We establish the existence and stability of cylindrical transonic shock solutions under three dimensional perturbations of the incoming flows and the exit pressure without any further restrictions on the background transonic shock solutions. The strength and position of the perturbed transonic shock are completely free and uniquely determined by the incoming flows and the exit pressure. The optimal regularity is obtained for all physical quantities, and the velocity, the Bernoulli's quantity, the entropy and the pressure share the same regularity. The approach is based on the deformation-curl decomposition to the steady Euler system introduced by the authors to decouple the hyperbolic and elliptic modes effectively. However, one of the key elements in application of the deformation-curl decomposition is to find a decomposition of the Rankine-Hugoniot conditions, which shows the mechanism of determining the shock front uniquely by an algebraic equation and also gives an unusual second order differential boundary conditions on the shock front for the first order deformation-curl system. After homogenizing the curl system and introducing a potential function, this unusual condition on the shock front becomes the Poisson equation with homogeneous Neumann boundary condition on the intersection of the shock front and the cylinder walls from which an oblique boundary condition for the potential function can be uniquely derived.
We establish a variational inequality formulation that captures the transonic shock for a steady compressible potential flow. Its critical point satisfies the transonic equation; moreover the associated jump conditions across its free boundary match the usual Rankine-Hugoniot jump conditions for a shock. By means of example we validate our formulation, and establish the necessary and sufficient condition for the existence of a transonic shock. Numerical results are also discussed.
This paper concerns with the existence of transonic shocks for steady Euler flows in a 3-D axisymmetric cylindrical nozzle, which are governed by the Euler equations with the slip boundary condition on the wall of the nozzle and a receiver pressure at the exit. Mathematically, it can be formulated as a free boundary problem with the shock front being the free boundary to be determined. In dealing with the free boundary problem, one of the key points is determining the position of the shock front. To this end, a free boundary problem for the linearized Euler system will be proposed, whose solut…
We prove global in time dynamical stability of steady transonic shock solutions in divergent quasi-one-dimensional nozzles. We assume neither the smallness of the relative slope of the nozzle nor the weakness of the shock. Key ingredients of the proof are an exponentially decaying energy estimate for a linearized problem together with methods from \cite{LRXX}.
In this paper, both structural and dynamical stabilities of steady transonic shock solutions for one-dimensional Euler-Poission system are investigated. First, a steady transonic shock solution with supersonic backgroumd charge is shown to be structurally stable with respect to small perturbations of the background charge, provided that the electric field is positive at the shock location. Second, any steady transonic shock solution with the supersonic background charge is proved to be dynamically and exponentially stable with respect to small perturbation of the initial data, provided the ele…
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 a…
We establish the local uniqueness of steady transonic shock solutions with spherical symmetry for the three-dimensional full Euler equations. These transonic shock-fronts are important for understanding transonic shock phenomena in divergent nozzles. From mathematical point of view, we show the uniqueness of solutions of a free boundary problem for a multidimensional quasilinear system of mixed-composite elliptic--hyperbolic type. To this end, we develop a decomposition of the Euler system which works in a general Riemannian manifold, a method to study a Venttsel problem of nonclassical nonloc…
In this paper, we are concerned with the instability problem of a 3-D transonic oblique shock wave for the steady supersonic flow past an infinitely long sharp wedge. The flow is assumed to be isentropic and irrotational. It was indicated in pages 317 of [9] that if a steady supersonic flow comes from minus infinity and hits a sharp symmetric wedge, then it follows from the Rankine-Hugoniot conditions and the physical entropy condition that there possibly appear a weak shock or a strong shock attached at the edge of the sharp wedge, which corresponds to a supersonic shock or a transonic shock,…
