2012/07/29 by Olivier Guès, Gues, Olivier, Metivier, Guy +6
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1207.6782
openalex publication_date 2012/07/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We initiate the study of noncharacteristic boundary layers in\nhyperbolic-parabolic problems with Neumann boundary conditions. More generally,\nwe study boundary layers with mixed Dirichlet--Neumann boundary conditions\nwhere the number of Dirichlet conditions is fewer than the number of hyperbolic\ncharacteristic modes entering the domain, that is, the number of boundary\nconditions needed to specify an outer hyperbolic solution. We have shown\npreviously that this situation prevents the usual WKB approximation involving\nan outer solution with pure Dirichlet conditions. It also rules out the usual\nmaximal estimates for the linearization of the hyperbolic-parabolic problem\nabout the boundary layer.\n Here we show that for linear, constant-coefficient, hyperbolic-parabolic\nproblems one obtains a reduced hyperbolic problem satisfying Neumann or mixed\nDirichlet--Neumann rather than Dirichlet boundary conditions. When this\nhyperbolic problem can be solved, a unique formal boundary-layer expansion can\nbe constructed. In the extreme case of pure Neumann conditions and totally\nincoming characteristics, we carry out a full analysis of the quasilinear case,\nobtaining a boundary-layer approximation to all orders with a rigorous error\nanalysis. As a corollary we characterize the small viscosity limit for this\nproblem. The analysis shows that although the associated linearized hyperbolic\nand hyperbolic--parabolic problems do not satisfy the usual maximal estimates\nfor Dirichlet conditions, they do satisfy analogous versions with losses.\n