifiques de l'Ecole normale superieure >VISCOUS BOUNDARY LAYERS IN HYPERBOLIC-PARABOLIC SYSTEMS WITH NEUMANN BOUNDARY CONDITIONS
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VISCOUS BOUNDARY LAYERS IN HYPERBOLIC-PARABOLIC SYSTEMS WITH NEUMANN BOUNDARY CONDITIONS

机译：具有NEUMANN边界条件的双曲抛物系统中的粘性边界层

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We initiate the study of noncharacteristic boundary layers in hyperbolic-parabolic problems with Neumann boundary conditions. More generally, we study boundary layers with mixed Dirichlet-Neumann boundary conditions where the number of Dirichlet conditions is fewer than the number of hyperbolic characteristic modes entering the domain, that is, the number of boundary conditions needed to specify an outer hyperbolic solution. We have shown previously that this situation prevents the usual WKB approximation involving an outer solution with pure Dirichlet conditions. It also rules out the usual maximal estimates for the linearization of the hyperbolic-parabolic problem about the boundary layer. Here we show that for linear, constant-coefficient, hyperbolic-parabolic problems one obtains a reduced hyperbolic problem satisfying Neumann or mixed Dirichlet-Neumann rather than Dirichlet boundary conditions. When this hyperbolic problem can be solved, a unique formal boundary-layer expansion can be constructed. In the extreme case of pure Neumann conditions and totally incoming characteristics, we carry out a full analysis of the quasilinear case, obtaining a boundary-layer approximation to all orders with a rigorous error analysis. As a corollary we characterize the small viscosity limit for this problem. The analysis shows that although the associated linearized hyperbolic and hyperbolic-parabolic problems do not satisfy the usual maximal estimates for Dirichlet conditions, they do satisfy analogous versions with losses.
机译：我们开始研究具有Neumann边界条件的双曲-抛物线问题的非特征边界层。更一般而言，我们研究混合Dirichlet-Neumann边界条件的边界层，其中Dirichlet条件的数量少于进入域的双曲特征模式的数量，即指定外部双曲解所需的边界条件的数量。先前我们已经表明，这种情况阻止了通常的WKB逼近，其中涉及具有纯Dirichlet条件的外部解。它还排除了关于边界层的双曲-抛物线问题线性化的通常最大估计。在这里，我们表明对于线性，常系数，双曲抛物问题，获得了满足Neumann或混合Dirichlet-Neumann而不是Dirichlet边界条件的简化双曲问题。当这个双曲问题可以解决时，可以构造一个独特的形式边界层扩展。在纯诺伊曼条件和完全传入特征的极端情况下，我们对准线性情况进行了全面分析，并通过严格的误差分析获得了所有阶的边界层近似值。作为推论，我们描述了该问题的小粘度极限。分析表明，尽管相关的线性双曲和双曲抛物线问题不满足Dirichlet条件的通常最大估计，但它们确实满足相似形式的损失。

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