On the Well-Posedness of Periodic Strong Transonic Shocks in Divergent Nozzles
School of Mathematical Sciences, Fudan University（复旦大学数学科学学院）；Shanghai Key Laboratory of Contemporary Applied Mathematics（上海当代应用数学重点实验室）；Department of mathematics, Shandong Normal University（山东师范大学数学系）
In this paper, we prove the structural stability of the transonic shocks for three dimensional axisymmetric Euler system with swirl velocity under the perturbations for the incoming supersonic flow, the nozzle boundary, and the exit pressure. Compared with the known results on the stability of transonic shocks, one of the major difficulties for the axisymmetric flows with swirls is that corner singularities near the intersection point of the shock surface and nozzle boundary and the artificial singularity near the axis appear simultaneously. One of the key points in the analysis for this paper…
Stability of steady transonic shocks with constant states in 3D rectangular nozzles under small external forces
School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）；School of Mathematics and Statistics, Hubei Key Laboratory of Mathematics & Intelligent Computation and Interdisciplinary Sciences, Wuhan University of Technology（武汉科技大学数学与统计学院）
This paper concerns the existence of transonic shock solutions to the 2-D steady compressible Euler system in an almost flat finite nozzle ( in the sense that it is a generic small perturbation of a flat one ), under physical boundary conditions proposed by Courant-Friedrichs in \cite{CourantFriedrichs1948}, in which the receiver pressure is prescribed at the exit of the nozzle. In the resulting free boundary problem, the location of the shock-front is one of the most desirable information one would like to determine. However, the location of the normal shock-front in a flat nozzle can be anyw…
We show that for steady compressible potential flow in a class of straight divergent nozzles with arbitrary cross-section, 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 shock-front 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 equa…
For unipolar hydrodynamic model of semiconductor device represented by Euler-Poisson equations, when the doping profile is supersonic, the existence of steady transonic shock solutions and C-smooth steady transonic solutions for Euler-Poisson Equations were established in [27] and [41], respectively. In this paper we further study the nonlinear structural stability and the linear dynamic instability of these steady transonic solutions. When the C^1-smooth transonic steady-states pass through the sonic line, they produce singularities for the system, and cause some essential difficulty in the p…
We prove that for the two-dimensional steady complete compressible Euler system, with given uniform upcoming supersonic flows, the following three fundamental flow patterns (special solutions) in gas dynamics involving transonic shocks are all unique in the class of piecewise $C^1$ smooth functions, under appropriate conditions on the downstream subsonic flows: $(\rmnum{1})$ the normal transonic shocks in a straight duct with finite or infinite length, after fixing a point the shock-front passing through; $(\rmnum{2})$ the oblique transonic shocks attached to an infinite wedge; $(\rmnum{3})$ a…
We are concerned with the stability of multidimensional (M-D) transonic shocks in steady supersonic flow past multidimensional wedges. One of our motivations is that the global stability issue for the M-D case is much more sensitive than that for the 2-D case, which requires more careful rigorous mathematical analysis. In this paper, we develop a nonlinear approach and employ it to establish the stability of weak shock solutions containing a transonic shock-front for potential flow with respect to the M-D perturbation of the wedge boundary in appropriate function spaces. To achieve this, we fi…
This paper concerns the existence of transonic shocks for steady exothermically reacting Euler flows in an almost flat nozzle with the small rate of the exothermic reaction. One of the key points is to quantitatively determine the position of the shock front in the nozzle. We focus on the contributions of the perturbation of the flat nozzle and the exothermic reaction in determining the position of the shock front by comparing the orders of $σ$ and $κ$, where $σ$ represents the scale of the perturbation of the flat nozzle and $κ$ the rate of the exothermic reaction. To this end, a free boundar…
