arXiv subjects

Moon-Jin Kang

Publications and source records attributed to Moon-Jin Kang.

At least 19 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.

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arXiv2026

Asymptotic behavior toward viscous shock for the outflow problem of barotropic Navier-Stokes equations

We study the time-asymptotic stability of viscous shock profile for the outflow problem of the barotropic Navier-Stokes equations on the half-line. We consider the case where the far-field state, as the right end state of the 2-Hugoniot curve, belongs to the subsonic region or the transonic curve. We employ the method of $a$-contraction with shifts to prove that, if both the shock strength and the initial perturbation are suitably small, and the viscous shock is far from the outflow boundary, then the solution asymptotically converges to the viscous shock up to a dynamical shift. We also prove that the speed of the time-dependent shift decays to zero as time goes to infinity, so that the shifted viscous shock still retains its original profile asymptotically. Since the outflow problem in the Lagrangian mass coordinate leads to a free-boundary problem due to the absence of a boundary condition for the fluid density, we consider the outflow problem in the original Eulerian coordinate instead. Although the method of $a$-contraction with shifts is technically more complicated in the Eulerian coordinate than in the Lagrangian one, this provides a more favorable framework by avoiding the difficulties arising from a free boundary. Note that this is the first result on the time-asymptotic stability of viscous shock for the outflow problem of the Navier-Stokes equations.

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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.

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arXiv2026

Hydrodynamic Limit of the Boltzmann Equation toward Generic Riemann Solutions with Shocks

We establish the hydrodynamic limit of the one-dimensional Boltzmann equation with hard-sphere collisions toward Riemann solutions of the compressible Euler system. The Riemann solutions covered by our result include generic superpositions of elementary waves: either two shock waves and a contact discontinuity, or a rarefaction wave, a contact discontinuity, and a shock wave. For suitably well-prepared initial data and sufficiently small wave strength, we prove that the corresponding Boltzmann solution exists globally in time and converges, as the Knudsen number vanishes, to the local Maxwellian associated with the Riemann solution in $L^2([0,T]\times\mathbb R_x\times\mathbb R^3_ξ)$ for any $T>0$. The proof combines the macro--micro decomposition with a kinetic adaptation of the $a$-contraction method, in which the propagation speeds of Boltzmann shocks are determined by Rakine-Hugoniot speed with dynamical modulation parameters. The resulting coercive control of the shock translation modes, together with layer analysis and the uniform bound of the Shifts, allows us to pass to the Knudsen limit in the full space-time domain. To the best of our knowledge, this is the first rigorous result on the hydrodynamic limit towards generic Riemann solutions containing shocks: either the shock--contact--shock case or the rarefaction--contact--shock case, in a global space-time energy norm without removing neighborhoods of either the initial time or the shock layers. In the special case of a single shock, the argument further yields a sharp quantitative description of the kinetic shock layer, up to the dynamically selected Shift.

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arXiv2026

Global Existence of Classical Solutions to Brenner-Navier-Stokes-Fourier System for Large Data

We study the 1D Brenner-Navier-Stokes-Fourier (BNSF) system, proposed as a refinement of the classical Navier--Stokes--Fourier model through the introduction of the volume velocity, distinct from the mass velocity describing convective transport. When formulated in the Lagrangian mass coordinates with the volume velocity, the discrepancy between the two velocities induces a dissipative structure in the mass conservation law. We prove the global existence of classical solutions for arbitrarily large initial data. More precisely, for initial data in $H^k(\mathbb{R})$ with $k\ge3$, with the specific volume and absolute temperature initially bounded away from zero, we construct global-in-time solutions that remain in the same regularity class. Our result accommodates arbitrarily large initial data. A major difficulty is to establish lower and upper bounds for the specific volume $v$. The additional dissipation yields an $L_t^2 L_x^2$ bound for $v_x$, which is further improved to an $L_t^\infty L_x^\infty$ bound of $v$ and $1/v$ via the parabolic De Giorgi method. We also apply the maximum principle to obtain a positive lower bound for the absolute temperature.

