Optimal Hölder Continuity for a Class of Degenerate Elliptic Problems with an Application to Subsonic-Sonic Flows

In this paper, we first study a class of elliptic equations with anisotropic boundary degeneracy. Besides establishing the existence, uniqueness and comparison principle, we obtain the optimal Hölder estimates for weak solutions by the estimates in the Campanato space. Based on such Hölder estimates, we then investigate subsonic-sonic flows with singularities at the sonic curves in a symmetric convergent nozzle with straight wall for an approximate model of the potential flow equation. It is proved that the perturbation problem of the symmetric subsonic-sonic flow is solvable and the symmetric subsonic-sonic flow is stable.     

A linear approximation for the regular reflection of a weak shock at a wedge satisfying sonic condition

The problem of shock reflection by a wedge, which the flow is dominated by the unsteady potential flow equation, is a important problem. In weak regular reflection, the flow behind the reflected shock is immediately supersonic and becomes subsonic further downstream. The reflected shock is transonic. Its position is a free boundary for the unsteady potential equation, which is degenerate at the sonic line in self-similar coordinates. Applying the special partial hodograph transformation used in [Zhouping Xin, Huicheng Yin, Transonic shock in a nozzle I, 2-D case, Comm. Pure Appl. Math. 57 (2004) 1-51; Zhouping Xin, Huicheng Yin, Transonic shock in a nozzle II, 3-D case, IMS, preprint (2003)], we derive a nonlinear degenerate elliptic equation with nonlinear boundary conditions in a piecewise smooth domain. When the angle, which between incident shock and wedge, is small, we can see that weak regular reflection as the disturbance of normal reflection as in [Shuxing Chen, Linear approximation of shock reflection at a wedge with large angle, Comm. Partial Differential Equations 21 (78) (1996) 1103-1118]. By linearizing the resulted nonlinear equation and boundary conditions with above viewpoint, we obtain a linear degenerate elliptic equation with mixed boundary conditions and a linear degenerate elliptic equation with oblique boundary conditions in a curved quadrilateral domain. By means of elliptic regularization techniques, delicate a priori estimate and compact arguments, we show that the solution of linearized problem with oblique boundary conditions is smooth in the interior and Lipschitz continuous up to the degenerate boundary.     

A linear approximation for the regular reflection of a weak shock at a wedge

The problem of shock reflection by a wedge in the flow dominated by the unsteady potential flow equation is an important problem. In weak regular reflection, the flow behind the reflected shock is immediately supersonic and becomes subsonic further downstream. The reflected shock is transonic. Its position is a free boundary for the unsteady potential equation, which is degenerate at the sonic line in self-similar coordinates. Applying the special partial hodograph transformation used in [Zhouping Xin, Huicheng Yin, Transonic shock in a nozzle I, 2-D case, Comm. Pure Appl. Math. LVII (2004) 1-51; Zhouping Xin, Huicheng Yin, Transonic shock in a nozzle II, 3-D case, IMS, preprint, 2003], we derive a nonlinear degenerate elliptic equation with nonlinear boundary conditions in a piecewise smooth domain. When the angle between incident shock and wedge is small, we can see the weak regular reflection as the disturbance of normal reflection as in [Chen Shuxing, Linear approximation of shock reflection at a wedge with large angle, Comm. Partial Differential Equations 21(78) (1996) 1103-1118]. By linearizing the resulted nonlinear equation and boundary conditions with the above viewpoint in [Chen Shuxing, Linear approximation of shock reflection at a wedge with large angle, Comm. Partial Differential Equations 21(78) (1996) 1103-1118], we obtain a linear degenerate elliptic equation with mixed boundary conditions in a curved quadrilateral domain. By means of elliptic regularization techniques, a delicate a priori estimate and compact arguments, we show that the solution of the linearized problem is smooth in the interior and Lipschitz continuous up to the degenerate boundary.     

Transonic shock in a nozzle I: 2D case

In this paper we establish the existence and uniqueness of a transonic shock for the steady flow through a general two‐dimensional nozzle with variable sections. The flow is governed by the inviscid potential equation, and is supersonic upstream, has no‐flow boundary conditions on the nozzle walls, and a given pressure at the exit of the exhaust section. The transonic shock is a free boundary dividing two regions of C flow in the nozzle. The potential equation is hyperbolic upstream where the flow is supersonic, and elliptic in the downstream subsonic region. In particular, our results show that there exists a solution to the corresponding free boundary problem such that the equation is always subsonic in the downstream region of the nozzle when the pressure in the exit of the exhaustion section is appropriately larger than that in the entry. This confirms exactly the conjecture of Courant and Friedrichs on the transonic phenomena in a nozzle [10]. Furthermore, the stability of the transonic shock is also proved when the upstream supersonic flow is a small steady perturbation for the uniform supersonic flow or the pressure at the exit has a small perturbation. The main ingredients of our analysis are a generalized hodograph transformation and multiplier methods for elliptic equation with mixed boundary conditions and corner singularities. © 2004 Wiley Periodicals, Inc.     