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 fun…
In this paper we are concerned with the existence of transonic shocks for 2-D steady isothermal Euler flows in a horizontal flat nozzle under vertical gravity. In particular, we focus on the contribution of the vertical gravity in determining the position of the shock front. For steady horizontal flows, the existence of normal shocks with the position of the shock front being arbitrary in the nozzle can be easily established. This paper will try to determine the position of the shock front as the state of the flow at the entrance of the nozzle and the pressure at the exit are slightly perturbe…
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 th…
We establish the stability of 3-D axisymmetric transonic shock solutions of the steady full Euler system in divergent nozzles under small perturbations of an incoming radial supersonic flow and a constant pressure at the exit of the nozzles. To study 3-D axisymmetric transonic shock solutions of the full Euler system, we use a stream function formulation of the full Euler system for a 3-D axisymmetric flow. We resolve the singularity issue arising in stream function formulations of the full Euler system for a 3-D axisymmetric flow. We develop a new scheme to determine a shock location of a tra…
Transonic shocks play a pivotal role in designation of supersonic inlets and ramjets. For the three-dimensional steady non-isentropic compressible Euler system with frictions, we had constructed a family of transonic shock solutions in rectilinear ducts with square cross-sections, and this paper is devoted to proving rigorously that almost all of these transonic shock solutions are stable, under multidimensional small perturbations of the upcoming supersonic flows and back pressures at the exits of ducts in suitable function spaces. This manifests that friction has a stabilization effect on tr…
For an upstream supersonic flow past a straight-sided cone in $\R^3$ whose vertex angle is less than the critical angle, a transonic (supersonic-subsonic) shock-front attached to the cone vertex can be formed in the flow. In this paper we analyze the stability of transonic shock-fronts in three-dimensional steady potential flow past a perturbed cone. We establish that the self-similar transonic shock-front solution is conditionally stable in structure with respect to the conical perturbation of the cone boundary and the upstream flow in appropriate function spaces. In particular, it is proved …
We are concerned with geometric properties of transonic shocks as free boundaries in two-dimensional self-similar coordinates for compressible fluid flows, which are not only important for the understanding of geometric structure and stability of fluid motions in continuum mechanics but also fundamental in the mathematical theory of multidimensional conservation laws. A transonic shock for the Euler equations for self-similar potential flow separates elliptic (subsonic) and hyperbolic (supersonic) phases of the self-similar solution of the corresponding nonlinear partial differential equation …
We establish the existence of an axisymmetric weak solution to the steady Euler system with a transonic shock, nonzero vorticity, and nonzero swirl in a three-dimensional cylinder. When prescribing the supersonic solution in the upstream region by axisymmetric functions with variable entropy and variable angular momentum density(=swirl), we construct such a solution by using a Helmholtz decomposition of the velocity field and the method of iteration. An iteration scheme is developed using a delicate decomposition of the Rankine-Hugoniot conditions on the transonic shock via Helmholtz decomposi…
We develop a method that works in general product Riemannian manifold to decompose the three-dimensional steady full compressible Euler system, which is of elliptic-hyperbolic composite-mixed type for subsonic flows. The method is applied to show stability of spherically symmetric subsonic flows and transonic shocks in space $\mathbb{R}^3$ under multidimensional perturbations of boundary conditions.
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 …
Given constant data of density $ρ_0$, velocity $-u_0{\bf e}_r$, pressure $p_0$ and electric force $-E_0{\bf e}_r$ for supersonic flow at the entrance, and constant pressure $p_{\rm ex}$ for subsonic flow at the exit, we prove that Euler-Poisson system admits a unique transonic shock solution in a two dimensional convergent nozzle, provided that $u_0>0$, $E_0>0$, and that $E_0$ is sufficiently large depending on $(ρ_0, u_0, p_0)$ and the length of the nozzle.
