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Shangkun Weng

Publications and source records attributed to Shangkun Weng.

At least 19 recordsLinked to original sources
arXiv2026

Three-dimensional steady transonic shocks for the relativistic Euler equations

We investigates the transonic shock problem for steady relativistic Euler equations in hemispherical shells. We show the existence and uniqueness of spherically symmetric transonic shock solutions via the shooting method and the monotonicity between the exit pressure and shock position. Furthermore, without any restrictions on background transonic shocks, we prove the existence and stability of transonic shocks under three dimensional perturbations of the exit pressure. The key techniques involve the decoupling the hyperbolic and elliptic components in steady relativistic Euler equations using the deformation tensor and vorticity. To address the coordinate singularities, the ``spherical projection coordinates" combining the spherical coordinates and the stereographic projection is employed. Subsequently, the Rankine-Hugoniot conditions are appropriately reformulated to determine the shock front and derive the boundary conditions on the shock front for a first-order nonlocal deformation-curl system.

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arXiv2025

Steady super-Alfv\'enic MHD shocks with aligned fields in two-dimensional almost flat nozzles

The Lorentz force induced by the magnetic field in MHD flow introduces a fundamental difference from pure gas dynamics by facilitating the anisotropic propagation of small disturbances, thus the type of steady MHD equations depends on not only the Mach number but also the Alfv\'en number. In the super-Alfv\'enic case, we derive an admissible condition for the locations of transonic shock fronts in terms of the nozzle wall profile and the exit total pressure (the kinetic plus magnetic pressure). Starting from this initial approximation, a nonlinear existence of super-Alfv\'enic transonic shock solution to steady MHD equations is established. Our admissible condition is slightly different from that first introduced by Fang-Xin in [Comm. Pure Appl. Math., 74 (2021), pp. 1493-1544], and because our formulation is based on the deformation-curl decomposition of the steady MHD equations, our admissible condition has the advantage that a direct generalization to three dimensional case is available at least at the level of the initial approximation of the shock position.

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arXiv2025

On three dimensional steady super-Alfv\'{e}nic magnetohydrodynamics shocks with aligned fields

The coupled motion between the hydrodynamic flow and magnetic field introduces significant complexity into the structure of the magnetohydrodynamic (MHD) equations. A key factor contributing to this complexity is the presence of Alfv\'en waves, which critically influences the character of the flow and makes the problem considerably more challenging. Within the framework where the magnetic field is everywhere parallel to the flow velocity, we give an effective decomposition of the steady MHD equations in terms of the deformation tensor and the modified vorticity, where the modification in the vorticity is to record the effect of the Lorentz force on the velocity field. The existence and structural stability of the super-Alfv\'{e}nic cylindrical transonic shock solutions for the steady MHD equations are established under three-dimensional perturbations of the incoming flow and the exit total pressure (kinetic plus magnetic).

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arXiv2025

Three dimensional spherical transonic shock in a hemispherical shell

The existence and stability of a spherical transonic shock in a hemispherical shell under the three dimensional perturbations of the incoming flows and the exit pressure is established without any further restrictions on the background transonic shock solutions. The perturbed transonic shock are completely free and its strength and position are uniquely determined by the incoming flows and the exit pressure. A key issue in the analysis is the ``spherical projection coordinates" (i.e. the composition of the spherical coordinates and the stereographic projection), which provides an appropriate setting for the spherical transonic shock problem in the sense that the transformed equations have a similar structure as the steady Euler equations and do not contain any coordinates singularities. Then we decompose the hyperbolic and elliptic modes in the steady Euler equations in terms of the deformation and vorticity. An elaborate reformulation of the Rankine-Hugoniot conditions yields an unusual second order differential boundary condition on the shock front to the first order nonlocal deformation-curl system, from which an oblique boundary condition can be derived after homogenizing the curl system and introducing the potential function. The analysis of the compatibility conditions at the intersection of the shock front and the shell boundary is crucial for the optimal regularity of all physical quantities.

