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Juhi Jang

Publications and source records attributed to Juhi Jang.

At least 19 recordsLinked to original sources
arXiv2026

Gradient blowup of smooth vacuum solutions to 1D compressible Euler equations

We consider the isentropic compressible Euler equations in the half-line which govern the motion of gaseous fluids in contact with stationary vacuum boundary. We construct a large class of solutions that are initially smooth and square-integrable, and which, in finite time, transition to $C^{1-μ}$ regularity for $μ\in [1/2,1)$ near the boundary, leading to the gradient blowup at the boundary. It is based on stability analysis of self-similar waiting time solutions \cite{JLN2025} recently constructed by the authors.

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arXiv2025

Nonlinear stability of the Larson-Penston collapse

We prove nonlinear stability of the Larson-Penston family of self-similarly collapsing solutions to the isothermal Euler-Poisson system. Our result applies to radially symmetric perturbations and it is the first full nonlinear stability result for radially imploding compressible flows. At the heart of the proof is the ground state character of the Larson-Penston solution, which exhibits important global monotonicity properties used throughout the proof. One of the key challenges is the proof of mode-stability for the non self-adjoint spectral problem which arises when linearising the dynamics around the Larson-Penston collapsing solution. To exclude the presence of complex growing modes other than the trivial one associated with time translation symmetry, we use a high-order energy method in low and high frequency regimes, for which the monotonicity properties are crucially exploited, and use rigorous computer-assisted techniques in the intermediate regime. In addition, the maximal dissipativity of the linearised operator is proven on arbitrary large backward light cones emanating from the singular point using the global monotonicity of the Larson-Penston solutions. Such a flexibility in linear analysis also facilitates nonlinear analysis and allows us to identify the exact number of derivatives necessary for the nonlinear stability statement. The proof is based on a two-tier high-order weighted energy method which ties bounds derived from the Duhamel formula to quasilinear top order estimates. To prove global existence we further use the Brouwer fixed point theorem to identify the final collapse time, which suppresses the trivial instability caused by the time-translation symmetry of the system.

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arXiv2025

Hydrostatic bubbles of compressible fluid in an incompressible fluid

We study stationary configurations of compressible barotropic fluids lying inside an incompressible fluid and acted upon by a constant gravitational field. Without gravity, it is a simple matter to construct solutions consisting of perfectly spherical bubbles of compressible fluid, but with gravity the stratification of the compressible fluid's density prevents the existence of solutions with such simple geometries. We construct smooth solutions with nontrivial gravity and nearly spherical, axisymmetric geometry by means of a bifurcation argument in an appropriate weighted Sobolev space, defined in terms of the Legendre operator. Along the way, we develop a number of essential linear and nonlinear functional analytic tools for this scale of spaces, which are of independent interest.

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arXiv2025

Separable motions for self-gravitating hyperelastic matter

In this paper, we prove the existence of separable solutions to the equations of motion for self-gravitating hyperelastic matter, under an appropriate class of constitutive assumptions on the strain-energy function. Our framework includes both global-in-time solutions which expand and also solutions which collapse to a point in finite time. Other authors have constructed expanding solutions in similar settings, but to the best of our knowledge, the collapsing solutions we construct are completely new.

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arXiv2025

Waiting Time Solutions in gas dynamics

In this article, we construct a continuum family of self-similar waiting time solutions for the one-dimensional compressible Euler equations for the adiabatic exponent $\ga\in(1,3)$ in the half-line with the vacuum boundary. The solutions are confined by a stationary vacuum interface for a finite time with at least $C^1$ regularity of the velocity and the sound speed up to the boundary. Subsequently, the solutions undergo the change of the behavior, becoming only Hölder continuous near the singular point, and simultaneously transition to the solutions to the vacuum moving boundary Euler equations satisfying the physical vacuum condition. When the boundary starts moving, a weak discontinuity emanating from the singular point along the sonic curve emerges. The solutions are locally smooth in the interior region away from the vacuum boundary and the sonic curve.