The convexity for compressible subsonic cavity flows

In this paper, we first show that the flow velocity takes its maximum on the free boundary, provided that the nozzle wall and obstacle satisfy some corresponding geometric assumptions. Second, the convexity of the free boundary to the compressible subsonic cavity flows will be established. Finally, the optimal regularity of the free boundary at the detachment point is obtained.     

Transonic nozzle flows and free boundary problems for the full Euler equations

We establish the existence and uniqueness of transonic flows with a transonic shock through a two-dimensional nozzle of slowly varying cross-sections. The transonic flow is governed by the steady, full Euler equations. Given an incoming smooth flow that is close to a constant supersonic state (i.e., smooth Cauchy data) at the entrance and the subsonic condition with nearly horizontal velocity at the exit of the nozzle, we prove that there exists a transonic flow whose downstream smooth subsonic region is separated by a smooth transonic shock from the upstream supersonic flow. This problem is approached by a one-phase free boundary problem in which the transonic shock is formulated as a free boundary. The full Euler equations are decomposed into an elliptic equation and a system of transport equations for the free boundary problem. An iteration scheme is developed and its fixed point is shown to exist, which is a solution of the free boundary problem, by combining some delicate estimates for the elliptic equation and the system of transport equations with the Schauder fixed point argument. The uniqueness of transonic nozzle flows is also established by employing the coordinate transformation of Euler-Lagrange type and detailed estimates of the solutions.     

Examples of steady subsonic flows in a convergent-divergent approximate nozzle

We construct special solutions of the full Euler system for steady compressible flows in a convergent-divergent approximate nozzle and study the stability of the purely subsonic flows. For a given pressure p₀ prescribed at the entry of the nozzle, as the pressure p₁ at the exit decreases, the flow patterns in the nozzle change continuously: there appear subsonic flow, subsonic-sonic flow, transonic flow and transonic shocks. Our results indicate that, to determine a subsonic flow in a two-dimensional nozzle, if the Bernoulli constant is uniform in the flow field, then this constant should not be prescribed if the pressure, density at the entry and the pressure at the exit of the nozzle are given; if the Bernoulli constant and both the pressures at the entrance and the exit are given, the average of the density at the entrance is then totally determined.     

Asymptotics of the solution to a stationary piecewise-smooth reaction-diffusion equation with a multiple root of the degenerate equation

A piecewise-smooth second-order singularly perturbed differential equation whose right-hand side is a nonlinear function with a discontinuity on some curve is investigated. This is a new class of problems in the case where the degenerate equation has a multiple root on the left-hand side of the curve which separates the domain and an isolated root on the right-hand side of that curve. The asymptotics of a solution with an internal layer near a point on the discontinuous curve and the transition point is constructed. The method to construct the internal layer function is proposed. The behavior of the solution in the internal layer consisting of four zones essentially differs from the case of isolated roots. For sufficiently small parameter values, the existence of a smooth solution with an internal layer from the multiple root of the degenerate equation to the isolated root in the neighborhood of a point on the discontinuous curve is proved. The method can be shown to be effective in the given example.     

Stability of transonic shock fronts in two-dimensional Euler systems

We study the stability of stationary transonic shock fronts under two-dimensional perturbation in gas dynamics. The motion of the gas is described by the full Euler system. The system is hyperbolic ahead of the shock front, and is a hyperbolic-elliptic composed system behind the shock front. The stability of the shock front and the downstream flow under two-dimensional perturbation of the upstream flow can be reduced to a free boundary value problem of the hyperbolic-elliptic composed system. We develop a method to deal with boundary value problems for such systems. The crucial point is to decompose the system to a canonical form, in which the hyperbolic part and the elliptic part are only weakly coupled in their coefficients. By several sophisticated iterative processes we establish the existence and uniqueness of the solution to the described free boundary value problem. Our result indicates the stability of the transonic shock front and the flow field behind the shock.

     

The regularity of the free boundary in a multidimensional filtration problem

We prove the regularity of the free boundary for a filtration problem with capillarity in more than one space dimension. The free boundary is the interface between the saturated region (in which the governing equation is elliptic) and the unsaturated region (where a degenerate parabolic equation is to be solved).This work was partially supported by National Project Equazioni di Evoluzione e Applicazioni Fisico Matematiche (M.U.R.S.T.).     

