This special issue and the two follow-up issues are dedicated to Professor Gui-Qiang G. Chen on the occasion of his 60th birthday. Professor Gui-Qiang G. Chen is internationally recognized as a leader in the analysis of partial differential equations (PDEs) and related disciplines in mathematics and science. He has made wide-ranging contributions, both original and significant, to an array of research areas in mathematical analysis, partial differential equations, mathematical physics, nonlinear science, and other disciplines, especially in the areas of nonlinear hyperbolic systems of conservation laws and the mathematical theory of shock waves, free boundary problems in the theory of supersonic and transonic flow, nonlinear degenerate and mixed-type PDEs and their applications, entropy analysis and weak convergence methods, singular limit problems for nonlinear PDEs, measure-theoretical analysis for discontinuous and singular entropy solutions, stability/instability analysis of characteristic discontinuities for nonlinear hyperbolic conservation laws, and convergence/stability analysis of shock-capturing methods and related numerical methods, among others.

We investigate the hypersonic limit for steady, uniform, and compressible polytropic gas passing a symmetric straight cone. By considering Radon measure solutions, we show that as the Mach number of the upstream flow tends to infinity, the measures associated with the weak entropy solution containing an attached shock ahead of the cone converge vaguely to the measures associated with a Radon measure solution to the conical hypersonic-limit flow. This justifies the Newtonian sine-squared pressure law for cones in hypersonic aerodynamics. For Chaplygin gas, assuming that the Mach number of the incoming flow is less than a finite critical value, we demonstrate that the vertex angle of the leading shock is independent of the conical body’s vertex angle and is totally determined by the incoming flow’s Mach number. If the Mach number exceeds the critical value, we explicitly construct a Radon measure solution with a concentration boundary layer.

We consider the initial-boundary value problem in the halfspace for the system of equations of ideal magneto-hydrodynamics (MHD) with a perfectly conducting wall boundary condition. We show the convergence of solutions to the solution of the equations of incompressible MHD as the Mach number goes to zero. Because of the characteristic boundary, where a loss of regularity in the normal direction to the boundary may occur, the convergence is shown in suitable anisotropic Sobolev spaces which take account of the singular behavior at the boundary.

Based on delicate construction of approximate initial data sequence, elaborate estimates on the viscous terms and elegant analysis of the convergence of momentum equation, we succeed in establishing the global existence of spherically symmetric finite-energy weak solution of the compressible Euler-Poisson equations for large initial data through justifying the inviscid limit of the solutions of Navier-Stokes-Poisson equations with a very general class of density-dependent viscosities. Our results can serve as an extension of [Chen et al., Comm. Pure Appl. Math. 77(6) (2024)].

In this work, we employ the relative energy method to obtain a weak-strong uniqueness principle for a Euler-Riesz system, as well as to establish its convergence in the high-friction limit towards a gradient flow equation. The main technical challenge in our analysis is addressed using a specific case of a Hardy-Littlewood-Sobolev inequality for Riesz potentials.

This paper studies an inverse problem of reconstructing the shape of the circular symmetric cone for the given leading shock front. Assuming that the attack angle is small and that the incoming flow has a large Mach Number, we show that the shape of the cone can be reconstructed near the vertex from the data of the given shock and establish an asymptotic expansion for the velocity and the slope of cone.

This paper is a review of wedge problems for supersonic relativistic Euler flows. We are mainly concerned with two-dimensional compressible supersonic relativistic Euler flows past Lipschitz wedges, sharp corners, or bending wedges from mathematical point of view. When the vertex angle of the upstream flow is less than the critical angle, a shock wave is generated from the wedge vertex. If the vertex angle is larger than $\pi$ and the angle of the flow is larger than the critical value, then a rarefaction wave will appear. In this paper, we employ modified wave front tracking method to establish the structural stability of such wave patterns under some small perturbations of both the upcoming supersonic flow and the tangent slope of the boundary. It is an initial-boundary value problem for two-dimensional steady compressible relativistic Euler system. Moreover, we study global non-relativistic limits of entropy solutions for relativistic Euler flows as well as the asymptotic behavior of the solutions at the infinity. In particular, we demonstrate the basic properties of nonlinear waves for the two-dimensional steady supersonic relativistic Euler flows, especially, the geometric structures of shock polar and rarefaction wave.