Narrow Results

In this paper, we first define the pseudo-circular evolute of a lightcone framed curve in the Lorentz–Minkowski 3-space, and show that the pseudo-circular evolute of a lightcone framed curve can be seen as a generalization of the circular evolute of a framed curve without type changing. We also define an involute of a lightcone framed curve, and study the duality relations between involutes and pseudo-circular evolutes. Moreover, we study the pseudo-circular evolutes and involutes with respect to a lightcone circle frame, and investigate the relationship with respect to a projection map. Meanwhile, examples are given for explaining the objects we investigated, which also show a certain physical significance of our investigation.

This paper concerns the structural stability of smooth cylindrically symmetric transonic flows in a concentric cylinder. Both cylindrical and axi-symmetric perturbations are considered. The governing system here is of mixed elliptic–hyperbolic and changes type and the suitable formulation of boundary conditions at the boundaries is of great importance. First, we establish the existence and uniqueness of smooth cylindrical transonic spiral solutions with nonzero angular velocity and vorticity which are close to the background transonic flow with small perturbations of the Bernoulli’s function and the entropy at the outer cylinder and the flow angles at both the inner and outer cylinders independent of the symmetric axis, and it is shown that in this case, the sonic points of the flow are nonexceptional and noncharacteristically degenerate, and form a cylindrical surface. Second, we also prove the existence and uniqueness of axi-symmetric smooth transonic rotational flows which are adjacent to the background transonic flow, whose sonic points form an axi-symmetric surface. The key elements in our analysis are to utilize the deformation-curl decomposition for the steady Euler system to deal with the hyperbolicity in subsonic regions and to find an appropriate multiplier for the linearized second-order mixed type equations which are crucial to identify the suitable boundary conditions and to yield the important basic energy estimates.

We study a quadratic system of conservation laws with an elliptic region. The second order terms in the fluxes correspond to type IV in Schaeffer and Shearer classification. There exists a special singularity for the EDOs associated to traveling waves for shocks. In our case, this singularity lies on the elliptic boundary. We prove that high amplitude Riemann solutions arise from Riemann data with arbitrarily small amplitude in the hyperbolic region near the special singularity. For such Riemann data there is no small amplitude solution. This behavior is related to the bifurcation of one of the codimension-3 nilpotent singularities of planar ODEs studied by Dumortier, Roussarie and Sotomaior.

We are concerned with free boundary problems arising from the analysis of multidimensional transonic shock waves for the Euler equations in compressible fluid dynamics. In this expository paper, we survey some recent developments in the analysis of multidimensional transonic shock waves and corresponding free boundary problems for the compressible Euler equations and related nonlinear partial differential equations (PDEs) of mixed type. The nonlinear PDEs under our analysis include the steady Euler equations for potential flow, the steady full Euler equations, the unsteady Euler equations for potential flow, and related nonlinear PDEs of mixed elliptic–hyperbolic type. The transonic shock problems include the problem of steady transonic flow past solid wedges, the von Neumann problem for shock reflection–diffraction, and the Prandtl–Meyer problem for unsteady supersonic flow onto solid wedges. We first show how these longstanding multidimensional transonic shock problems can be formulated as free boundary problems for the compressible Euler equations and related nonlinear PDEs of mixed type. Then we present an effective nonlinear method and related ideas and techniques to solve these free boundary problems. The method, ideas, and techniques should be useful to analyze other longstanding and newly emerging free boundary problems for nonlinear PDEs.