arXiv subjects

Hyangdong Park

Publications and source records attributed to Hyangdong Park.

8 recordsLinked to original sources

arXiv2026

A Two-Phase Free Boundary Problem for Axisymmetric Subsonic Euler Flows with Contact Discontinuities

We study a free boundary problem for the three-dimensional steady compressible Euler equations in an infinitely long circular cylinder. The free boundary is a contact discontinuity separating an axisymmetric rotational subsonic flow and an axisymmetric potential subsonic flow, neither of which is prescribed a priori. The pressure continuity condition couples two unknown Euler phases through an unknown interface, leading to a genuinely two-phase free boundary problem. Using a Helmholtz decomposition, we reformulate the pressure continuity condition as nonlinear boundary conditions for the Helmholtz variables and develop a coupled iteration framework that simultaneously determines the free boundary and the two flow fields. Uniform estimates independent of the truncation length allow us to pass to the infinite-length limit and construct global solutions. We further establish the downstream asymptotic behavior of solutions, showing that the contact discontinuity becomes asymptotically cylindrical and that the radial velocity vanishes at infinity. To the best of our knowledge, this provides the first existence result and asymptotic characterization for a genuinely two-phase free boundary problem involving a contact discontinuity between an unknown rotational subsonic flow and an unknown potential subsonic flow in a three-dimensional infinitely long cylinder.

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arXiv2025

Three-dimensional Supersonic flows for the steady Euler-Poisson system in divergent nozzles

We are concerned with the unique existence of an axisymmetric supersonic solution with nonzero vorticity and nonzero angular momentum density for the steady Euler-Poisson system in three-dimensional divergent nozzles when prescribing the velocity, strength of electric field, and the entropy at the entrance. We first reformulate the problem via the method of the Helmholtz decomposition for three-dimensional axisymmetric flows and obtain a solution to the reformulated problem by the iteration method. Furthermore, we deal carefully with singularity issues related to the polar angle on the axis of the divergent nozzle.

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arXiv2024

Transonic shocks for three-dimensional axisymmetric flows in divergent nozzles

We prove the stability of three-dimensional axisymmetric solutions to the steady Euler system with transonic shocks in divergent nozzles under perturbations of the exit pressure and the supersonic solution in the upstream region. We first derive a free boundary problem with the newly introduced formulation of the Euler system for three-dimensional axisymmetric flows in divergent nozzles via the method of Helmholtz decomposition. We then construct an iteration scheme and use the Schauder fixed point theorem and weak implicit function theorem to solve the problem.

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arXiv2024

Supersonic flows of the Euler-Poisson system in three-dimensional cylinders

In this paper, we prove the unique existence of three-dimensional supersonic solutions to the steady Euler-Poisson system in cylindrical nozzles when prescribing the velocity, entropy, and the strength of electric field at the entrance. We first establish the unique existence of irrotational supersonic solutions in a cylindrical nozzle with an arbitrary cross section by extending the results of \cite{bae2021three} with an aid of weighted Sobolev norms. Then, we establish the unique existence of three-dimensional axisymmetric supersonic solutions to the Euler-Poisson system with nonzero vorticity in a circular cylinder. In particular, we construct a three-dimensional solution with a nonzero angular momentum density (or equivalently a nonzero swirl). Therefore this is truly a three dimensional flow in the sense that the Euler-Poisson system cannot be reduced to a two dimensional system via a stream function formulation. The main idea is to reformulate the system into a second order hyperbolic-elliptic coupled system and two transport equations via the method of Helmholtz decomposition, and to employ the method of iterations. Several technical issues, including the issue of singularities on the axis of symmetry and the issue of corner singularities in a Lipschitz domain, are carefully addressed.

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arXiv2021

Three-dimensional supersonic flows of Euler-Poisson system for potential flow

We prove the unique existence of supersonic solutions of the Euler- Poisson system for potential flow in a three-dimensional rectangular cylinder when prescribing the velocity and the strength of electric field at the entrance. Overall, the main framework is similar to [1], but there are several technical differences to be taken care of vary carefully. And, it is our main goal to treat all the technical differences occurring when one considers a three dimensional supersonic solution of the steady Euler-Poisson system.

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arXiv2019

Transonic shocks for 3-D axisymmetric compressible inviscid flows in cylinders

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 decomposition.

math.AP↗

arXiv2019

Contact discontinuities for 2-D inviscid compressible flows in infinitely long nozzles

We prove the existence of a subsonic weak solution $({\bf u}, ρ, p)$ to steady Euler system in a two-dimensional infinitely long nozzle when prescribing the value of the entropy $(= \frac{p}{ρ^γ})$ at the entrance by a piecewise $C^2$ function with a discontinuity at a point. Due to the variable entropy condition with a discontinuity at the entrance, the corresponding solution has a nonzero vorticity and contains a contact discontinuity $x_2=g_D(x_1)$. We construct such a solution via Helmholtz decomposition. The key step is to decompose the Rankine-Hugoniot conditions on the contact discontinuity via Helmholtz decomposition so that the compactness of approximated solutions can be achieved. Then we apply the method of iteration to obtain a piecewise smooth subsonic flow with a contact discontinuity and nonzero vorticity. We also analyze the asymptotic behavior of the solution at far field.

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arXiv2019

Contact discontinuities for 3-D axisymmetric inviscid compressible flows in infinitely long cylinders

We prove the existence of a subsonic axisymmetric weak solution $({\bf u},ρ,p)$ with ${\bf u}=u_x{\bf e}_x+u_r{\bf e}_r+u_θ{\bf e}_θ$ to steady Euler system in a three-dimensional infinitely long cylinder $\mathcal{N}$ when prescribing the values of the entropy $(=\frac{p}{ρ^γ})$ and angular momentum density $(=ru_θ)$ at the entrance by piecewise $C^2$ functions with a discontinuity on a curve on the entrance of $\mathcal{N}$. Due to the variable entropy and angular momentum density (=swirl) conditions with a discontinuity at the entrance, the corresponding solution has a nonzero vorticity, nonzero swirl, and contains a contact discontinuity $r=g_D(x)$. We construct such a solution via Helmholtz decomposition. The key step is to decompose the Rankine-Hugoniot conditions on the contact discontinuity via Helmholtz decomposition so that the compactness of approximated solutions can be achieved. Then we apply the method of iteration to obtain a piecewise smooth subsonic flow with a contact discontinuity, nonzero vorticity, and nonzero angular momentum density. We also analyze the asymptotic behavior of the solution at far field.

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