217 research outputs found
Global Solutions of Shock Reflection by Large-Angle Wedges for Potential Flow
When a plane shock hits a wedge head on, it experiences a
reflection-diffraction process and then a self-similar 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 elliptic-hyperbolic 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 large-angle wedges for potential flow. The techniques
and ideas developed here will be useful for other nonlinear problems involving
similar difficulties.Comment: 108 page
Transonic Flows with Shocks Past Curved Wedges for the Full Euler Equations
We establish the existence, stability, and asymptotic behavior of transonic
flows with a transonic shock past a curved wedge for the steady full Euler
equations in an important physical regime, which form a nonlinear system of
mixed-composite hyperbolic-elliptic type. To achieve this, we first employ the
coordinate transformation of Euler-Lagrange type and then exploit one of the
new equations to identify a potential function in Lagrangian coordinates. By
capturing the conservation properties of the Euler system, we derive a single
second-order nonlinear elliptic equation for the potential function in the
subsonic region so that the transonic shock problem is reformulated as a
one-phase free boundary problem for a second-order nonlinear elliptic equation
with the shock-front as a free boundary. One of the advantages of this approach
is that, given the shock location or quivalently the entropy function along the
shock-front downstream, all the physical variables can expressed as functions
of the gradient of the potential function, and the downstream asymptotic
behavior of the potential function at the infinite exit can be uniquely
determined with uniform decay rate.
To solve the free boundary problem, we employ the hodograph transformation to
transfer the free boundary to a fixed boundary, while keeping the ellipticity
of the second-order equations, and then update the entropy function to prove
that it has a fixed point. Another advantage in our analysis here is in the
context of the real full Euler equations so that the solutions do not
necessarily obey Bernoulli's law with a uniform Bernoulli constant, that is,
the Bernoulli constant is allowed to change for different fluid trajectories.Comment: 35 pages, 2 figures in Discrete and Continuous Dynamical Systems, 36
(2016
Prandtl-Meyer Reflection Configurations, Transonic Shocks, and Free Boundary Problems
We are concerned with the Prandtl-Meyer reflection configurations of unsteady
global solutions for supersonic flow impinging upon a symmetric solid wedge.
Prandtl (1936) first employed the shock polar analysis to show that there are
two possible steady configurations: the steady weak/strong shock solutions,
when a steady supersonic flow impinges upon the wedge whose angle is less than
the detachment angle, and then conjectured that the steady weak shock solution
is physically admissible. The fundamental issue of whether one or both of the
steady wea/strong shocks are physically admissible has been vigorously debated
over the past eight decades. On the other hand, the Prandtl-Meyer reflection
configurations are core configurations in the structure of global entropy
solutions of the 2-D Riemann problem, while the Riemann solutions themselves
are local building blocks and determine local structures, global attractors,
and large-time asymptotic states of general entropy solutions. In this sense,
we have to understand the reflection configurations in order to understand
fully the global entropy solutions of 2-D hyperbolic systems of conservation
laws, including the admissibility issue for the entropy solutions. In this
monograph, we address this longstanding open issue and present our analysis to
establish the stability theorem for the steady weak shock solutions as the
long-time asymptotics of the Prandtl-Meyer reflection configurations for
unsteady potential flow for all the physical parameters up to the detachment
angle. To achieve these, we first reformulate the problem as a free boundary
problem involving transonic shocks and then obtain appropriate monotonicity
properties and uniform a priori estimates for admissible solutions, which allow
us to employ the Leray-Schauder degree argument to complete the theory for all
the physical parameters up to the detachment angle.Comment: 192 pages; 17 figures; To appear in the AMS series "Memoirs of the
American Mathematical Society", 202
Loss of Regularity of Solutions of the Lighthill Problem for Shock Diffraction for Potential Flow
We are concerned with the suitability of the main models of compressible
fluid dynamics for the Lighthill problem for shock diffraction by a convex
corned wedge, by studying the regularity of solutions of the problem, which can
be formulated as a free boundary problem. In this paper, we prove that there is
no regular solution that is subsonic up to the wedge corner for potential flow.
This indicates that, if the solution is subsonic at the wedge corner, at least
a characteristic discontinuity (vortex sheet or entropy wave) is expected to be
generated, which is consistent with the experimental and computational results.
Therefore, the potential flow equation is not suitable for the Lighthill
problem so that the compressible Euler system must be considered. In order to
achieve the non-existence result, a weak maximum principle for the solution is
established, and several other mathematical techniques are developed. The
methods and techniques developed here are also useful to the other problems
with similar difficulties.Comment: 20 pages, 4 figures, To appear in: SIAM Journal of Mathematical
Analysis, 202
Multidimensional transonic shock waves and free boundary problems
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
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