arXiv subjects

Namhyun Eun

Publications and source records attributed to Namhyun Eun.

9 recordsLinked to original sources

arXiv2026

Large-Time Behavior towards Composite Waves of Degenerate Shock and Rarefaction Wave for Modified KdV-Burgers Equation

In this paper, we study the modified Korteweg-de Vries-Burgers (mKdVB) equation \[u_t + (u^3)_x = μu_{xx} - κu_{xxx}\] with $μ>0$ and $κ>0$. Since the flux is non-convex, the (inviscid) Riemann problem may be solved by a composite wave of a degenerate Oleinik shock and a rarefaction wave. We prove the global existence of solutions and the time-asymptotic stability of the corresponding viscous-dispersive composite wave, up to a time-dependent shift, under small $H^1$ perturbations. The shock strength $δ_S$ and the rarefaction strength $δ_R$ need not be small and are restricted only by the explicit ratio $δ_R\le\frac{5}{18}δ_S$ and the quantity $κδ_S^2/μ^2$ governing the monotonicity of the shock. The stability estimate is uniform with respect to the dispersion strength $κ$. As a key ingredient of the stability analysis, we establish structural properties of the degenerate viscous-dispersive shock profile in the monotone regime, including two-sided pointwise bounds on its derivative and the rates at which it converges to its two end states. These decay rates coincide with those of the corresponding purely viscous profile, with constants independent of the dispersion coefficient. In the absence of dispersion, namely, when $κ=0$, the time-asymptotic stability of the corresponding viscous composite wave was established by Huang, Wang and Zhang [26]. The present work extends their result to the viscous-dispersive setting and relaxes the restriction on the rarefaction strength. The proof relies on the method of $a$-contraction with shifts (for viscous conservation laws) developed by Kang and Vasseur [32,33,35].

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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

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

Traveling Wave Solutions to a Large Class of Brenner-Navier-Stokes-Fourier Systems

The Brenner-Navier-Stokes-Fourier (BNSF) system, introduced by Howard Brenner, was developed to address some deficiencies in the classical Navier-Stokes-Fourier system, based on the concept of volume velocity. We consider the one-dimensional BNSF system in Lagrangian mass coordinates, incorporating temperature-dependent transport coefficients, which yields a more physically realistic framework. We establish the existence and uniqueness of monotone traveling wave solutions (or viscous shocks) to the BNSF system with any positive $C^2$ dissipation coefficients, provided that the shock amplitude is sufficiently small. We utilize geometric singular perturbation theory as in the constant coefficient case [13]; however, due to the arbitrary nonlinearities of the coefficients, we employ the implicit function theorem, which grants robustness to our approach. This work is motivated by [12], which proves a contraction property of any large solutions to the BNSF system around the traveling wave solutions. Thus, we also derive some quantitative estimates on the traveling wave solutions that play a fundamental role in [12].

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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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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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