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arXiv2024

Nonradial stability of expanding Goldreich-Weber stars

Goldreich-Weber solutions constitute a finite-parameter of expanding and collapsing solutions to the mass-critical Euler-Poisson system. Two subclasses of this family correspond to compactly supported density profiles suitably modulated by the dynamic radius of the star that expands at the self-similar rate $λ(t)_{t\to\infty}\sim t^{\frac23}$ and linear rate $λ(t)_{t\to\infty}\sim t$ respectively. We prove two results: any linearly expanding Goldreich-Weber star is nonlinearly stable, while any given self-similarly expanding Goldreich-Weber star is codimension-4 nonlinearly stable against irrotational perturbations. The codimension-4 condition in the latter result is optimal and reflects the presence of 4 unstable directions in the linearised dynamics in self-similar coordinates, which are induced by the conservation of the energy and the momentum. This result can be viewed as a codimension-1 nonlinear stability of the moduli space of self-similarly expanding Goldreich-Weber stars against irrotational perturbations.

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arXiv2024

On the mass transfer in the 3D Pitaevskii model

We examine a micro-scale model of superfluidity derived by Pitaevskii in 1959 which describes the interacting dynamics between superfluid He-4 and its normal fluid phase. This system consists of the nonlinear Schrödinger equation and the incompressible, inhomogeneous Navier-Stokes equations, coupled to each other via a bidirectional nonlinear relaxation mechanism. The coupling permits mass/momentum/energy transfer between the phases, and accounts for the conversion of superfluid into normal fluid. We prove the existence of weak solutions in $\mathbb{T}^3$ for a power-type nonlinearity, beginning from small initial data. The main challenge is to control the inter-phase mass transfer in order to ensure the strict positivity of the normal fluid density, while obtaining time-independent a priori estimates.

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arXiv2024

Converging/diverging self-similar shock waves: from collapse to reflection

We solve the continuation problem for the non-isentropic Euler equations following the collapse of an imploding shock wave. More precisely, we prove that the self-similar Güderley imploding shock solutions for a perfect gas with adiabatic exponent $γ\in(1,3]$ admit a self-similar extension consisting of two regions of smooth flow separated by an outgoing spherically symmetric shock wave of finite strength. In addition, for $γ\in(1,\frac53]$, we show that there is a unique choice of shock wave that gives rise to a globally defined self-similar flow with physical state at the spatial origin.

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arXiv2024

Small-data global existence of solutions for the Pitaevskii model of superfluidity

We investigate a micro-scale model of superfluidity derived by Pitaevskii in 1959 to describe the interacting dynamics between the superfluid and normal fluid phases of Helium-4. The model involves the nonlinear Schrödinger equation (NLS) and the Navier-Stokes equations (NSE), coupled to each other via a bidirectional nonlinear relaxation mechanism. Depending on the nature of the nonlinearity in the NLS, we prove global/almost global existence of solutions to this system in $\mathbb{T}^2$ -- strong in wavefunction and velocity, and weak in density.

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arXiv2023

On self-similar converging shock waves

In this paper, we rigorously prove the existence of self-similar converging shock wave solutions for the non-isentropic Euler equations for $γ\in (1,3]$. These solutions are analytic away from the shock interface before collapse, and the shock wave reaches the origin at the time of collapse. The region behind the shock undergoes a sonic degeneracy, which causes numerous difficulties for smoothness of the flow and the analytic construction of the solution. The proof is based on continuity arguments, nonlinear invariances, and barrier functions.

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arXiv2023

Existence of Rotating Magnetic Stars

We consider a star as a compressible fluid subject to gravitational and magnetic forces. This leads to an Euler-Poisson system coupled to a magnetic field, which may be regarded as an MHD model together with gravity. The star executes steadily rotating motion about a fixed axis. We prove, for the first time, the existence of such stars provided that the rotation speed and the magnetic field are sufficiently small.

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arXiv2022

Asymptotic stability and sharp decay rates to the linearly stratified Boussinesq equations in horizontally periodic strip domain

We consider an initial boundary value problem of the multi-dimensional Boussinesq equations in the absence of thermal diffusion with velocity damping or velocity diffusion under the stress free boundary condition in horizontally periodic strip domain. We prove the global-in-time existence of classical solutions in high order Sobolev spaces satisfying high order compatibility conditions around the linearly stratified equilibrium, the convergence of the temperature to the asymptotic profile, and sharp decay rates of the velocity field and temperature fluctuation in all intermediate norms based on spectral analysis combined with energy estimates. To the best of our knowledge, our results provide first sharp decay rates for the temperature fluctuation and the vertical velocity to the linearly stratified Boussinesq equations in all intermediate norms.

