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Myoungjean Bae

Publications and source records attributed to Myoungjean Bae.

At least 19 recordsLinked to original sources
arXiv2026

Contact discontinuities for 3-D axisymmetric flows with nonzero swirl

In this paper, we prove the existence and far-field behavior of an axisymmetric solution $(ρ,\mathbf{u},p)$ to the steady Euler system in an infinitely long cylindrical nozzle with a perturbed wall $r=R(x_1)$, containing a contact discontinuity $r=g_D(x_1)$. The background flow carries a nonzero swirl component $β_0\mathbf{e}_θ$, where $β_0$ is not necessarily small. Consequently, the background entropy, Bernoulli function, and angular momentum density depend on the radial variable, and their radial derivatives are not necessarily small.

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arXiv2026

Global Uniqueness of Subsonic Flows for the Steady Euler-Poisson System

We prove the global uniqueness of multidimensional subsonic flows for the steady Euler--Poisson system in a bounded nozzle in the sense that uniqueness holds without restricting solutions to be small perturbations of a background state. The proof is based on a convexity property of the set of subsonic states and energy estimates.

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arXiv2025

Classical solutions to a mixed-type PDE with a Keldysh-type degeneracy and accelerating transonic solutions to the Euler-Poisson system

In this paper, we first prove the existence of classical solutions to a class of Keldysh-type equations. Next, we apply this existence result to prove the structural stability of one-dimensional smooth transonic solutions to the steady Euler-Poisson system. Most importantly, the solutions constructed in this paper are classical solutions to the Euler-Poisson system, thus their sonic interfaces are not weak discontinuities in the sense that all the flow variables, such as density, velocity and pressure, are at least $C^1$ across the interfaces.

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arXiv2024

A review on two types of sonic interfaces

In this paper, two examples of sonic interfaces are presented. The first example shows the case of sonic interfaces as weak discontinuities in self-similar shock configurations of unsteady Euler system. The second example shows the case of sonic interfaces as regular interfaces in accelerating transonic flows governed by the steady Euler-Poisson system with self-generated electric forces. And, we discuss analytic differences of the two examples, and introduce an open problem on decelerating transonic solution to the steady Euler-Poisson system.

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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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arXiv2024

Prandtl-Meyer Reflection Configurations, Transonic Shocks, and Free Boundary Problems

We are concerned with the Prandtl-Meyer reflection configurations of unsteady global solutions for supersonic flow impinging upon a symmetric solid wedge. Prandtl (1936) first employed the shock polar analysis to show that there are two possible steady configurations: the steady weak and strong shock solutions, when a steady supersonic flow impinges upon the wedge whose angle is less than the detachment angle, and then conjectured that the steady weak shock solution is physically admissible. The fundamental issue of whether one or both of the steady weak/strong shocks are physically admissible has been vigorously debated over the past eight decades and has not yet been settled definitively. On the other hand, the Prandtl-Meyer reflection configurations are core configurations in the structure of global entropy solutions of the 2-D Riemann problem, while the Riemann solutions themselves are local building blocks and determine local structures, global attractors, and large-time asymptotic states of general entropy solutions. In this sense, we have to understand the reflection configurations to understand fully the global entropy solutions of 2-D hyperbolic systems of conservation laws, including the admissibility issue for the entropy solutions. In this monograph, we address this longstanding open issue and present our analysis to establish the stability theorem for the steady weak shock solutions as the long-time asymptotics of the Prandtl-Meyer reflection configurations for unsteady potential flow for all the physical parameters up to the detachment angle. To achieve these, we first reformulate the problem as a free boundary problem involving transonic shocks and then obtain appropriate monotonicity properties and uniform a priori estimates for admissible solutions, which allow us to employ the Leray-Schauder degree argument to complete the theory.

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arXiv2023

The steady Euler-Poisson system and accelerating flows with transonic $C^1$-transitions

In this paper, we prove the existence of two-dimensional solutions to the steady Euler-Poisson system with continuous transonic transitions across sonic interfaces of codimension 1. First, we establish the well-posedness of a boundary value problem for a linear second order system that consists of an elliptic-hyperbolic mixed type equation with a degeneracy occurring on an interface of codimension 1, and an elliptic equation weakly coupled together. Then we apply the Schauder fixed point theorem to prove the existence of two-dimensional solutions to the potential flow model of the steady Euler-Poisson system with continuous transonic transitions across sonic interfaces. With the aid of Helmholtz decomposition, established in [6], we extend the existence result to the full Euler-Poisson system for the case of nonzero vorticity. Most importantly, the solutions constructed in this paper are classical solutions to Euler-Poisson system, thus their sonic interfaces are not weak discontinuities in the sense that all the flow variables are $C^1$ across the interfaces.

