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Persistence of the Steady Normal Shock Structure for the Unsteady Potential Flow
SIAM Journal on Mathematical Analysis ( IF 2 ) Pub Date : 2020-12-01 , DOI: 10.1137/20m1315439 Beixiang Fang , Wei Xiang , Feng Xiao
SIAM Journal on Mathematical Analysis ( IF 2 ) Pub Date : 2020-12-01 , DOI: 10.1137/20m1315439 Beixiang Fang , Wei Xiang , Feng Xiao
SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 6033-6104, January 2020.
This paper is devoted to the study of the stability of the steady normal shock structure in potential flows under an unsteady perturbation. The dynamic stability problem is formulated as the well-posedness problem of an initial boundary value problem of a nonlinear wave equation in a cornered space domain with a free boundary. The corner singularity is the essential difficulty and there is no result available even for the linear problem without the symmetry assumptions, i.e., “even” or “odd” traces vanish on the solid boundary, which allows extension from the cornered space domain to the half-space domain, as in the previous works. In this paper, we first obtain an existence result for the initial boundary value problem of linear hyperbolic equations of second order in a cornered space domain without such symmetry assumptions. The key idea is based on the construction of a new auxilliary problem, which allows us to reduce the linear problem to a new one that can be even extended to a half-space domain such that the existence of $H_\eta^2$-solutions can be established. However, due to the lack of the symmetry assumptions, the low regularity of the extended coefficients block us from obtaining the higher regularity of the solutions in the extended domain, which is necessary for the iteration to the nonlinear problem. In order to deal with it, new hyperbolic type and elliptic type estimates in the cornered space domain are established carefully. The results on the general linear problems can be applied to the linearized problem that we are concerned with in this paper. Due to the loss of regularity in the estimates of the linearized problem, a modified Nash--Moser iteration is developed.
中文翻译：
非稳态势流的稳态法向冲击结构的持久性
SIAM数学分析杂志，第52卷，第6期，第6033-6104页，2020年1月。
本文致力于非稳态扰动下稳态法向激波结构在势流中的稳定性研究。将动态稳定性问题表述为具有自由边界的弯角空间域中非线性波动方程初始边界值问题的适定性问题。拐角奇点是基本困难，没有对称性假设，即使对于线性问题也没有结果可用，即，“偶数”或“奇数”迹线在固体边界上消失，这允许从拐角空间域扩展到一半-space域，如先前的作品中所述。本文首先在没有这种对称性假设的情况下，获得了转角空间域中二阶线性双曲型方程组初始边值问题的存在性结果。关键思想是基于构造一个新的辅助问题，这使我们可以将线性问题简化为一个新问题，该问题甚至可以扩展到半空间域，从而存在$ H_ \ eta ^ 2 $-可以建立解决方案。然而，由于缺乏对称性假设，扩展系数的低规则性使我们无法获得扩展域中解的较高规则性，这对于迭代非线性问题是必需的。为了解决这个问题，在角空间域中仔细建立了新的双曲型和椭圆型估计。一般线性问题的结果可以应用于本文所关注的线性化问题。由于线性化问题的估计中缺少规则性，
更新日期：2020-12-02
This paper is devoted to the study of the stability of the steady normal shock structure in potential flows under an unsteady perturbation. The dynamic stability problem is formulated as the well-posedness problem of an initial boundary value problem of a nonlinear wave equation in a cornered space domain with a free boundary. The corner singularity is the essential difficulty and there is no result available even for the linear problem without the symmetry assumptions, i.e., “even” or “odd” traces vanish on the solid boundary, which allows extension from the cornered space domain to the half-space domain, as in the previous works. In this paper, we first obtain an existence result for the initial boundary value problem of linear hyperbolic equations of second order in a cornered space domain without such symmetry assumptions. The key idea is based on the construction of a new auxilliary problem, which allows us to reduce the linear problem to a new one that can be even extended to a half-space domain such that the existence of $H_\eta^2$-solutions can be established. However, due to the lack of the symmetry assumptions, the low regularity of the extended coefficients block us from obtaining the higher regularity of the solutions in the extended domain, which is necessary for the iteration to the nonlinear problem. In order to deal with it, new hyperbolic type and elliptic type estimates in the cornered space domain are established carefully. The results on the general linear problems can be applied to the linearized problem that we are concerned with in this paper. Due to the loss of regularity in the estimates of the linearized problem, a modified Nash--Moser iteration is developed.
中文翻译：
非稳态势流的稳态法向冲击结构的持久性
SIAM数学分析杂志，第52卷，第6期，第6033-6104页，2020年1月。
本文致力于非稳态扰动下稳态法向激波结构在势流中的稳定性研究。将动态稳定性问题表述为具有自由边界的弯角空间域中非线性波动方程初始边界值问题的适定性问题。拐角奇点是基本困难，没有对称性假设，即使对于线性问题也没有结果可用，即，“偶数”或“奇数”迹线在固体边界上消失，这允许从拐角空间域扩展到一半-space域，如先前的作品中所述。本文首先在没有这种对称性假设的情况下，获得了转角空间域中二阶线性双曲型方程组初始边值问题的存在性结果。关键思想是基于构造一个新的辅助问题，这使我们可以将线性问题简化为一个新问题，该问题甚至可以扩展到半空间域，从而存在$ H_ \ eta ^ 2 $-可以建立解决方案。然而，由于缺乏对称性假设，扩展系数的低规则性使我们无法获得扩展域中解的较高规则性，这对于迭代非线性问题是必需的。为了解决这个问题，在角空间域中仔细建立了新的双曲型和椭圆型估计。一般线性问题的结果可以应用于本文所关注的线性化问题。由于线性化问题的估计中缺少规则性，
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