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11
Global solutions to shock reflection by a largeangle wedges for potential flow
, 2006
"... When a plane shock hits a wedge head on, it experiences a reflectiondiffraction process and then a selfsimilar reflected shock moves outward as the original shock moves forward in time. Experimental, computational, and asymptotic analysis has shown that various patterns of shock reflection may occu ..."
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Cited by 20 (9 self)
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When a plane shock hits a wedge head on, it experiences a reflectiondiffraction process and then a selfsimilar reflected shock moves outward as the original shock moves forward in time. Experimental, computational, and asymptotic analysis has shown that various patterns of shock reflection may occur, including regular and Mach reflection. However, most of the fundamental issues for shock reflection have not been understood, including the global structure, stability, and transition of the different patterns of shock reflection. Therefore, it is essential to establish the global existence and structural stability of solutions of shock reflection in order to understand fully the phenomena of shock reflection. On the other hand, there has been no rigorous mathematical result on the global existence and structural stability of shock reflection, including the case of potential flow which is widely used in aerodynamics. Such problems involve several challenging difficulties in the analysis of nonlinear partial differential equations such as mixed equations of elliptichyperbolic type, free boundary problems, and corner singularity where an elliptic degenerate curve meets a free boundary. In this paper we develop a rigorous mathematical approach to overcome these difficulties involved and establish a global theory of existence and stability for shock reflection by largeangle wedges for potential flow. The techniques and ideas developed here will be useful for other nonlinear problems involving similar difficulties. 1.
Uniqueness of transonic shock solutions in a duct for steady potential
, 2009
"... Abstract. We study the uniqueness of solutions with a transonic shock in a duct in a class of transonic shock solutions, which are not necessarily small perturbations of the background solution, for steady potential flow. We prove that, for given uniform supersonic upstream flow in a straight duct, ..."
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Cited by 4 (4 self)
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Abstract. We study the uniqueness of solutions with a transonic shock in a duct in a class of transonic shock solutions, which are not necessarily small perturbations of the background solution, for steady potential flow. We prove that, for given uniform supersonic upstream flow in a straight duct, there exists a unique uniform pressure at the exit of the duct such that a transonic shock solution exists in the duct, which is unique modulo translation. For any other given uniform pressure at the exit, there exists no transonic shock solution in the duct. This is equivalent to establishing a uniqueness theorem for a free boundary problem of a partial differential equation of second order in a bounded or unbounded duct. The proof is based on the maximum/comparison principle and a judicious choice of special transonic shock solutions as a comparison solution.
Transonic regular reflection for the nonlinear wave system
 Journal of Hyperbolic Differential Equations
"... Abstract. We consider Riemann data for the nonlinear wave system which result in a regular reflection with a subsonic state behind the reflected shock. The problem in selfsimilar coordinates leads to a system of mixed type and a free boundary value problem for the reflected shock and the solution i ..."
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Cited by 1 (0 self)
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Abstract. We consider Riemann data for the nonlinear wave system which result in a regular reflection with a subsonic state behind the reflected shock. The problem in selfsimilar coordinates leads to a system of mixed type and a free boundary value problem for the reflected shock and the solution in the subsonic region. We show existence of a solution in a neighborhood of the reflection point. 1.
GLOBAL UNIQUENESS OF TRANSONIC SHOCKS IN DIVERGENT NOZZLES FOR STEADY POTENTIAL FLOWS
, 903
"... Abstract. We show that for steady compressible potential flow in a class of straight divergent nozzles with arbitrary crosssection, if the flow is supersonic and spherically symmetric at the entry, and the given pressure (velocity) is appropriately large (small) and also spherically symmetric at th ..."
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Cited by 1 (1 self)
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Abstract. We show that for steady compressible potential flow in a class of straight divergent nozzles with arbitrary crosssection, if the flow is supersonic and spherically symmetric at the entry, and the given pressure (velocity) is appropriately large (small) and also spherically symmetric at the exit, then there exists uniquely one transonic shock in the nozzle. In addition, the shockfront and the supersonic flow ahead of it, as well as the subsonic flow behind of it, are all spherically symmetric. This is a global uniqueness result of free boundary problems of elliptic–hyperbolic mixed type equations. The proof depends on the maximum principles and judicious choices of comparison functions.
