arXiv subjects

Hobin Lee

Publications and source records attributed to Hobin Lee.

9 recordsLinked to original sources

arXiv2026

Asymptotic stability of viscous shock for the outflow problem of 3D Navier-Stokes system

We establish the asymptotic stability of planar viscous shock waves for the outflow problem of the three-dimensional barotropic compressible Navier--Stokes equations in a half-space, with periodic boundary conditions imposed in the transverse directions. For a weak shock located sufficiently far from the boundary, we prove that, under small perturbations in $H^{2}$, the outflow problem admits a unique global-in-time solution and that the solution converges uniformly to the viscous shock, up to a dynamical shift, whose velocity time-asymptotically decays. This provides the first shock-stability result for the multidimensional outflow problem. Our proof combines the $a$-contraction method and higher-order energy estimates adapted to the half-space boundary. The boundary trace remaining in the zeroth-order estimate is controlled by differentiating the perturbation system in the tangential directions and exploiting the favorable sign of the outflow flux. The highest-order normal derivatives are then recovered from the time and tangential derivatives through the momentum equation, while the boundary terms generated by the dynamical shift are controlled using the exponential decay of the viscous shock tail.

math.AP↗

arXiv2026

Long-time dynamics toward a generic composite wave for the inflow problem of the Navier--Stokes--Fourier system

We study the time-asymptotic stability of solutions to the inflow problem for the one-dimensional Navier--Stokes--Fourier system on the half-line. We consider the most generic wave pattern: the superposition of a degenerate boundary layer, a rarefaction, a viscous contact wave, and a viscous shock. More precisely, if the boundary data belongs to the subsonic region, and the initial perturbation and strengths of the boundary layer, viscous contact wave, and viscous shock are sufficiently small, then the solution to the inflow problem converges to the corresponding superposition, up to a time-dependent shift for a shock. The rarefaction wave, however, is allowed to have arbitrarily large strength. To control the viscous shock, we employ the method of $a$-contraction with shifts. A notable feature of our analysis is that this method can be applied even when the rarefaction wave has large amplitude. In particular, this resolves, in a generic setting, the open problem of the stability of inflow wave patterns containing a viscous shock for Navier--Stokes--Fourier system.

math.AP↗

arXiv2025

Stability of viscous shock for the Navier-Stokes-Fourier system: outflow and impermeable wall problems

We investigate the time-asymptotic stability of solutions to the one-dimensional Navier-Stokes-Fourier system in the half-space, focusing on the outflow and impermeable wall problems. When the prescribed boundary and far-field conditions form an outgoing viscous shock, we prove that the solution converges to the viscous shock profile, up to a dynamical shift, provided that the initial perturbation and the shock amplitude are sufficiently small. In order to obtain our results, we employ the method of $a$-contraction with shifts. Although the impermeable wall problem is technically simpler to analyze in Lagrangian mass coordinates, the outflow problem leads to a free boundary in that framework. Therefore, we use Eulerian coordinates to provide a unified approach to both problems. This is the first result on the time-asymptotic stability of viscous shocks for initial-boundary value problems of the Navier-Stokes-Fourier system for the outflow and impermeable wall cases.

math.AP↗

arXiv2025

Time-asymptotic stability of composite wave of viscous shocks and viscous contact wave for Navier-Stokes-Fourier equations

We investigate the nonlinear time-asymptotic stability of the composite wave consisting of two viscous shocks and a viscous contact discontinuity for the one-dimensional compressible Navier-Stokes-Fourier (NSF) equations. Specifically, we establish that if the composite wave strength and the perturbations are sufficiently small, the NSF system admits a unique global-in-time strong solution, which converges uniformly in space as time tends to infinity, towards the corresponding composite wave, up to dynamical shifts in the positions of the two viscous shocks. Notably, the strengths of the two viscous shocks can be chosen independently. Our proof relies upon the $a$-contraction method with time-dependent shifts and suitable weight functions.

math.AP↗

arXiv2024

The method of $a$-contraction with shifts used for long-time behavior toward viscous shock