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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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arXiv2020

Detached shock past a blunt body

In $\R^2$, a symmetric blunt body $W_b$ is fixed by smoothing out the tip of a symmetric wedge $W_0$ with the half-wedge angle $θ_w\in (0, \fracπ{2})$. We first show that if a horizontal supersonic flow of uniform state moves toward $W_0$ with a Mach number $M_{\infty}>1$ sufficiently large, %depending on $θ_w$, then there exist two shock solutions, {\emph{a weak shock solution and a strong shock solution}}, with the shocks being straight and attached to the tip of the wedge $W_0$. Such shock solutions are given by a shock polar analysis, and they satisfy entropy conditions. The main goal of this work is to construct a detached shock solution of the steady Euler system for inviscid compressible irrotational flow in $\R^2\setminus W_b$. In particular, we seek a shock solution with the far-field state being the strong shock solution obtained from the shock polar analysis. Furthermore, we prove that the detached shock forms a convex curve around the blunt body $W_b$ if the Mach number of the incoming supersonic flow is sufficiently large, and if the boundary of $W_b$ is convex.

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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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arXiv2019

Structural stability of Supersonic solutions to the Euler-Poisson system

The well-posedness for the supersonic solutions of the Euler-Poisson system for hydrodynamical model in semiconductor devices and plasmas is studied in this paper. We first reformulate the Euler-Poisson system in the supersonic region into a second order hyperbolic-elliptic coupled system together with several transport equations. One of the key ingredients of the analysis is to obtain the well-posedness of the boundary value problem for the associated linearized hyperbolic-elliptic coupled system, which is achieved via a delicate choice of multiplier to gain energy estimate. The nonlinear structural stability of supersonic solution in the general situation is established by combining the iteration method with the estimate for hyperbolic-elliptic system and the transport equations together.

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arXiv2017

Radial transonic shock solutions of Euler-Poisson system in convergent nozzles

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.

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arXiv2016

3-D axisymmetric subsonic flows with nonzero swirl for the compressible Euler-Poisson system

We address the structural stability of 3-D axisymmetric subsonic flows with nonzero swirl for the steady compressible Euler-Poisson system in a cylinder supplemented with non small boundary data. A special Helmholtz decomposition of the velocity field is introduced for 3-D axisymmetric flow with a nonzero swirl(=angular momentum density) component. With the newly introduced decomposition, a quasilinear elliptic system of second order is derived from the elliptic modes in Euler-Poisson system for subsonic flows. Due to the nonzero swirl, the main difficulties lie in the solvability of a singular elliptic equation which concerns the angular component of the vorticity in its cylindrical representation, and in analysis of streamlines near the axis $r=0$.

math.AP↗
arXiv2014

Two dimensional subsonic flows with self-gravitation in bounded domain

We investigate two dimensional steady Euler-Poisson system which describe the motion of compressible self-gravitating flows. The unique existence and stability of subsonic flows in a duct of finite length are obtained when prescribing the entropy at the entrance and the pressure at the exit. After introducing the stream function, the Euler-Poisson system can be decomposed into several transport equations and a second order nonlinear elliptic system. We discover an energy estimate for the associated elliptic system which is a key ingredient to prove the unique existence and stability of subsonic flow.

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arXiv2013

Subsonic solutions for steady Euler-Poisson system in two dimensional nozzles

In this paper, we prove the existence and stability of subsonic flows for steady full Euler-Poisson system in a two dimensional nozzle of finite length when imposing the electric potential difference on non-insulated boundary from a fixed point at the entrance, and prescribing the pressure at the exit of the nozzle. The Euler-Poisson system for subsonic flow is a hyperbolic-elliptic coupled nonlinear system. One of the crucial ingredient of this work is the combination of Helmholtz decomposition for the velocity field and stream function formulation together. In terms of the Helmholtz decomposition, the Euler-Poisson system is rewritten as a second order nonlinear elliptic system of three equations and transport equations for entropy and pseudo-Bernoulli's invariant. The associated elliptic system in a Lipschitz domain with nonlinear boundary conditions is solved with the help of the estimates developed in [2] based on its nice structure. The transport equations are resolved via the flow map induced by the stream function formulation. Furthermore, the delicate estimates for the flow map give the uniqueness of the solutions.

math.AP↗
arXiv2013

Subsonic flow for multidimensional Euler-Poisson system

We establish unique existence and stability of subsonic potential flow for steady Euler-Poisson system in a multidimensional nozzle of a finite length when prescribing the electric potential difference on non-insulated boundary from a fixed point at the exit, and prescribing the pressure at the exit of the nozzle. The Euler-Poisson system for subsonic potential flow can be reduced to a nonlinear elliptic system of second order. In this paper, we develop a technique to achieve a priori $C^{1,\alp}$ estimates of solutions to a quasi-linear second order elliptic system with mixed boundary conditions in a multidimensional domain with Lipschitz continuous boundary. Particularly, we discovered a special structure of the Euler-Poisson system which enables us to obtain $C^{1,\alp}$ estimates of velocity potential and electric potential functions, and this leads us to establish structural stability of subsonic flows for the Euler-Poisson system under perturbations of various data.

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