Shock ReflectionDiffraction Phenomena and Multidimensional Conservation Laws
, 2009
"... Abstract. When a plane shock hits a wedge head on, it experiences a reflectiondiffraction process, and then a selfsimilar reflected shock moves outward as the original shock moves forward in time. The complexity of reflectiondiffraction configurations was first reported by Ernst Mach in 1878, and ..."
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Cited by 1 (1 self)
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Abstract. When a plane shock hits a wedge head on, it experiences a reflectiondiffraction process, and then a selfsimilar reflected shock moves outward as the original shock moves forward in time. The complexity of reflectiondiffraction configurations was first reported by Ernst Mach in 1878, and experimental, computational, and asymptotic analysis has shown that various patterns of shock reflectiondiffraction configurations may occur, including regular reflection and Mach reflection. In this paper we start with various shock reflectiondiffraction phenomena, their fundamental scientific issues, and their theoretical roles as building blocks and asymptotic attractors of general solutions in the mathematical theory of multidimensional hyperbolic systems of conservation laws. Then we describe how the global problem of shock reflectiondiffraction by a wedge can be formulated as a free boundary problem for nonlinear conservation laws of mixedcomposite hyperbolicelliptic type. Finally we discuss some recent developments in attacking the shock reflectiondiffraction problem, including the existence, stability, and regularity of global regular reflectiondiffraction solutions. The approach includes techniques to handle free boundary problems, degenerate elliptic equations, and corner singularities, which is highly motivated by experimental, computational, and asymptotic results. Further trends and open problems in this direction are also addressed. 1.
SUBSONIC FLOWS FOR THE FULL EULER EQUATIONS IN HALF PLANE
"... Abstract. We study the subsonic flows governed by full Euler equations in the half plane bounded below by a piecewise smooth curve asymptotically approaching x1axis. Nonconstant conditions in the far field are prescribed to ensure the real Euler flows. The Euler system is reduced to a single ellipt ..."
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Abstract. We study the subsonic flows governed by full Euler equations in the half plane bounded below by a piecewise smooth curve asymptotically approaching x1axis. Nonconstant conditions in the far field are prescribed to ensure the real Euler flows. The Euler system is reduced to a single elliptic equation for the stream function. The existence, uniqueness and asymptotic behaviors of the solutions for the reduced equation are established by Schauder fixed point argument and some delicate estimates. The existence of subsonic flows for the original Euler system is proved based on the results for the reduced equation, and their asymptotic behaviors in the far field are also obtained. 1.
SUBSONIC FLOWS FOR THE FULL EULER EQUATIONS IN HALF PLANE
, 710
"... Abstract. We study the subsonic flows governed by full Euler equations in the half plane bounded below by a piecewise smooth curve asymptotically approaching x1axis. Nonconstant conditions in the far field are prescribed to ensure the real Euler flows. The Euler system is reduced to a single ellipt ..."
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Abstract. We study the subsonic flows governed by full Euler equations in the half plane bounded below by a piecewise smooth curve asymptotically approaching x1axis. Nonconstant conditions in the far field are prescribed to ensure the real Euler flows. The Euler system is reduced to a single elliptic equation for the stream function. The existence, uniqueness and asymptotic behaviors of the solutions for the reduced equation are established by Schauder fixed point argument and some delicate estimates. The existence of subsonic flows for the original Euler system is proved based on the results for the reduced equation, and their asymptotic behaviors in the far field are also obtained. 1.
OVERVIEW, PROBLEMS, AND PERSPECTIVE
"... Multidimensional conservation laws: overview, problems, and ..."
LONG NOZZLES FOR THE FULL EULER EQUATIONS
"... Abstract. We are concerned with global steady subsonic flows through general infinitely long nozzles for the full Euler equations. The problem is formulated as a boundary value problem in the unbounded domain for a nonlinear elliptic equation of second order in terms of the stream function. It is es ..."
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Abstract. We are concerned with global steady subsonic flows through general infinitely long nozzles for the full Euler equations. The problem is formulated as a boundary value problem in the unbounded domain for a nonlinear elliptic equation of second order in terms of the stream function. It is established that, when the oscillation of the entropy and Bernoulli functions at the upstream is sufficiently small in C 1,1 and the mass flux is in a suitable regime, there exists a unique global subsonic solution in a suitable class of general nozzles. The assumptions are required to prevent from the occurrence of supersonic bubbles inside the nozzles. The asymptotic behavior of subsonic flows at the downstream and upstream, as well as the critical mass flux, have been clarified. 1.