Existence and regularity of solution of mixed boundary value problem of a Busemann-equation with nonlinear absorb term

The Busemann-equation is a classical equation coming from fluid dynamics. The well-posed problem and regularity of solution of Busemann-equation with nonlinear term are interesting and important. The Busemann-equation is elliptic in y>0 and is degenerate at the line y=0 in R². With a special nonlinear absorb term, we study a nonlinear degenerate elliptic equation with mixed boundary conditions in a piecewise smooth domain. By means of elliptic regularization technique, a delicate prior estimate and compact argument, we show that the solution of mixed boundary value problem of Busemann-equation is smooth in the interior and Lipschitz continuous up to the degenerate boundary on some conditions. The result is better than the result of classical boundary degenerate elliptic equation.     

Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type

We establish 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 a second-order, nonlinear equation of mixed elliptic-hyperbolic type for the velocity potential. The transonic shock problem can be formulated into the following free boundary problem: The free boundary is the location of the transonic shock which divides the two regions of smooth flow, and the equation is hyperbolic in the upstream region where the smooth perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem. Our results indicate that there exists a unique solution of the free boundary problem such that the equation is always elliptic in the downstream region and the free boundary is smooth, provided that the hyperbolic phase is close to a uniform flow. We prove that the free boundary is stable under the steady perturbation of the hyperbolic phase. We also establish the existence and stability of multidimensional transonic shocks near spherical or circular transonic shocks.

     

Optimizing buffer size for the retrial queue: two state space collapse results in heavy traffic

We study a single server queueing model with admission control and retrials. In the heavy traffic limit, the main queue and retrial queue lengths jointly converge to a degenerate two-dimensional diffusion process. When this model is considered with holding and rejection costs, formal limits lead to a free boundary curve that determines a threshold on the main queue length as a function of the retrial queue length, above which arrivals must be rejected. However, it is known to be a notoriously difficult problem to characterize this curve. We aim instead at optimizing the threshold on the main queue length independently of the retrial queue length. Our main result shows that in the small and large retrial rate limits, this problem is governed by the Harrison–Taksar free boundary problem, which is a Bellman equation in which the free boundary consists of a single point. We derive the asymptotically optimal buffer size in these two extreme cases, as the scaling parameter and the retrial rate approach their limits.     

Steady transonic shocks and free boundary problems in infinite cylinders for the Euler equations

We establish the existence and stability of multidimensional transonic shocks (hyperbolic‐elliptic shocks) for the Euler equations for steady compressible potential fluids in infinite cylinders. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for velocity, can be written as a second order nonlinear equation of mixed elliptic‐hyperbolic type for the velocity potential. The transonic shock problem in an infinite cylinder can be formulated into the following free boundary problem: The free boundary is the location of the multidimensional transonic shock which divides two regions of C^1,α flow in the infinite cylinder, and the equation is hyperbolic in the upstream region where the C^1,α perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem in unbounded domains. Our results indicate that there exists a solution of the free boundary problem such that the equation is always elliptic in the unbounded downstream region, the uniform velocity state at infinity in the downstream direction is uniquely determined by the given hyperbolic phase, and the free boundary is C^1,α, provided that the hyperbolic phase is close in C^1,α to a uniform flow. We further prove that, if the steady perturbation of the hyperbolic phase is C^2,α, the free boundary is C^2,α and stable under the steady perturbation. © 2003 Wiley Periodicals Inc.     

Regularity of solutions to regular shock reflection for potential flow

The shock reflection problem is one of the most important problems in mathematical fluid dynamics, since this problem not only arises in many important physical situations but also is fundamental for the mathematical theory of multidimensional conservation laws that is still largely incomplete. However, most of the fundamental issues for shock reflection have not been understood, including the regularity and transition of different patterns of shock reflection configurations. Therefore, it is important to establish the regularity of solutions to shock reflection in order to understand fully the phenomena of shock reflection. On the other hand, for a regular reflection configuration, the potential flow governs the exact behavior of the solution in C ^1,1 across the pseudo-sonic circle even starting from the full Euler flow, that is, both of the nonlinear systems are actually the same in a physically significant region near the pseudo-sonic circle; thus, it becomes essential to understand the optimal regularity of solutions for the potential flow across the pseudo-sonic circle (the transonic boundary from the elliptic to hyperbolic region) and at the point where the pseudo-sonic circle (the degenerate elliptic curve) meets the reflected shock (a free boundary connecting the elliptic to hyperbolic region). In this paper, we study the regularity of solutions to regular shock reflection for potential flow. In particular, we prove that the C ^1,1-regularity is optimal for the solution across the pseudo-sonic circle and at the point where the pseudo-sonic circle meets the reflected shock. We also obtain the C ^2,α regularity of the solution up to the pseudo-sonic circle in the pseudo-subsonic region. The problem involves two types of transonic flow: one is a continuous transition through the pseudo-sonic circle from the pseudo-supersonic region to the pseudo-subsonic region; the other a jump transition through the transonic shock as a free boundary from another pseudo-supersonic region to the pseudo-subsonic region. The techniques and ideas developed in this paper will be useful to other regularity problems for nonlinear degenerate equations involving similar difficulties.     

