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Steady transonic shocks and free boundary problems in infinite cylinders for the Euler equations
"... We are concerned with the existence and stability of multidimensional transonic shocks for the Euler equations for steady potential compressible fluids. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for the velocity, can be written as the following secondorde ..."
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Cited by 48 (19 self)
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We are concerned with the existence and stability of multidimensional transonic shocks for the Euler equations for steady potential compressible fluids. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for the velocity, can be written as the following secondorder nonlinear equation of mixed
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 23 (11 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.
On transonic shocks in twodimensional variablearea ducts for steady Euler system
 SIAM J. Math. Anal
"... Abstract. This paper concerns transonic shocks in compressible inviscid flow passing a twodimensional variablearea duct for the complete steady Euler system. The flow is supersonic at the entrance of the duct, whose boundaries are slightly curved. The condition of impenetrability is posed on the b ..."
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Cited by 11 (8 self)
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Abstract. This paper concerns transonic shocks in compressible inviscid flow passing a twodimensional variablearea duct for the complete steady Euler system. The flow is supersonic at the entrance of the duct, whose boundaries are slightly curved. The condition of impenetrability is posed on the boundaries. After crossing a nearly flat shock front, which passes through a fixed point on the boundary of the duct, the flow becomes subsonic. We show that to ensure the stability of such shocks, pressure should not be completely given at the exit: it only should be given with freedom one, that is, containing an unknown constant to be determined by the upstream flow and the profile of the duct. Careful analysis shows that this is due to the requirement of conservation of mass in the duct. We used Lagrangian transformation and characteristic decomposition to write the Euler system as a 2 × 2 system, which is valid for general smooth flows. Due to such a simplification, we can employ the theory of boundary value problems for elliptic equations to discuss wellposedness or illposedness of transonic shock problems in variablearea duct for various conditions giving at the exit. Key words. Euler system, transonic shocks, free boundary problem, hyperbolicelliptic composite system, illposed problem
Stability of Cylindrical Transonic Shocks for TwoDimensional Steady Compressible Euler Flows
, 2006
"... Abstract. For given supersonic flow passing a divergent nozzle, if the pressure at the exit of the nozzle (back pressure) is sufficiently large, then a transonic shock may appear in the nozzle. The pressure increases across the shock front, while the supersonic flow jumps down to subsonic. This pape ..."
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Cited by 7 (6 self)
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Abstract. For given supersonic flow passing a divergent nozzle, if the pressure at the exit of the nozzle (back pressure) is sufficiently large, then a transonic shock may appear in the nozzle. The pressure increases across the shock front, while the supersonic flow jumps down to subsonic. This paper is devoted to analyze such phenomena by establishing the stability of a class of cylindrical symmetric transonic shocks for two–dimensional complete compressible steady Euler system. This result also partly explains the effectiveness of the popular quasi–one–dimensional model of nozzle flows used in aerodynamics. Mathematically, this is to solve a nonlinear free boundary problem for an elliptic–hyperbolic composite system, with the circular transonic shock front as the free boundary. We accomplish this by finding the (locally) unique fixed point of an appropriately defined boundary profile updating mapping. To define this mapping, we encounter a series of nonclassical boundary value problems on an annulus, which involve a new type of nonlocal elliptic problem, and integral–like solvability conditions to determine the position of the free boundary. This reflects an interesting new feature of boundary value problems of elliptic–hyperbolic composite systems. 1.
A remark on determination of transonic shocks in divergent nozzles for steady compressible Euler flows, Nonlinear Analysis
 Real World Appl
"... In this paper we construct a class of transonic shock in a divergent nozzle which is a part of an angular sector (for twodimensional case) or a cone (for threedimensional case) which does not contain the vertex. The state of the compressible
ow depends only on the distance from the vertex of the ..."
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Cited by 6 (6 self)
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In this paper we construct a class of transonic shock in a divergent nozzle which is a part of an angular sector (for twodimensional case) or a cone (for threedimensional case) which does not contain the vertex. The state of the compressible
ow depends only on the distance from the vertex of the angular sector or the cone. It is supersonic at the entrance, while for appropriately given large pressure at the exit, a transonic shock front appears in the nozzle and the
ow becomes subsonic after passing it. The position and strength of the shock is automatically adjusted according to the pressure given at the exit. We demonstrate these phenomena by using the two dimensional and three dimensional full steady compressible Euler systems. The idea involved is to solve discontinuous solutions of a class of twopoint boundary value problems for systems of ordinary dierential equations. Results established in this paper may be used to analyze transonic shocks in general nozzles.
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 2 (1 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.
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.
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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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.