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arXiv2025

The steady inviscid compressible self-similar flows and the stability analysis

We investigate the steady inviscid compressible self-similar flows which depends only on the polar angle in spherical coordinates. It is shown that besides the purely supersonic and subsonic self-similar flows, there exists purely sonic flows, Beltrami flows with a nonconstant proportionnality factor and smooth transonic self-similar flows with large vorticity. For a constant supersonic incoming flow past an infinitely long circular cone, a conic shock attached to the tip of the cone will form, provided the opening angle of the cone is less than a critical value. We introduce the shock polar for the radial and polar components of the velocity and show that there exists a monotonicity relation between the shock angle and the radial velocity, which seems to be new and not been observed before. If a supersonic incoming flow is self-similar with nonzero azimuthal velocity, a conic shock also form attached to the tip of the cone. The state at the downstream may change smoothly from supersonic to subsonic, thus the shock can be supersonic-supersonic, supersonic-subsonic and even supersonic-sonic where the shock front and the sonic front coincide. We further investigate the structural stability of smooth self-similar irrotational transonic flows and analyze the corresponding linear mixed type second order equation of Tricomi type. By exploring some key properties of the self-similar solutions, we find a multiplier and identify a class of admissible boundary conditions for the linearized mixed type second-order equation. We also prove the existence and uniqueness of a class of smooth transonic flows with nonzero vorticity which depends only on the polar and azimuthal angles in spherical coordinates.

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arXiv2024

Some three dimensional smooth transonic flows for the steady Euler equations with an external force

We establish the existence and uniqueness of some smooth accelerating transonic flows governed by the three dimensional steady compressible Euler equations with an external force in cylinders with arbitrary cross sections, which include both irrotational flows and Beltrami flows with nonuniform proportionality factors. One of the key ingredients in the analysis of smooth transonic irrotational flows is the well-posedness theory of classical solutions in $H^4$ to a linear elliptic-hyperbolic mixed second order differential equation of Keldysh type in cylinders with mixed boundary conditions. This is achieved by extending the problem to an auxiliary linear elliptic-hyberbolic-elliptic mixed problem in a longer cylinder where the governing equation becomes elliptic at the exit of the new cylinder, so that one can use the multiplier method and the cut-off techniques to derive the $H^2$ and higher order estimates in transonic regions. It is further shown that the energy estimate can be closed in the $H^4$ framework. For smooth transonic Beltrami flows, we solve a transport equation for the proportionality factor and a type-changing enlarged deformation-curl system with mixed boundary conditions. The compatibility conditions for the $H^4$ estimate to the enlarged deformation-curl system near the intersection between the entrance and the cylinder wall play a crucial role in the analysis.

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arXiv2024

Subsonic flows with a contact discontinuity in a two-dimensional finitely long curved nozzle

In this paper, we establish the existence and uniqueness of subsonic flows with a contact discontinuity in a two-dimensional finitely long slightly curved nozzle by prescribing the entropy, the Bernoulli's quantity and the horizontal mass flux distribution at the entrance and the flow angle at the exit. The problem is formulated as a nonlinear boundary value problem for a hyperbolic-elliptic mixed system with a free boundary. The Lagrangian transformation is employed to straighten the contact discontinuity and the Euler system is reduced to a second-order nonlinear elliptic equation for the stream function. One of the key points in the analysis is to solve the associated linearized elliptic boundary value problem with mixed boundary conditions in a weighted Hölder space. Another one is to employ the implicit function theorem to locate the contact discontinuity.

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arXiv2024

Structural stability of three dimensional transonic shock flows with an external force

We establish the existence and uniqueness of the transonic shock solution for steady isentropic Euler system with an external force in a rectangular cylinder under the three-dimensional perturbations for the incoming supersonic flow, the exit pressure and the external force. The external force has a stabilization effect on the transonic shocks in flat nozzles and the transonic shock is completely free, we do not require it passing through a fixed point. By utilizing the deformation-curl decomposition to decouple the hyperbolic and elliptic modes in the steady Euler system effectively and reformulating the Rankine-Hugoniot conditions, the transonic shock problem is reduced to a deformation-curl first order system for the velocity field with nonlocal terms supplementing with an unusual second order differential boundary condition on the shock front, an algebraic equation for determining the shock front and two transport equations for the Bernoulli's quantity and the first component of the vorticity.

