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Register a new userStability of Transonic Shocks in Steady Supersonic Flow past Multidimensional Wedges
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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 first formulate the stability problem as a free boundary problem for nonlinear elliptic equations. Then we introduce the partial hodograph transformation to reduce the free boundary problem into a fixed boundary value problem near a background solution with fully nonlinear boundary conditions for second-order nonlinear elliptic equations in an unbounded domain. To solve this reduced problem, we linearize the nonlinear problem on the background shock solution and then, after solving this linearized elliptic problem, develop a nonlinear iteration scheme that is proved to be contractive.
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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 Bernoullis law with a uniform Bernoulli constant, that is, the Bernoulli constant is allowed to change for different fluid trajectories.
We are concerned with the structural stability of conical shocks in the three-dimensional steady supersonic flows past Lipschitz perturbed cones whose vertex angles are less than the critical angle. The flows under consideration are governed by the steady isothermal Euler equations for potential flow with axisymmetry so that the equations contain a singular geometric source term. We first formulate the shock stability problem as an initial-boundary value problem with the leading conical shock-front as a free boundary, and then establish the existence and asymptotic behavior of global entropy solutions of bounded variation (BV) of the problem. To achieve this, we first develop a modified Glimm scheme to construct approximate solutions via self-similar solutions as building blocks in order to incorporate with the geometric source term. Then we introduce the Glimm-type functional, based on the local interaction estimates between weak waves, the strong leading conical shock, and self-similar solutions, as well as the estimates of the center changes of the self-similar solutions. To make sure the decreasing of the Glimm-type functional, we choose appropriate weights by careful asymptotic analysis of the reflection coefficients in the interaction estimates, when the Mach number of the incoming flow is sufficiently large. Finally, we establish the existence of global entropy solutions involving a strong leading conical shock-front, besides weak waves, under the conditions that the Mach number of the incoming flow is sufficiently large and the weighted total variation of the slopes of the generating curve of the Lipschitz perturbed cone is sufficiently small. Furthermore, the entropy solution is shown to approach asymptotically the self-similar solution that is determined by the incoming flow and the asymptotic tangent of the cone boundary at infinity.
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 that the slope of the shock-front tends asymptotically to the slope of the unperturbed self-similar shock-front downstream at infinity.
We are concerned with the two-dimensional steady supersonic reacting Euler flow past Lipschitz bending walls that are small perturbations of a convex one, and establish the existence of global entropy solutions when the total variation of both the initial data and the slope of the boundary is sufficiently small. The flow is governed by an ideal polytropic gas and undergoes a one-step exothermic chemical reaction under the reaction rate function that is Lipschtiz and has a positive lower bound. The heat released by the reaction may cause the total variation of the solution to increase along the flow direction. We employ the modified wave-front tracking scheme to construct approximate solutions and develop a Glimm-type functional by incorporating the approximate strong rarefaction waves and Lipschitz bending walls to obtain the uniform bound on the total variation of the approximate solutions. Then we employ this bound to prove the convergence of the approximate solutions to a global entropy solution that contains a strong rarefaction wave generated by the Lipschitz bending wall. In addition, the asymptotic behavior of the entropy solution in the flow direction is also analyzed.
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 flat Mach configuration containing one supersonic shock, two transonic shocks, and a contact discontinuity, after fixing the point the four discontinuities intersect. These special solutions are constructed traditionally under the assumption that they are piecewise constant, and they have played important roles in the studies of mathematical gas dynamics. Our results show that the assumption of piecewise constant can be replaced by some more weaker assumptions on the downstream subsonic flows, which are sufficient to uniquely determine these special solutions. Mathematically, these are uniqueness results on solutions of free boundary problems of a quasi-linear system of elliptic-hyperbolic composite-mixed type in bounded or unbounded planar domains, without any assumptions on smallness. The proof relies on an elliptic system of pressure $p$ and the tangent of the flow angle $w=v/u$ obtained by decomposition of the Euler system in Lagrangian coordinates, and a newly developed method for the $L^{infty}$ estimate that is independent of the free boundaries, by combining the maximum principles of elliptic equations, and careful analysis of shock polar applied on the (maybe curved) shock-fronts.
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