Study of the stability in the problem on flowing around a wedge. The case of strong wave

Abstract

We consider the flow of an inviscid nonheatconducting gas in the thermodynamical equilibrium state around a plane infinite wedge and study the stationary solution to this problem, the so-called strong shock wave; the flow behind the shock front is subsonic.We find a solution to a mixed problem for a linear analog of the initial problem, prove that the solution trace on the shock wave is the superposition of direct and reflected waves, and, the main point, justify the Lyapunov asymptotical stability of the strong shock wave provided that the angle at the wedge vertex is small, the uniform Lopatinsky condition is fulfilled, the initial data have a compact support, and the solvability conditions take place if needed (their number depends on the class in which the generalized solution is found).

Stability condition for strong shock waves in the problem of flow around an infinite plane wedge

“…In this work, the ideas formulated in [13,14] are developed; whereas mixed problems for the wave equation in coordinate domains with boundary conditions of the first order are studied in detail in the monograph [26] and the article [27].…”

“…By analogy with (2.1), the following lemma is true. 14]). The function V (z, s) is the solution to the problem: we seek an analytical function in the domain which lies above the curve Ω :…”

“…In this work, we continue the study of the stability of the solution with the strong shock started in [13,14] and formulate some stability conditions on the initial data for a mixed linear problem.…”

Устойчивость Сверхзвукового Обтекания Клина Со Слабой Ударной Волной

“…Во-первых, в работе [14] доказана корректность линейной смешанной задачи по крайней мере для случая малых углов при вершине кли-на. Во-вторых, в работах [15]- [17] для финитных начальных данных и при выполнении дополнительного интегрального условия в вершине клина найдено точное обобщенное решение задачи (угол при вершине клина вновь достаточно мал).…”

Justification of the Courant–Friedrich's hypothesis in the case of a weak shock. Part I. Presentation of solution to linear problem

“…Firstly, in [14] the well-posedness of the linearized initial-boundary value problem has been proved at least for the case of small angles at the wedge's vertex. Secondly, in [15][16][17] an implicit generalized solution of the linearized problem has been found for compactly supported initial data and under the fulfillment of an additional integral condition at the wedge's vertex (again the angle at the wedge's vertex is assumed small enough). For the first time one has managed to realize that the boundary singularity influences on the character of the solution itself.…”