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We prove the stability of a Mach configuration, which occurs in shock reflection off an obstacle or shock interaction in compressible flow. The compressible flow is described by a full, steady Euler system of gas dynamics. The unperturbed Mach configuration is composed of three straight shock lines and a slip line carrying contact discontinuity. Among four regions divided by these four lines in the neighborhood of the intersection, two are supersonic regions, and other two are subsonic regions. We prove that if the constant states in the supersonic regions are slightly perturbed, then the structure of the whole configuration holds, while the other two shock fronts and the slip line, as well as the flow field in the subsonic regions, are also slightly perturbed. Such a conclusion asserts the existence and stability of the general Mach configuration in shock theory.In order to prove the result, we reduce the problem to a free boundary value problem, where two unknown shock fronts are free boundaries, while the slip line is transformed to a fixed line by a Lagrange transformation. In the region where the solution is to be determined, we have to deal with an elliptic-hyperbolic composed system. By decoupling this system and combining the technique for both hyperbolic equations and elliptic equations, we establish the required estimates, which are crucial in the proof of the existence of a solution to the free boundary value problem.

Sulforaphane is a common antioxidant selectively abundant in cruciferous plants, which exhibits effective anti-cancer actions in control of tumorigenesis or progression of various cancers. A recent study has shown that sulforaphane attenuates the EGFR signaling pathway in non-small cell lung cancer (NSCLC), suggesting its potential anti-metastatic effects. In this study we assessed the involvement of sulforaphane and miR-616-5p in epithelial-mesenchymal transition (EMT) and NSCLC metastasis. Sulforaphane suppressed the cell proliferation in human NSCLC cell lines H1299, 95C and 95D with IC 50 values of 9.52±1.23, 9.04±1.90 and 17.35±2.03 μmol/L, respectively. At low concentrations (1-5 μmol/L), sulforaphane dose-dependently inhibited the migration and invasion of 95D and H1299 cells with relatively high metastatic potential. The anti-metastatic action of sulforaphane was confirmed in 95D and H1299 cell xenografts in vivo. In fresh NSCLC tissue samples from 179 patients, miR-616-5p levels were upregulated in late-stage NSCLCs, and strongly correlated with risk of NSCLC recurrence and metastasis. Consistent with the clinic observation, miR-616-5p levels in the 3 NSCLC cell lines were correlated with their metastatic ability, and were decreased by sulforaphane treatment. Silencing miR-616-5p markedly suppressed the migration and invasion of 95D cells in vitro and NSCLC metastasis in vivo. Further studies revealed that miR-616-5p directly targeted GSK3β and decreased its expression, whereas sulforaphane decreased miR-616-5p levels by histone modification, and followed by inactivation of the GSK3β/β-catenin signaling pathway and inhibition of EMT, which was characterized by loss of epithelial markers and acquisition of a mesenchymal phenotype in NSCLC cells. Our findings suggest that sulforaphane is a potential adjuvant chemotherapeutic agent for the prevention of NSCLC recurrence and metastasis, and miR-616-5p can be clinically utilized as a biomarker or therapeutic target to inhibit metastasis.

In this paper we study the stability of transonic shocks in steady supersonic flow past a wedge. We take the potential flow equation as the mathematical model to describe the compressible flow. It is known that in generic case such a problem admits two possible location of shock, connecting the flow ahead it and behind it. They can be distinguished as supersonic-supersonic shock and supersonic-subsonic shock (or transonic shock). Both these possible shocks satisfy the Rankine-Hugoniot conditions and entropy condition. In this paper we prove that the transonic shock is also stable under perturbation of the coming flow provided the pressure at infinity is well controlled.

Abstract. We study the stability of stationary transonic shock fronts under two-dimensional perturbation in gas dynamics. The motion of the gas is described by the full Euler system. The system is hyperbolic ahead of the shock front, and is a hyperbolic-elliptic composed system behind the shock front. The stability of the shock front and the downstream flow under two-dimensional perturbation of the upstream flow can be reduced to a free boundary value problem of the hyperbolic-elliptic composed system. We develop a method to deal with boundary value problems for such systems. The crucial point is to decompose the system to a canonical form, in which the hyperbolic part and the elliptic part are only weakly coupled in their coefficients. By several sophisticated iterative processes we establish the existence and uniqueness of the solution to the described free boundary value problem. Our result indicates the stability of the transonic shock front and the flow field behind the shock.

Abstract. This paper is devoted to the study of a transonic shock in threedimensional steady compressible flow passing a duct with a general section. The flow is described by the steady full Euler system, which is purely hyperbolic in the supersonic region and is of elliptic-hyperbolic type in the subsonic region. The upstream flow at the entrance of the duct is a uniform supersonic one adding a three-dimensional perturbation, while the pressure of the downstream flow at the exit of the duct is assigned apart from a constant difference. The problem to determine the transonic shock and the flow behind the shock is reduced to a free boundary value problem of an elliptic-hyperbolic system. The new ingredients of our paper contain the decomposition of the elliptichyperbolic system, the determination of the shock front by a pair of partial differential equations coupled with the three-dimensional Euler system, and the regularity analysis of solutions to the boundary value problems introduced in our discussion.

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