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arXiv2022

opPINN: Physics-Informed Neural Network with operator learning to approximate solutions to the Fokker-Planck-Landau equation

We propose a hybrid framework opPINN: physics-informed neural network (PINN) with operator learning for approximating the solution to the Fokker-Planck-Landau (FPL) equation. The opPINN framework is divided into two steps: Step 1 and Step 2. After the operator surrogate models are trained during Step 1, PINN can effectively approximate the solution to the FPL equation during Step 2 by using the pre-trained surrogate models. The operator surrogate models greatly reduce the computational cost and boost PINN by approximating the complex Landau collision integral in the FPL equation. The operator surrogate models can also be combined with the traditional numerical schemes. It provides a high efficiency in computational time when the number of velocity modes becomes larger. Using the opPINN framework, we provide the neural network solutions for the FPL equation under the various types of initial conditions, and interaction models in two and three dimensions. Furthermore, based on the theoretical properties of the FPL equation, we show that the approximated neural network solution converges to the a priori classical solution of the FPL equation as the pre-defined loss function is reduced.

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arXiv2022

Existence of Rotating Stars with Variable Entropy

We model a rotating star as a compressible fluid subject to gravitational forces. In almost all the mathematical literature the entropy is considered to be constant. Here we allow it to be variable. We consider a star that steadily rotates differentially around a fixed axis, say the $z$-axis. We prove the existence of a family of such stars with small angular velocity $ω$ and small entropy variation $s$ and with an equation of state $p=Ke^sρ^γ$. Our analysis reduces to a hyperbolic equation for the modified entropy coupled to an elliptic equation for the modified density, together with a mass constraint. Due to the variable entropy and the consequent loss of both regularity and variational structure, all the methods in the previous literature fail. We develop a new ad hoc perturbative strategy that allows us to construct rotating stars that bifurcate from the non-rotating ones.

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arXiv2021

Naked singularities in the Einstein-Euler system

In 1990, based on numerical and formal asymptotic analysis, Ori and Piran predicted the existence of self-similar spacetimes, called relativistic Larson-Penston solutions, that can be suitably flattened to obtain examples of spacetimes that dynamically form naked singularities from smooth initial data, and solve the radially symmetric Einstein-Euler system. Despite its importance, a rigorous proof of the existence of such spacetimes has remained elusive, in part due to the complications associated with the analysis across the so-called sonic hypersurface. We provide a rigorous mathematical proof. Our strategy is based on a delicate study of nonlinear invariances associated with the underlying non-autonomous dynamical system to which the problem reduces after a self-similar reduction. Key technical ingredients are a monotonicity lemma tailored to the problem, an ad hoc shooting method developed to construct a solution connecting the sonic hypersurface to the so-called Friedmann solution, and a nonlinear argument to construct the maximal analytic extension of the solution. Finally, we reformulate the problem in double-null gauge to truncate the self-similar profile and thus obtain an asymptotically flat spacetime with an isolated naked singularity.

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arXiv2021

Gravitational Collapse for Polytropic Gaseous Stars: Self-similar Solutions

In the supercritical range of the polytropic indices $γ\in(1,\frac43)$ we show the existence of smooth radially symmetric self-similar solutions to the gravitational Euler-Poisson system. These solutions exhibit gravitational collapse in the sense that the density blows-up in finite time. Some of these solutions were numerically found by Yahil in 1983 and they can be thought of as polytropic analogues of the Larson-Penston collapsing solutions in the isothermal case $γ=1$. They each contain a sonic point, which leads to numerous mathematical difficulties in the existence proof.

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arXiv2021

Mach limits in analytic spaces

We address the Mach limit problem for the Euler equations in the analytic spaces. We prove that, given analytic data, the solutions to the compressible Euler equations are uniformly bounded in a suitable analytic norm and then show that the convergence toward the incompressible Euler solution holds in the analytic norm. We also show that the same results hold more generally for Gevrey data with the convergence in the Gevrey norms.

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