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arXiv2024

Smooth axisymmetric transonic irrotational flows to the steady Euler equations with an external force

For a class of external forces, we prove the existence and uniqueness of smooth transonic flows to the one dimensional steady Euler system with an external force, which is subsonic at the inlet and flows out at supersonic speed after smoothly accelerating through the sonic point. We then investigate the structural stability of the one-dimensional smooth transonic flows with positive acceleration under axisymmetric perturbations of suitable boundary conditions, and establish the first existence and uniqueness result for smooth axisymmetric transonic irrotational flows. The key point lies on the analysis of a linear second order elliptic-hyperbolic mixed differential equation of Keldysh type with a singular term. Some weighted Sobolev spaces $H_r^m(D) (m=2,3,4)$ are introduced to deal with the singularities near the axis. Compared with the stability analysis in the two dimensional case by Weng and Xin (arXiv:2309.07468), there are several interesting new observations about the structure of the linear mixed type equation with a singular term which play crucial roles in establishing the $H^4_r(D)$ estimate.

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arXiv2024

Smooth helically symmetric transonic flows with nonzero vorticity in a concentric cylinder

This paper concerns the structural stability of smooth cylindrical symmetric transonic flows in a concentric cylinder under helically symmetric perturbation of suitable boundary conditions. The deformation-curl decomposition developed by the second author and his collaborator is utilized to effectively decouple the elliptic-hyperbolic mixed structure in the steady compressible Euler equation. A key parameter in the helical symmetry is the step (denoted by $σ$), which denotes the magnitude of the translation along the symmetry axis after rotating one full turn. It is shown that the step determines the type of the first order partial differential system satisfied by the radial and vertical velocity. There exists a critical number $σ_{*}$ depending only on the background transonic flows, such that if $0<σ<σ_{*}$, one can prove the existence and uniqueness of smooth helically symmetric transonic flows with nonzero vorticity.

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arXiv2024

Existence and stability of cylindrical transonic shock solutions under three dimensional perturbations

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.

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arXiv2024

Subsonic Euler flows in a three-dimensional finitely long cylinder with arbitrary cross section

This paper concerns the well-posedness of subsonic flows in a three-dimensional finitely long cylinder with arbitrary cross section. We establish the existence and uniqueness of subsonic flows in the Sobolev space by prescribing the normal component of the momentum, the vorticity, the entropy, the Bernoulli's quantity at the entrance and the normal component of the momentum at the exit. One of the key points in the analysis is to utilize the deformation-curl decomposition for the steady Euler system introduced in \cite{WX19} to deal with the hyperbolic and elliptic modes. Another one is to employ the separation of variables to improve the regularity of solutions to a deformation-curl system near the intersection between the entrance and exit with the cylinder wall.

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arXiv2023

Smooth transonic flows with nonzero vorticity to a quasi two dimensional steady Euler flow model

This paper concerns studies on smooth transonic flows with nonzero vorticity in De Laval nozzles for a quasi two dimensional steady Euler flow model which is a generalization of the classical quasi one dimensional model. First, the existence and uniqueness of smooth transonic flows to the quasi one-dimensional model, which start from a subsonic state at the entrance and accelerate to reach a sonic state at the throat and then become supersonic are proved by a reduction of degeneracy of the velocity near the sonic point and the implicit function theorem. These flows can have positive or zero acceleration at their sonic points and the degeneracy types near the sonic point are classified precisely. We then establish the structural stability of the smooth one dimensional transonic flow with positive acceleration at the sonic point for the quasi two dimensional steady Euler flow model under small perturbations of suitable boundary conditions, which yields the existence and uniqueness of a class of smooth transonic flows with nonzero vorticity and positive acceleration to the quasi two dimensional model. The positive acceleration of the one dimensional transonic solutions plays an important role in searching for an appropriate multiplier for the linearized second order mixed type equations. A deformation-curl decomposition for the quasi two dimensional model is utilized to deal with the transonic flows with nonzero vorticity.

