Riemann问题的奇异极限：一般扩散、松弛和边界条件
Singular limits for the Riemann problem. General diffusion, relaxation, and boundary conditions
AI总结：
研究具有Riemann初值和一般扩散矩阵的非线性双曲型系统的自相似逼近，证明光滑自相似解的存在性，并在零扩散极限下收敛到双曲系统的Riemann解，同时扩展到松弛近似和半空间边值问题。
摘要：
我们考虑一维空间中具有Riemann初值和一般扩散矩阵的非线性双曲型系统的自相似逼近。假设系统的矩阵是严格双曲的，且扩散矩阵接近单位矩阵。不需要真正的非线性假设。我们证明存在一个光滑的自相似解，其总变差有界，且一致依赖于扩散参数。在零扩散极限下，解收敛到与双曲系统相关的Riemann问题的解。对于松弛近似和半空间中的边值问题，对于相同的正则化，建立了类似的结果。
英文摘要：
We consider self-similar approximations of nonlinear hyperbolic systems in one space dimension with Riemann initial data and general diffusion matrix. We assume that the matrix of the system is strictly hyperbolic and the diffusion matrix is close to the identity. No genuine nonlinearity assumption is required. We show the existence of a smooth, self-similar solution which has bounded total variation, uniformly in the diffusion parameter. In the zero-diffusion limit, the solutions converge to a solution of the Riemann problem associated with the hyperbolic system. A similar result is established for the relaxation approximation and the boundary-value problem in a half-space for the same regularizations.
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