We revisit the method of $a$-contraction with shifts used for long-time behavior of barotropic Navier-Stokes flows perturbed from a Riemann shock. For the usage of the method of $a$-contraction with shifts, we do not employ the effective velocity $h$ variable even for higher order estimates. This approach would be important when handling the barotropic Navier-Stokes system with other effects, for example, such as capillary effect and boundary effect.

math.AP↗

arXiv2024

Long-time behavior toward composite wave of shocks for 3D barotropic navier-stokes system

We consider the barotropic Navier-Stokes system in three space dimensions with periodic boundary condition in the transversal direction. We show the long-time behavior of the 3D barotropic Navier-Stokes flow perturbed from a composition of two shock waves with suitably small amplitudes. We prove that the perturbed Navier-Stokes flow converges, uniformly in space, towards a composition of two planar viscous shock waves as time goes to infinity, up to dynamical shifts. This is the first result on time-asymptotic stability of composite wave of two shocks for multi-D Navier-Stokes system. The main part of proof is based on the method of a-contraction with shifts.

math.AP↗

arXiv2024

Asymptotic behavior toward viscous shock for impermeable wall and inflow problem of barotropic Navier-Stokes equations

We consider the compressible barotropic Navier-Stokes equations in a half-line and study the time-asymptotic behavior toward the outgoing viscous shock wave. Precisely, we consider the two boundary problems: impermeable wall and inflow problems, where the velocity at the boundary is given as a constant state. For both problems, when the asymptotic profile determined by the prescribed constant states at the boundary and far-fields is a viscous shock, we show that the solution asymptotically converges to the shifted viscous shock profiles uniformly in space, under the condition that initial perturbation is small enough in $H^1$ norm. Since our method works on the physical variables, we do not require that the anti-derivative variables belong to $L^2$ space as in \cite{HMS03,MM99}. Moreover, for the inflow case, we remove the assumption $\gamma\le 3$ in \cite{HMS03}. Our results are based on the method of $a$-contraction with shifts, as the first extension of the method to the boundary value problems.

math.AP↗

arXiv2024

Long-time behavior towards viscous-dispersive shock for Navier-Stokes equations of Korteweg type

We consider the so-called Naiver-Stokes-Korteweg(NSK) equations for the dynamics of compressible barotropic viscous fluids with internal capillarity. We handle the time-asymptotic stability in 1D of the viscous-dispersive shock wave that is a traveling wave solution to NSK as a viscous-dispersive counterpart of a Riemann shock. More precisely, we prove that when the prescribed far-field states of NSK are connected by a single Hugoniot curve, then solutions of NSK tend to the viscous-dispersive shock wave as time goes to infinity. To obtain the convergence, we extend the theory of $a$-contraction with shifts, used for the Navier-Stokes equations, to the NSK system. The main difficulty in analysis for NSK is due to the third-order derivative terms of the specific volume in the momentum equation. To resolve the problem, we introduce an auxiliary variable that is equivalent to the derivative of the specific volume.

math.AP↗

arXiv2018

Electric-magnetic duality as a quantum operator and more symmetries of $U(1)$ gauge theory

We promote the Noether charge of the electric-magnetic duality symmetry of $U(1)$ gauge theory, "$G$" to a quantum operator. We construct ladder operators, $D_{(\pm)a}^\dagger(k)$ and $D_{(\pm)a}(k)$ which create and annihilate the simultaneous quantum eigen states of the quantum Hamiltonian(or number) and the electric-magnetic duality operators respectively. Therefore all the quantum states of the $U(1)$ gauge fields can be expressed by a form of $|E,g\rangle$, where $E$ is the energy of the state, the $g$ is the eigen value of the quantum operator $G$, where the $g$ is quantized in the unit of 1. We also show that 10 independent bilinears comprised of the creation and annihilation operators can form $SO(2,3)$ which is as demonstrated in the Dirac's paper published in 1962. The number operator and the electric-magnetic duality operator are the members of the $SO(2,3)$ generators. We note that there are two more generators which commute with the number operator(or Hamiltonian). We prove that these generators are indeed symmetries of the $U(1)$ gauge field theory action.

hep-th↗