Transport equation with boundary conditions of hysteresis type

This paper is an advanced extension of the work reported in (Nonlinear Anal. 2005; 63 :1467–1473). A transport equation that describes the propagation of a substance in a moving fluid or gas is considered. The equation contains the transient, convection, and diffusion terms. The problem is formulated in a bounded domain provided with an inlet and an outlet for the fluid or gas flow. The crucial point of the problem setting is a hysteresis‐type condition posed on an active part of the boundary. This condition reflects the nondecreasing accumulation with saturation of the transported substance at each point of the active boundary part. We prove the existence and uniqueness of solutions to this problem, study the regularity properties of solutions, and perform numerical simulations that clarify the behavior of the model. Comparing with the results of (Nonlinear Anal. 2005; 63 :1467–1473), the advancement of this work consists in accounting for the motion of the fluid or gas and posing inlet and outlet boundary conditions. Copyright © 2009 John Wiley & Sons, Ltd.     

Viscosity solutions of a class of degenerate quasilinear parabolic equations with Dirichlet boundary condition

This paper is concerned with viscosity solutions for a class of degenerate quasilinear parabolic equations in a bounded domain with homogeneous Dirichlet boundary condition. The equation under consideration arises from a number of practical model problems including reaction–diffusion processes in a porous medium. The degeneracy of the problem appears on the boundary and possibly in the interior of the domain. The goal of this paper is to establish some comparison properties between viscosity upper and lower solutions and to show the existence of a continuous viscosity solution between them. An application of the above results is given to a porous-medium type of reaction–diffusion model which demonstrates some distinctive properties of the solution when compared with the corresponding semilinear problem.     

一类具非局部边界条件的四阶非线性微分方程的对称正解

本文考虑形如的非线性四阶微分方程非局部边值问题,这里a,b∈L~1[0,1],g:(0,1)→[0,∞)在(0,1)上连续、对称,且可能在t=0和t=1处奇异．f:[0,1]×[0,∞)→[0,∞)连续且对所有x∈[0,∞],f(·,x)在[0,1]上对称．在某些适当的增长性条件下,应用Krasnoselskii不动点定理证明了对称正解的存在性和多重性．     

Assessment of turbulence modeling for gas flow in two-dimensional convergent–divergent rocket nozzle

In the present study, the turbulent gas flow dynamics in a two-dimensional convergent–divergent rocket nozzle is numerically predicted and the associated physical phenomena are investigated for various operating conditions. The nozzle is assumed to have impermeable and adiabatic walls with a flow straightener in the upstream side and is connected to a plenum surrounding the nozzle geometry and extended in the downstream direction. In this integrated component model, the inlet flow is assumed a two-dimensional, steady, compressible, turbulent and subsonic. The physics based mathematical model of the considered flow consists of conservation of mass, momentum and energy equations subject to appropriate boundary conditions as defined by the physical problem stated above. The system of the governing equations with turbulent effects is solved numerically using different turbulence models to demonstrate their numerical accuracy in predicting the characteristics of turbulent gas flow in such complex geometry. The performance of the different turbulence models adopted has been assessed by comparing the obtained results of the static wall pressure and the shock position with the available experimental and numerical data. The dimensionless shear stress at the nozzle wall and the separation point are also computed and the flow field is illustrated. The various implemented turbulence models have shown different behavior of the turbulent characteristics. However, the shear-stress transport (SST) k–ω model exhibits the best overall agreement with the experimental measurements. In general, the proposed numerical procedure applied in the present paper shows good capability in predicting the physical phenomena and the flow characteristics encountered in such kinds of complex turbulent flow.     

Pointwise Green's Function Estimates Toward Stability for Degenerate Viscous Shock Waves

ABSTRACT

We consider degenerate viscous shock waves arising in systems of two conservation laws, where degeneracy describes viscous shock waves for which the asymptotic endstates are sonic to the hyperbolic system (the shock speed is equal to one of the characteristic speeds). In particular, we develop detailed pointwise estimates on the Green's function associated with the linearized perturbation equation, sufficient for establishing that spectral stability implies nonlinear stability. The analysis of degenerate viscous shock waves involves several new features, such as algebraic (nonintegrable) convection coefficients, loss of analyticity of the Evans function at the leading eigenvalue, and asymptotic time decay of perturbations intermediate between that of the Lax case and that of the undercompressive case.