We establish existence, uniqueness and stability of transonic shocks for steady compressible non-isentropic potential flow system in a multidimensional divergent nozzle with an arbitrary smooth cross-section, for a prescribed exit pressure. The proof is based on solving a free boundary problem for a system of partial differential equations consisting of an elliptic equation and a transport equation. In the process, we obtain unique solvability for a class of transport equations with velocity fields of weak regularity(non-Lipschitz), an infinite dimensional weak implicit mapping theorem which d…
We develop a new approach and employ it to establish the global existence and nonlinear structural stability of attached weak transonic shocks in steady potential flow past three-dimensional wedges; in particular, the restriction that the perturbation is away from the wedge edge in the previous results is removed. One of the key ingredients is to identify a "good" direction of the boundary operator of a boundary condition of the shock along the wedge edge, based on the non-obliqueness of the boundary condition for the weak shock on the edge. With the identification of this direction, an additi…
Shock waves are steep wave fronts that are fundamental in nature, especially in high-speed fluid flows. When a shock hits an obstacle, or a flying body meets a shock, shock reflection/diffraction phenomena occur. In this paper, we show how several longstanding shock reflection/diffraction problems can be formulated as free boundary problems, discuss some recent progress in developing mathematical ideas, approaches, and techniques for solving these problems, and present some further open problems in this direction. In particular, these shock problems include von Neumann's problem for shock refl…
The existence, uniqueness, and asymptotic behavior of steady transonic flows past a curved wedge, involving transonic shocks, governed by the two-dimensional full Euler equations are established. The stability of both weak and strong transonic shocks under the perturbation of both the upstream supersonic flow and the wedge boundary is proved. The problem is formulated as a one-phase free boundary problem, in which the transonic shock is treated as a free boundary. The full Euler equations are decomposed into two algebraic equations and a first-order elliptic system of two equations in Lagrangi…
We analyze rarefaction wave interactions of self-similar transonic irrotational flow in gas dynamics for the two dimensional Riemann problems. We establish the existence result of the supersonic solution to the prototype nonlinear wave system for the sectorial Riemann data, and study the formation of the sonic boundary and the transonic shock. The transition from the sonic boundary to the shock boundary inherits at least two types of degeneracies (1) the system is sonic, and in addition (2) the angular derivative of the solution becomes zero where the sonic and shock boundaries meet.
For the stationary hydrodynamic model for semiconductors with sonic boundary, represented by Euler-Poisson equations, it possesses the various physical solutions including interior subsonic solutions/interior supersonic solutions/shock transonic solutions/$C^1$-smooth transonic solutions. However, the structural stability for these physical solutions is challenging and has remained open as we know. In this paper, we investigate the structural stability of interior subsonic solutions when the doping profiles are restricted in the subsonic region. The main result is proved by using the local (we…
When a plane shock hits a two-dimensional 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. The experimental, computational, and asymptotic analysis has indicated that various patterns occur, including regular reflection and Mach reflection. The von Neumann conjectures on the transition from regular to Mach reflection involve the existence, uniqueness, and stability of regular shock reflection-diffraction configurations, generated by concave cornered wedges for compressible flow. In…
For the three-dimensional steady non-isentropic compressible Euler system with friction, we show existence of a class of symmetric subsonic, supersonic and transonic-shock solutions in a straight duct with constant square-section. Such flows are called Fanno flow in engineering. We formulate a boundary value problem for subsonic flows, and study their stability under multidimensional small perturbations of boundary conditions. Since the subsonic Euler system is of elliptic-hyperbolic composite-mixed type, this is achieved by using the framework established in [L. Liu; G. Xu; H. Yuan: Stability…
When an upstream steady uniform supersonic flow impinges onto a symmetric straight-sided wedge, governed by the Euler equations, there are two possible steady oblique shock configurations if the wedge angle is less than the detachment angle -- the steady weak shock with supersonic or subsonic downstream flow (determined by the wedge angle that is less or larger than the sonic angle) and the steady strong shock with subsonic downstream flow, both of which satisfy the entropy condition. The fundamental issue -- whether one or both of the steady weak and strong shocks are physically admissible so…
We are concerned with a two-dimensional ($2$-D) Riemann problem for compressible flows modeled by the pressure gradient system that is a $2$-D hyperbolic system of conservation laws. The Riemann initial data consist of four constant states in four sectorial regions such that two shock waves and two vortex sheets are generated between the adjacent states. This Riemann problem can be reduced to a boundary value problem in the self-similar coordinates with the Riemann initial data as its asymptotic boundary data, along with two sonic circles determined by the Riemann initial data, for a nonlinear…