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arXiv2026

$L^2$-contraction of Shock Waves for KdV-Burgers Equation

The KdV-Burgers equation is a canonical model describing the interplay between nonlinearity, viscosity and dispersion, and it admits viscous-dispersive shocks as traveling wave solutions. In this paper, we establish an $L^2$-contraction property for viscous-dispersive shocks under arbitrarily large perturbations, up to a time-dependent shift. This yields time-asymptotic stability and uniform estimates with respect to the strengths of viscosity and dispersion. We present the proof for the monotone shocks, and introduce the companion work in [6] on the stability and structural properties of oscillatory shocks.

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arXiv2026

Uniform Stability of Oscillatory Shocks for KdV-Burgers Equation

We study viscous-dispersive shock waves with infinite oscillations of the Korteweg-de Vries-Burgers (KdVB) equation. First, we establish detail structures of the shock waves, including the rates at which the local extrema converge to the left end state towards the left far field. Then, by exploiting the structural properties of the shocks, we show the $L^2$-contraction property of the shock profiles under arbitrarily large perturbations, up to time-dependent shifts. This property implies both time-asymptotic stability and uniform stability with respect to the viscosity and dispersion coefficients. This uniformity yields the existence of zero viscosity-dispersion limits, on which Riemann shocks are orbitally stable.

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arXiv2026

Convergence to Superposition of Boundary Layer, Rarefaction and Shock for the Inflow Problem of the 1D Navier--Stokes Equations

We establish the asymptotic stability of solutions to the inflow problem for the one-dimensional barotropic Navier--Stokes equations in half space. When the boundary value is located at the subsonic regime, all the possible thirteen asymptotic patterns are classified in \cite{M01}. We consider the most complicated pattern, the superposition of the boundary layer solution, the 1-rarefaction wave, and the viscous 2-shock waves. In this superposition, the boundary layer is degenerate and large. We prove that, if the strengths of the rarefaction wave and shock wave are small, and if the initial data is a small perturbation of the superposition, then the solution asymptotically converges to the superposition up to a dynamical shift for the shock. As a corollary, our result implies the asymptotic stability for the simpler case where the superposition consists of the degenerate boundary layer solution and the viscous 2-shock. Therefore, we complete the study of the asymptotic stability of the inflow problem for the 1D barotropic Navier--Stokes equations for subsonic boundary values.

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arXiv2026

Contraction of viscous-dispersive shocks: Zero viscosity-capillarity limits

We prove the contraction property of any large solution perturbed from a viscous-dispersive shock wave of the Navier--Stokes--Korteweg (NSK) system. The contraction holds up to a dynamical shift, since the contraction is measured by the relative entropy that is locally $L^2$. We use the contraction property to show the global existence of large solution perturbed from a viscous-dispersive shock wave. To prove the contraction property, we first employ the effective velocity to transform the NSK system into the system of two degenerate parabolic equations, then apply the method of $a$-contraction with shifts. The contraction property does not depend on the strengths of viscosity and capillarity. Based on this uniformity, we show the existence of zero viscosity-capillarity limits of solutions to the NSK system, on which Riemann shocks are unique and stable up to shifts.

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arXiv2025

From Navier-Stokes to BV solutions of the barotropic Euler equations

In the realm of mathematical fluid dynamics, a formidable challenge lies in establishing inviscid limits from the Navier-Stokes equations to the Euler equations, wherein physically admissible solutions can be discerned. The pursuit of solving this intricate problem, particularly concerning singular solutions, persists in both compressible and incompressible scenarios. This article focuses on small $BV$ solutions to the barotropic Euler equation in one spatial dimension. Our investigation demonstrates that these solutions are inviscid limits for solutions to the associated compressible Navier-Stokes equation. Moreover, we extend our findings by establishing the well-posedness of such solutions within the broader class of inviscid limits of Navier-Stokes equations with locally bounded energy initial values.

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arXiv2025

Long-Time Behavior towards Shock Profiles for the Navier-Stokes-Poisson System

We study the stability of shock profiles in one spatial dimension for the isothermal Navier-Stokes-Poisson (NSP) system, which describes the dynamics of ions in a collision-dominated plasma. The NSP system admits a one-parameter family of smooth traveling waves, called shock profiles, for a given far-field condition satisfying the Lax entropy condition. In this paper, we prove that if the initial data is sufficiently close to a shock profile in $H^2$-norm, then the global solution of the Cauchy problem tends to the smooth manifold formed by the parametrized shock profiles as time goes to infinity. This is achieved using the method of $a$-contraction with shifts, which does not require the zero mass condition.

