arXiv subjects

Wanyong Shim

Publications and source records attributed to Wanyong Shim.

6 recordsLinked to original sources

arXiv2026

Stable $C^{7/9}$ cusp formation for the Novikov equation

We establish stable cusp formation for the Novikov equation, a cubically nonlinear Camassa--Holm-type equation. We identify an open set of smooth initial data for which the first gradient blow-up produces a cusp with sharp Hölder regularity $C^{7/9}$. This result shows that, in nonlocal wave-breaking problems, the sharp regularity of the cusp is not determined by the nonlocal or nonlinear structure alone. While the conserved $H^1$-type quantity excludes the $C^{1/3}$ cusp associated with Burgers-type gradient blow-up, the precise Hölder exponent is selected by the coupling between the nonlocal term and the algebraic structure of the nonlinearity. In the Novikov equation, this coupling yields the exponent $7/9$, rather than the $3/5$ exponent known for the Camassa--Holm and Hunter--Saxton equations. The main difficulty is that the naive high-frequency limit retains the cubic character of the equation and therefore does not exhibit a self-similar leading flow. We overcome this by introducing a Galilean-type change of variables around a nonzero background, which reveals a quadratic Hunter--Saxton-type leading equation. Its self-similar profiles determine the $C^{7/9}$ cusp, while the nonlocal and cubic remainders are controlled perturbatively in modulated similarity variables.

math.AP↗

arXiv2026

$L^2$-contraction and asymptotic stability of Cahn--Hilliard fronts

We study the stability of transition fronts for the one-dimensional Cahn--Hilliard equation. More precisely, we prove that for any Hölder initial datum sufficiently close to a front in the $L^2$ norm, the corresponding solution of the Cahn--Hilliard equation exists globally and converges, up to a dynamical shift, to the front as time tends to infinity. For the proof, we develop an $L^2$-stability framework adapted to Cahn--Hilliard fronts. The key ingredient is a nontrivial second-order Poincaré-type inequality that reveals a coercive structure in the indefinite quadratic energy form associated with the linearized operator about the front, once the dynamical shift is taken into account. This yields an $L^2$-contraction estimate and, via a far-field semigroup argument, asymptotic orbital stability of the front in the unweighted $L^2$ topology. The stability analysis requires only the $L^2$-smallness of the initial perturbation; no higher-order smallness, spatial localization, or moment assumptions are imposed.

math.AP↗

arXiv2026

Asymptotic self-similar blow-up for the regularized Saint-Venant equations

We investigate singularity formation in the regularized Saint--Venant (rSV) equations, a conservative, non-dispersive shallow water system that is formally regarded as a Hamiltonian regularization of the isentropic Euler equations. While it is known that smooth solutions to the rSV system can develop gradient blow-up in finite time, the precise structure of such singularities has not been rigorously characterized. In this work, we establish stability of self-similar blow-up profiles of the Hunter--Saxton equation within the rSV framework, using a nonlinear bootstrap argument in dynamically rescaled coordinates. Our analysis captures the detailed space-time dynamics of solutions near the singularity, and proves their sharp $C^{3/5}$ Hölder regularity at the singular time. This regularity differs from the $C^{1/3}$ Hölder regularity of the cubic-root singularities found in the compressible Euler and inviscid Burgers equations. This contrast highlights the structural influence of the Hamiltonian regularization on singularity formation. To illuminate this effect, we also show that the same $C^{3/5}$ blow-up profile emerges in the regularized Burgers equation, a scalar analogue of the rSV system.

math.AP↗

arXiv2026

Spectral stability of shock profiles for the Navier-Stokes-Poisson system

We investigate the spectral stability of small-amplitude shock profiles for the one-dimensional isothermal Navier-Stokes-Poisson system, which describes ion dynamics in a collision-dominated plasma. Specifically, we establish (i) bounds on the essential spectrum, (ii) bounds on the point spectrum, and (iii) simplicity of the zero eigenvalue for the linearized operator about the profile in $L^2$. The result in (i) shows that the zero eigenvalue arising from translation invariance is embedded in the essential spectrum. Consequently, the standard Evans function approach cannot be applied directly to prove (iii). To resolve this, we employ an Evans-function framework that extends into regions of the essential spectrum, thereby enabling us to compute the derivative of the Evans function at the origin. Our result establishes that this derivative admits a factorization into two factors: one associated with transversality of the connecting profile and the other with hyperbolic stability of the corresponding shock of the quasi-neutral Euler system. We further show that both factors are nonzero, which implies simplicity of the zero eigenvalue.

math.AP↗

arXiv2025

Asymptotic stability of composite waves of shock profile and rarefaction for the Navier-Stokes-Poisson system

We study the stability of composite waves consisting of a shock profile and a rarefaction wave for the one-dimensional isothermal Navier--Stokes--Poisson (NSP) system, which describes the ion dynamics in a collision-dominated plasma. More precisely, we prove that if the initial data are sufficiently close in the $H^2$ norm to the Riemann data corresponding to a solution consisting of a shock and a rarefaction wave of the associated quasi-neutral Euler system, then the solution to the Cauchy problem for the NSP system converges, up to a dynamical shift, to a superposition of the corresponding shock profile and the rarefaction wave as time tends to infinity. Our proof is based on the method of $a$-contraction with shifts, which has recently been applied to the Navier--Stokes equations to establish the asymptotic stability of composite waves. To adapt this method to the NSP system, we employ a modulated relative functional introduced in our previous work on the stability of single shock profiles.

math.AP↗

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.

math.AP↗