math.AP↗
arXiv2023

Smooth Subsonic and Transonic Flows with Nonzero Angular Velocity and Vorticity to steady Euler-Poisson system in a Concentric Cylinder

In this paper, both smooth subsonic and transonic flows to steady Euler-Poisson system in a concentric cylinder are studied. We first establish the existence of cylindrically symmetric smooth subsonic and transonic flows to steady Euler-Poisson system in a concentric cylinder. On one hand, we investigate the structural stability of smooth cylindrically symmetric subsonic flows under three-dimensional perturbations on the inner and outer cylinders. On the other hand, the structural stability of smooth transonic flows under the axi-symmetric perturbations are examined. There is no any restrictions on the background subsonic and transonic solutions. A deformation-curl-Poisson decomposition to the steady Euler-Poisson system is utilized in our work to deal with the hyperbolic-elliptic mixed structure in subsonic region. It should be emphasized that there is a special structure of the steady Euler-Poisson system which yields a priori estimates and uniqueness of a second order elliptic system for the velocity potential and the electrostatic potential.

math.AP↗
arXiv2023

Subsonic flows with a contact discontinuity in a finitely long axisymmetric cylinder

This paper concerns the structural stability of subsonic flows with a contact discontinuity in a finitely long axisymmetric cylinder. We establish the existence and uniqueness of axisymmetric subsonic flows with a contact discontinuity by prescribing the horizontal mass flux distribution, the swirl velocity, the entropy and the Bernoulli's quantity at the entrance and the radial velocity at the exit. It can be formulated as a free boundary problem with the contact discontinuity to be determined simultaneously with the flows. Compared with the two-dimensional case, a new difficulty arises due to the singularity near the axis. One of the key points in the analysis is the introduction of an invertible modified Lagrangian transformation which can overcome this difficulty and straighten the contact discontinuity. Another one is to utilize the deformation-curl decomposition for the steady Euler system introduced in \cite{WX19} to effectively decouple the hyperbolic and elliptic modes. Finally, the contact discontinuity will be located by using the implicit function theorem.

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arXiv2023

Supersonic flows with a contact discontinuity to the two-dimensional steady rotating Euler system

This paper concerns the structural stability of supersonic flows with a contact discontinuity in a finitely long curved nozzle for the two-dimensional steady compressible rotating Euler system. Concerning the effect of Coriolis force, we first establish the existence of supersonic shear flows with a contact discontinuity in the flat nozzle. Then we consider the stability of these background supersonic shear flows with a contact discontinuity when the incoming supersonic flow and the upper and lower nozzle walls are suitably perturbed. The problem can be formulated as an initial boundary value problem with a contact discontinuity as a free boundary. To deal with the free boundary value problem, the Lagrangian transformation is introduced to straighten and fix the contact discontinuity. The rotating Euler system is reduced to a first order hyperbolic system for the Riemann invariants. We design an iteration scheme and derive some estimates for the solution to the hyperbolic system. Finally, by using the inverse Lagrangian transformation, we prove the original free boundary problem admits two layers of smooth supersonic flows separated by a smooth contact discontinuity.

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arXiv2022

Structural Stability of Transonic Shock Flows with an External Force

This paper is devoted to the structural stability of a transonic shock passing through a flat nozzle for two-dimensional steady compressible flows with an external force. We first establish the existence and uniqueness of one dimensional transonic shock solutions to the steady Euler system with an external force by prescribing suitable pressure at the exit of the nozzle when the upstream flow is a uniform supersonic flow. It is shown that the external force helps to stabilize the transonic shock in flat nozzles and the shock position is uniquely determined. Then we are concerned with the structural stability of these transonic shoc solutions when the exit pressure is suitably perturbed. One of the new ingredients in our analysis is to use the deformation-curl decomposition to the steady Euler system developed in \cite{WengX2019} to deal with the transonic shock problem.

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