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arXiv2025

Stability of a Riemann Shock in a Physical Class: From Brenner-Navier-Stokes-Fourier to Euler

The stability of an irreversible singularity, such as a Riemann shock to the full Euler system, in the absence of any technical conditions on perturbations, remains a major open problem even within mono-dimensional framework. A natural approach to justify such stability is to consider vanishing dissipation (or viscosity) limits of physical viscous flows. We prove the existence of vanishing dissipation limits, on which a Riemann shock of small amplitude is stable (up to a time-dependent shift) and unique. Thus, a Riemann weak shock is rigid (not turbulent) under physical disturbances. We adopt the Brenner-Navier-Stokes-Fourier system, based on the bi-velocity theory, as a physical viscous model. The key ingredient of the proof is the uniform stability of the viscous shock with respect to the viscosity strength. The uniformity is ensured by contraction estimates of any large perturbations around the shock. The absence of any restrictions on size of initial perturbations forces us to handle extreme values of density and temperature, which constitutes the most challenging part of our analysis. We use the method of a-contraction with shifts, but we improve it by introducing a more delicate analysis of the localizing effect given by viscous shock derivatives. This improvement possesses a degree of robustness that renders it applicable to a wide range of models. This is the first resolution for the challenging open problem on the "unconditional" stability and uniqueness of Riemann shock solutions to the full Euler system in a class of vanishing physical dissipation limits.

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arXiv2025

Stability of Riemann Shocks for isothermal Euler by Inviscid limits of global-in-time large Navier-Stokes flows

In this paper, we study the isothermal gas dynamics. We first establish the global existence of strong solutions to the one-dimensional isothermal Navier-Stokes system for smooth initial data without any smallness conditions, assuming that the initial density has strictly positive lower bound. The existence result allows for possibly degenerate viscosity coefficients and admits different asymptotic states at the far fields. We then prove a contraction property for the strong solutions perturbed from viscous shocks, yielding uniform estimates with respect to the viscosity coefficients. This covers any large perturbations, and consequently, we establish the inviscid limits and their stability estimate. In other words, we demonstrate the stability of Riemann shocks to the one-dimensional isothermal Euler system in the class of vanishing viscosity limits of the associated Navier-Stokes system.

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arXiv2025

Decay of large solutions around shocks to multi-D viscous conservation law with strictly convex flux

We consider a planar viscous shock for a scalar viscous conservation law with a strictly convex flux in multi-dimensional setting, where the transversal direction is periodic. We first show the contraction property for any solutions evolving from a large bounded initial perturbation in $L^2$ of the viscous shock. The contraction holds up to a dynamical shift, and it is measured by a weighted relative entropy. This result for the contraction extends the existing result in 1D \cite{Kang19} to the multi-dimensional case. As a consequence, if the large bounded initial $L^2$-perturbation is also in $L^1$, then the large perturbation decays of rate $t^{-1/4}$ in $L^2$, up to a dynamical shift that is uniformly bounded in time. This is the first result for the quantitative estimate converging to a planar shock under large perturbations.

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arXiv2025

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 $γ\le 3$ 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.

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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.

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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.

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arXiv2024

Traveling Wave Solutions to Brenner-Navier-Stokes-Fourier system

As a continuum model for compressible fluid flows, Howard Brenner proposed the so-called Brenner-Navier-Stokes-Fourier(BNSF) system that improves some flaws of the Navier-Stokes-Fourier(NSF) system. For BNSF system, the volume velocity concept is introduced and is far different from the mass velocity of NSF, since the density of a compressible fluid is inhomogeneous. Although BNSF was introduced more than ten years ago, the mathematical study on BNSF is still in its infancy. We consider the BNSF system in the Lagrangian mass coordinates. We prove the existence and uniqueness of monotone traveling wave solutions to the BNSF system. We also present some quantitative estimates for them.

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